Introduction to Schrödinger’s Wave Mechanics1

Carl K. Darrow, New York.

At a time when some particular domain of physical phenomena arouses especially keen interest and is undergoing especially intensive development, attracting the attention of many brilliant representatives of theoretical physics, someone’s ingenious intuition sometimes succeeds in giving the known laws such a new foundation, in presenting them in such a new light, that this new aspect of familiar phenomena displaces all earlier ones with extraordinary speed. At the same time the new theory often does not even give greater agreement with experiment than the old one; it may lead to no new predictions; its mathematical conclusions may be identical with those of the old theory; the former symbols may recur in essentially the same equations, under new names. In other cases the theory may possess all the enumerated merits which usually prove decisive when one theory replaces another, and yet owe its victory to none of them. The new theory triumphs because it seems more natural, more intelligible, or more beautiful—all these words mean that the theory satisfies some demand (or does not contradict some prejudice) deeply rooted in the mind of its contempor—

methods. Later a wave of success may befall a theory not because some essential shortcomings of it have been revealed, but simply because the next generation of physicists does not share the sympathies and prejudices of the preceding one. The kinetic theory of gases was greeted with delight by a generation that wanted to believe in the existence of atoms; Maxwell’s electromagnetic theory—with a generation that did not wish to acknowledge action at a distance. Quantum theory always had to struggle against the skepticism of those who did not wish to reconcile themselves to the abandonment of continuity in natural phenomena. The theory to be discussed in the present article has, within a few months, won broad recognition for itself in the scientific world because it promised to fulfill a long-suppressed, but ineradicable, desire of the majority of contemporary physicists.

Since the wave mechanics of de Broglie–Schrödinger is a new way of explaining a wide domain of already known phenomena, an attempt at a complete survey in a single article of everything that this theory can explain would be both useless and hardly feasible. Perhaps in a few years we shall find the most convincing proofs in favor of the correctness of the new theory in some new or still little-known physical phenomena, such as the recently discovered interference of electrons; at present, a sufficient proof may be considered the possibility of explaining by means of wave mechanics those basic facts which served as the material for the creation of the Bohr model of the atom. I shall recall that the most important and most essential facts in the field that interests us in the present article are the following: atoms exist in special, so-called stationary states; they emit or absorb radiant energy upon transition from one stationary state to another; in this process the frequency of the oscillations is determined by the difference between the energies of the atom before and after the process. Further, between the energies of different stationary states in certain atoms and molecules there exist regular relations expressed by one or another empirical formula. These, in brief, are the facts to be explained.

Bohr showed that the values of the energy possessed by a hydrogen atom in its various stationary states can be explained theoretically with the aid of the following assumptions: first, the atom consists of a nucleus and one electron, with a known mass ratio and with charges equal in magnitude but opposite in sign. Second, both particles—the nucleus and the electron—are assumed to revolve about their common center of gravity, in accordance with the laws of classical mechanics, but without radiating energy. Third, from the infinite number of orbits possible according to the laws of classical mechanics, a group of ellipses satisfying certain requirements is singled out, and these ellipses are declared to be the only “permitted” orbits for the electron; to each such orbit there corresponds a special “stationary state” of the atom. Conversely, every stationary state may be regarded as corresponding to a special permitted orbit.

The first of these assumptions, since the time of its appearance, has not disappeared from physicists’ view. Wave mechanics has likewise retained it, although in a somewhat hidden form. Somewhat less firm are the second and third of Bohr’s basic propositions. These assertions in essence remain—and will always remain—valid within the limits within which they alone were valid from the very beginning. This means: if we accept the first two propositions of Bohr, then one can say with certainty that for every empirically established stationary state of the atom it will be possible to find a suitable elliptical orbit, i.e. one possessing the proper energy. What is essential, however, is not this, but whether we are in a position to indicate simple and clear criteria that make it possible to single out the family of permitted ellipses from the infinite multitude of all possible orbits, and to indicate the features satisfied by all permitted orbits—and by no others. At first glance this seems possible. However, closer investigation showed that the characteristic features established for distinguishing permitted orbits from all the rest are by no means in all cases suitable for this purpose. The prestige of the permitted ellipses thereby declined somewhat. True, the introduction of a rotating about its own—

…of the electron’s own axis (spinning electron) improved the situation to a considerable degree; however, even this improvement of the model could not save it from growing mistrust—especially on the part of those who had never believed, as one should, in its reality.

As regards other atoms and molecules, here too the situation was analogous. Bohr and his followers considered atoms as systems of a greater or lesser number of electrons surrounding a nucleus. Diatomic molecules were regarded as systems of two nuclei bound by common electrons and capable, on the one hand, of rotating about a common center of gravity and, on the other, of vibrating, like two rings of a spring, in the direction of the line joining the centers of the two atoms. This image is preserved also in wave mechanics, but the ideas of permitted amplitudes of vibration and of rates of rotation of the atoms and of the allowed orbits of the electrons have at present been taken under suspicion just as strongly as the idea of allowed elliptical orbits in the hydrogen atom.

The loss of confidence in the real existence of elliptical orbits only sharpened attention to another essential shortcoming of the original Bohr model. This model gave no explanation at all for the fact that, when an atom passes from one stationary state with energy \(E_i\) to another, with energy \(E_j\), it absorbs (or emits) a light wave with a frequency exactly determined by the equation

\[ \nu=\frac{1}{h}(E_i-E_j), \]

i.e. equal to the quotient obtained by dividing the difference of the energies by Planck’s constant. Neither in the initial nor in the final state of the atom does anything oscillate or rotate in it with a frequency equal to the frequency of the wave emitted in the transition (exceptions to this rule have no fundamental significance). Thus, the frequency of the wave emitted by the atom has nothing in common with the period of revolution or oscillation of the constituent parts of the atom—an idea that could not but be called enigmatic, since it conflicted with all observations in the domain of both sound and electric waves.

If it were possible to introduce into the atomic model the representation of some kind of vibrator or rotator, with a frequency of periodic motion measured by the quotient of the energy of the corresponding stationary state divided by Planck’s constant, then this “something” would oscillate, before emission, with frequency \(E_i/h\), and after emission with frequency \(E_j/h\); and the frequency of the emitted wave would prove to be the beat frequency obtained in the interference of the two oscillations. This is a very seductive possibility, and wave mechanics opens the way to its practical use. If it proves possible to obtain the correct values for the energies of stationary states by imposing certain requirements on this oscillating “something” in place of electronic orbits, then we shall obtain a representation of the atom which explains everything that the model with elliptical orbits is capable of explaining, plus the interpretation, mentioned above, of the frequency of the oscillations emitted by the atom, and perhaps something else in addition. This is the progress promised to us by the development of wave mechanics.

Before passing to the exposition itself, I should like to conclude the present introduction with two cautions. First, it is necessary to point out the circumstance that wave mechanics has several different aspects, and that it can be approached from several different sides. The path that I have chosen in the present article is not entirely identical either with the path chosen by de Broglie, or with those adopted by Schrödinger in their original works. Secondly, it must be said in advance that wave mechanics is not yet complete. It has been successfully applied to a number of important problems; but there still exist many phenomena which, for their explanation, require an extension of the theory, and this extension is still the subject of dispute among a number of theorists. The new mechanics has not yet solidified into any final form; it remains flexible, and the work of many theorists—and perhaps also many experimentalists—will still be needed in order to give it its definitive form. In the present article I attempt to set forth only the first foundations of the theory; to outline only the basic considerations of Schrödinger and de Broglie.

CLASSICAL AND WAVE MECHANICS.

The basic principles of classical Newtonian mechanics can be expressed in various forms, each of which is especially adapted for solving certain problems. The most widely known formulation is that given by Newton himself. Unfortunately, for the problems that interest us in the present article, this is not the most convenient one; rather, another form of expression of the fundamental principles of mechanics is. I shall derive this form from the Newtonian one, using a particularly simple example and proceeding from Cartesian coordinates.

Let us imagine a particle with mass \(m\) and charge \(e\), moving in an electrostatic field whose potential is a function of the coordinates: \(U(x,y,z)\).

The impulse (quantity of motion) of the particle is a vector with components \(m\dot{x}\), \(m\dot{y}\), and \(m\dot{z}\). We shall call them the impulses in the directions of the coordinates \(x\), \(y\), and \(z\), and denote them by the letters \(p_x\), \(p_y\), and \(p_z\). The force acting on the particle is equal to the product of the charge \(e\) by the gradient of the potential, taken with the opposite sign. The gradient of the potential is a vector with components \(dU/dx\), \(dU/dy\), \(dU/dz\).

Newton’s formulation of the fundamental law of mechanics (force is the time derivative of the impulse)¹ gives:²

\[ -e\,\frac{dU}{dx}=\frac{dp_x}{dt}\quad p_x;\qquad -e\,\frac{dU}{dy}=\frac{dp_y}{dt}\quad p_y; \]

\[ -e\,\frac{dU}{dz}=\frac{dp_z}{dt}\quad p_z \tag{1} \]

¹ The formulation “force = mass × acceleration,” as is known, does not belong to Newton himself. It is, however, identical with the Newtonian formulation when the mass is constant; for from \(K=\dfrac{d(mv)}{dt}\), when \(m=\mathrm{const}\), it follows that \(K=m\dfrac{dv}{dt}\).

² The sign \(=\), as usual, denotes a relation that is an asserted equality; the sign \(\equiv\) denotes a self-evident identity, i.e. we thereby symbolize only another designation of the same quantity.

Multiplying both sides by \(\dot{x}\left(\equiv \dfrac{dx}{dt}\right)\), \(\dot{y}\), \(\dot{z}\) and adding all three equations, we obtain:

\[ p_x \dot{x} + p_y \dot{y} + p_z \dot{z} = - e \left( \frac{dU}{dx}\cdot \frac{dx}{dt} + \frac{dU}{dy}\cdot \frac{dy}{dt} + \frac{dU}{dz}\cdot \frac{dz}{dt} \right). \]

Or, since \(p_x = m \dfrac{dx}{dt}\), etc.:

\[ \frac{d}{dt}\,\frac{1}{2}m\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right) = - e \left( \frac{dU}{dx}\cdot \frac{dx}{dt} + \frac{dU}{dy}\cdot \frac{dy}{dt} + \frac{dU}{dz}\cdot \frac{dz}{dt} \right) \tag{2} \]

On the left-hand side of this equality stands the rate of change of the kinetic energy, which we shall, as usual, denote by \(T\). Let us now give an interpretation of the right-hand side.

For this purpose we introduce the quantity \(V\). By this symbol we denote the product of the potential \(U\) at the place where the particle is located at the given moment by its charge \(e\). This product is the potential energy of the particle, and the right-hand side of equation (2) gives the rate of change of this quantity with time. Thus the whole equation may be rewritten in the form

\[ \frac{d(T+V)}{dt}=0 \]

or

\[ T+V=\mathrm{const}\equiv E. \tag{3} \]

We call \(E\) the total energy, and equation (3) expresses the law of conservation of energy as applied to a closed system particle \(+\) field.

For the further development I shall make use of an even more special case; namely, I shall assume that we are dealing with the motion of a particle of mass \(m\) and charge \(e\) in the field of a “nucleus” attracting the particle with a force inversely proportional to the square of the distance. We imagine the nucleus as an immobile center of attraction carrying a charge equal in magnitude and opposite in sign to the charge of the particle. Using Cartesian coordinates, with the origin coinciding with the nucleus, we have:

\[ V = -\frac{e^{2}}{\sqrt{x^{2}+y^{2}+z^{2}}}. \]

K. K. DARROW

In polar coordinates \(r, \theta, \varphi\) (transformation formulas \(x = r \sin \theta \cos \varphi;\ y = r \sin \theta \sin \varphi;\ z = r \cos \theta\)) the potential energy is expressed by the equation:

\[ V = -\frac{e^2}{r}. \]

It is clear that the expression for the potential energy in polar coordinates is in this case much simpler than in rectangular coordinates. The reverse is true with respect to the kinetic energy. The proper choice of coordinates is, in solving many physical problems, a question of primary importance. For some time yet we shall carry on the discussion in parallel in both coordinate systems—rectangular and polar. The fundamental equations (3) acquire, in the case we are considering, the form, in rectangular coordinates:

\[ \frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}) - \frac{e^{2}}{\sqrt{x^{2}+y^{2}+z^{2}}} = E, \tag{4a} \]

in polar coordinates:

\[ \frac{1}{2}m\left(\dot{r}^{2}+r^{2}\dot{\theta}^{2}+r^{2}\sin^{2}\theta\,\dot{\varphi}^{2}\right) - \frac{e^{2}}{r} = E. \tag{4b} \]

In these equations the potential energy is expressed as a function of the coordinates (\(x, y, z\) or \(r, \theta, \varphi\)). The kinetic energy is represented as a function of the coordinates and velocities (\(\dot{x}, \dot{y}, \dot{z}\) or \(\dot{r}, \dot{\theta}, \dot{\varphi}\)). Of decisive importance for the success of the subsequent derivations is the representation of the kinetic energy as a function of the coordinates and momenta, instead of velocities. We have already become acquainted with the expression of the momenta in a rectangular coordinate system; these were the quantities \(m\dot{x}\), \(m\dot{y}\), and \(m\dot{z}\). It is easy to note that these quantities are the derivatives of the kinetic energy

\[ \left[ T=\frac{1}{2}\left(m\dot{x}^{2}+m\dot{y}^{2}+m\dot{z}^{2}\right) \right] \]

with respect to the velocities:

\[ p_x=\frac{dT}{d\dot{x}},\qquad p_y=\frac{dT}{d\dot{y}},\qquad p_z=\frac{dT}{d\dot{z}}. \tag{5} \]

In an analogous manner the momenta are also defined in other coordinate systems: the kinetic energy is expressed in the form

functions of the velocities, and is then differentiated with respect to the latter. In polar coordinates we obtain, in this way:

\[ p_r=\frac{dT}{d\dot r}=m\dot r \qquad p_\vartheta=\frac{dT}{d\dot\vartheta}=mr^2\dot\vartheta, \tag{6} \]

\[ p_\varphi=\frac{dT}{d\dot\varphi}=mr^2\sin^2\vartheta\cdot\dot\varphi. \tag{7} \]

Expressing in equations (4a) and (4b) the kinetic energy in the form of a function of the coordinates and momenta, we arrive at the equations:

\[ \frac{1}{2m}(p_x^2+p_y^2+p_z^2)-\frac{e^2}{\sqrt{x^2+y^2+z^2}}=E \tag{7a} \]

\[ \frac{1}{2m}\left(p_r^2+\frac{1}{r^2}p_\vartheta^2+\frac{1}{r^2\sin^2\vartheta}p_\varphi^2\right)-\frac{e^2}{r}=E. \tag{7b} \]

When the kinetic and potential energies are expressed as functions of the coordinates and momenta, the problem of the kind we are considering may be regarded as prepared for solution by the methods of classical mechanics that interest us.

In order to take the next step, we shall pass to consideration of the function \(L=T-V\), i.e. the difference between the kinetic and potential energies of the particle during its motion in the force field:

\[ L=T-V=2T-E. \tag{8} \]

In particular, we are interested in the integral of this function with respect to time:

\[ W=\int Ldt=\int 2Tdt-Et \tag{9} \]

(\(E\) does not depend on time).

We substitute into formula (9) the expression for the kinetic energy in rectangular or polar (or any other) coordinates and obtain:

\[ W=m\int(\dot x^2+\dot y^2+\dot z^2)\,dt-Et = \]

\[ =m\int(\dot x\,dx+\dot y\,dy+\dot z\,dz)-Et \tag{10a} \]

\[ W=m\int(\dot r^2+r^2\dot\vartheta^2+r^2\sin^2\vartheta\cdot\dot\varphi^2)\,dt-Et = \]

\[ =m\int(\dot r\,dr+r^2\dot\vartheta\,d\vartheta+r^2\sin^2\vartheta\cdot\dot\varphi\,d\varphi)-Et. \tag{10b} \]

From equations (10a) and (10b) it is seen that

\[ p_x=\frac{dW}{dx}\qquad p_y=\frac{dW}{dy}\qquad p_z=\frac{dW}{dz} \tag{11a} \]

\[ p_r=\frac{dW}{dr}\qquad p_\vartheta=\frac{dW}{d\vartheta}\qquad p_\zeta=\frac{dW}{d\zeta}. \tag{11b} \]

In general, the momenta corresponding to one or another system of coordinates are the derivatives of the function \(W\) with respect to the coordinates.

Substituting these expressions for the momenta into the fundamental equation (7), we obtain:

\[ \frac{1}{2m}\left[\left(\frac{dW}{dx}\right)^2+ \left(\frac{dW}{dy}\right)^2+ \left(\frac{dW}{dz}\right)^2\right]+V(x,y,z)=E. \tag{12} \]

The expression

\[ \sqrt{\left(\frac{dW}{dx}\right)^2+ \left(\frac{dW}{dy}\right)^2+ \left(\frac{dW}{dz}\right)^2} \]

is nothing other than the absolute value of the gradient of the function \(W\), which, as is customary in vector analysis, we denote by placing the prefix \(\operatorname{grad}\) before the symbol of the function. We may thus write, instead of (12):

\[ (\operatorname{grad} W)^2=2m(E-V). \tag{13} \]

This equation determines the behavior of the derivatives of the function \(W\) with respect to the coordinates; complementing it is the relation following from (9):1

\[ \frac{\partial W}{\partial t}=-E, \tag{13a} \]

which determines the derivative of \(W\) with respect to time.

We have now reached the point where the paths of wave mechanics diverge from the paths of classical mechanics.

Following the path of classical mechanics, we would now have had to proceed to integrating the equations and to certain other transformations, as a result of which we would obtain equations representing the trajectories or orbits, but

which the particle we are considering should move. In the particular case which we have chosen, i.e. in the case of a central force acting inversely proportional to the square of the distance, these orbits would prove to be ellipses. In each particular case the magnitude and form of the ellipse would be determined in accordance with the value of the constant \(E\), and also with the values adopted for the other constants occurring in the course of the integration. As for the function \(W\), having played its role as a quantity facilitating the calculation of the orbits, it would disappear from the final result. As the physical reality there would remain an electron revolving in an ellipse around the nucleus, or a planet revolving around the sun.

Wave mechanics proceeds otherwise. It is based on the observation that equations (18) and (18a) together represent a family of waves propagating in space with velocity

\[ u=\frac{E}{\sqrt{2m(E-V)}}. \]

In order to discover this hidden meaning of equations (18) and (18a), let us imagine that at a given instant \(t_0\) the function \(W\) has a certain prescribed value \(W_0\) at all points of some surface \(S_0\); for example, the value \(W_0=1\) at the instant \(t_0=1\) at all points of a spherical surface with radius \(r_0=1\) around the origin of coordinates. We shall show that after a short time, at the instant \(t_0+dt\), there again exists a surface, at all points of which \(W\) has the value \(W_0\); only this is no longer the surface \(S_0\), but another surface \(S_1\), situated in such a way that the shortest distance from a point on the initial surface \(S_0\) to the new surface \(S_1\) is equal to1

\[ u\,dt=\frac{E}{\sqrt{2m(E-V)}}\,dt. \]

It is possible to prove this assertion without any difficulty. Let us suppose that at the instant \(t_0\) we are at the point \(P_0\) and

K. K. Darrow

we move with velocity \(u\) in the direction perpendicular to the surface \(S_0\). At the time \(t_0+dt\) we shall be at a point where the value of \(W\) is determined by the formula:

\[ \begin{aligned} W &= W_0+dW = W_0+\frac{\partial W}{\partial s}\,ds+\frac{\partial W}{\partial t}\,dt \\ &= W_0+\operatorname{grad} W\,ds+\left(\frac{\partial W}{\partial t}\right)dt \\ &= W_0+u\,\operatorname{grad} W\,dt-E\,dt, \end{aligned} \tag{14} \]

since during the interval of time \(dt\) we advance by the distance \(ds=u\,dt\) along the normal to the surface \(S_0\), i.e. in the direction in which \(W\) changes at the rate \(\operatorname{grad} W\); at the same time, according to equation (13a), at every point of space \(W\) changes with the passage of time at the rate \(-E\); thus, in general, by the instant of our arrival at \(S_1\), \(W\) increases to the value represented in equation (14).1 If, however, we now assume that the velocity of our motion is

\[ u=\frac{E}{\sqrt{2m(E-V)}} \tag{15} \]

and substitute in equation (14) the value of \(\operatorname{grad} W\) from equation (13), then \(u\,\operatorname{grad} W-E\) in equation (14) turns out to be equal to zero; in other words, at all points of space through which we pass, at the instant of our passage there will prevail one and the same value \(W_0\); we, so to speak, carry this value with us. The same may also be expressed in other words by saying that \(W\) propagates in space in the form of a wave front moving everywhere with the velocity indicated in equation (15).

\[ dW=\frac{\partial W}{\partial s}\,ds+\frac{\partial W}{\partial t}\,dt. \tag{a} \]

But the rate of change of \(W\) in the direction of the normal to the “level surface” \(S_0\), by definition, is nothing other than the gradient of \(W\). Next, \(ds=u\,dt\), and by (13a) \(\dfrac{\partial W}{\partial t}=-E\). Making use of this, we may write (a) in the following form:

\[ dW=u\,\operatorname{grad} W\,dt-E\,dt. \]

The reader will now quite naturally ask—if he has not done so already—the following natural question: what, in reality, is this function \(W\), which at first played only an auxiliary role, but has now unexpectedly acquired such central significance? The reader looks back, tries to grasp the intuitive meaning of the quantity \(W\), to form for himself a concrete conception of it. Unfortunately, I cannot do much to help him satisfy this very natural desire. I can only point out that \(W\) is precisely that same quantity which, under the name of “action,” plays so essential a role in the formulation of the mechanical principle of least action. This circumstance is hardly likely to make our conception of this function more intuitive; but at least our respect for it, and our belief in its important significance, are somewhat increased. I may further emphasize that, since no one has ever seen particles moving inside an atom, the conception of waves streaming, in a medium inaccessible to our observation, around the nucleus is no less “direct” than the conception of electrons, inaccessible to our direct observation, revolving in ellipses around the nucleus. (True, one may object to this that the revolution of planets around the sun visually illustrates the conception of electrons revolving around the nucleus, whereas no one has yet seen in the sky anything like the moving wave fronts of the function \(W\).) I may finally note that for many practical applications—in particular, for the prediction of the energies of stationary states—it is not important what the function \(W\) represents “in itself.” This is just as immaterial as it is immaterial, for the solution of a quadratic equation, whether the person who wrote down the equation had distance or time in mind when denoting the unknown quantity by the letter \(x\) or \(t\). In the practical use of wave mechanics one may simply begin with equation (20), taking it as the basis of the theory without further explanation or justification. In reality, however, there must be a deep inner connection between the new and the old mechanics, which

with such a mechanical method of introducing equation (20) will remain quite unnoticed. I could have confined myself here to Schrödinger’s own attempt to give the function \(W\) a real explanation (these attempts will be discussed in greater detail in the last part of the article). However, I should like the reader to form an idea of the function \(W\) independently, in the course of becoming acquainted with the foundations of wave mechanics.

We suggested that the reader regard equations (18) and (13a) as the description of a family of wave fronts moving forward with the velocity

\[ \frac{E}{\sqrt{2m(E - V)}}. \]

It is easy to notice that the description of the wave thus given is incomplete. In equations (13) and (13a) there is no indication either of the “wavelength” or of the “frequency” of the periodic process which constitutes the motion of the wave. If we were somehow to determine this frequency, then in equations of type (13) there would be no room left for it. These equations correspond approximately to the simple assertion that the crests of waves arising on the surface of water as a result of a falling stone spread out circularly with a definite velocity; or that sound signals from a very remote source of sound may be regarded as plane waves moving with a velocity of \(340\) m/sec. But in order actually to describe water or sound waves in detail, one must also specify their frequency and intensity; consequently, one must seek a more general wave equation. The same applies to the waves of the function \(W\).

In the study of ordinary oscillatory phenomena, such as the vibrations of stretched strings, membranes, etc., one usually makes use of the wave equation in the following general form:

\[ u^2\left(\frac{d^2\psi}{dx^2} + \frac{d^2\psi}{dy^2} + \frac{d^2\psi}{dz^2}\right) = u^2\Delta\psi = \frac{d^2\psi}{dt^2}. \tag{16} \]

In this equation \(\psi\) denotes the quantity which propagates in wave-like fashion, for example, in the case of mechanical vibrations—elongation, in the case of electrical oscillations—the field intensity, and so on. Equation (16) asserts that the acceleration with which this quantity changes at a definite

INTRODUCTION TO SCHRÖDINGER’S WAVE MECHANICS

at a given place in space (the right-hand side of our equation), is proportional to the “curvature” at the given point (the expression in parentheses on the left-hand side);1 for example, the acceleration with which a point of a displaced string tends toward its equilibrium position is proportional to the curvature of the string at that point, etc. Calculation further shows that the proportionality factor \(u^2\) is nothing other than the velocity with which the wave propagates—e.g. along the string. In equation (16) we use the usual abbreviated notation, according to which the sum of the second derivatives of the function \(\psi\) with respect to the three coordinates is denoted by the symbol \(\Delta\psi\). This sum is called the “Laplace operator” (not to be confused with the gradient, which is the sum of the squares of the first derivatives).

In considering ordinary mechanical oscillatory processes, one usually appends to equation (16) another equation:

\[ \frac{d^2\psi}{dt^2}=-4\pi^2\nu^2\psi, \tag{17} \]

where \(\nu\) denotes the frequency of oscillation. (This equation says that the acceleration is proportional to the displacement from the equilibrium position; the oscillation is thereby treated as harmonic, which is generally permissible for oscillations of small amplitude.)

Combining (16) and (17), we obtain:

\[ \Delta\psi+\frac{4\pi^2\nu^2}{u^2}\psi =\Delta\psi+\frac{4\pi^2}{\lambda^2}\psi =\Delta\psi+k^2\psi=0 . \tag{18} \]

Here \(\lambda\) is the wavelength \(\left(=\frac{u}{\nu}\right)\), and \(k^2=\frac{4\pi^2}{\lambda^2}\) has been introduced for brevity (\(k^2\), and not \(k\), in order, as is usually done, to show that the coefficient of \(\psi\) is essentially positive).

In the next part of the article we shall deal in greater detail with the application of equations (16), (17), and (18) to special mecha-

netic problems. Now let us turn to the wave propagation of the function \(W\), which we have partly—only partly—described by equation (13). We shall assume that the oscillation underlying this wave process is likewise subject to laws (16) and (17), and that equation (18) is therefore also valid for it. The velocity of propagation of the wave \(u\) is known from equation (15); in order to free the wave equation from unknowns, it remains to make an assumption concerning the frequency \(\nu\). We shall put

\[ \nu=\frac{E}{h}. \tag{19} \]

This assumption is neither useless nor arbitrary. On the contrary, it is the original and, in the highest degree, bold hypothesis to which we are indebted to de Broglie. According to this hypothesis, to every motion with energy \(E\)—and likewise to the simple translational motion of the electron—there corresponds a frequency \(\nu\), determined by the relation \(E=h\nu\).

Substituting in (18) \(u\) from (15) and \(\nu\) from (19), we obtain the fundamental equation of wave mechanics:

\[ \Delta\psi+\frac{8\pi^{2}m}{h^{2}}(E-V)\psi=0. \tag{20} \]

This is the wave equation of de Broglie and Schrödinger. In this form we shall use it throughout the further exposition. This form is sufficient for deriving the principal features of the structure of the atom, for example for a general explanation of the structure of the spectrum of hydrogen (without the fine structure), and even for explaining the diffraction experiments with electron beams (Davisson and Germer, G. P. Thomson). In short, this formula is sufficient for introducing into the world the wave mechanics. However, there is no doubt that it cannot be the full and final formula of this theory, since it needs supplementation in at least two respects.

The first obvious shortcoming of formula (20) is that it is based on classical Newtonian mechanics, and not on relativistic mechanics. Thus we must

to expect that this formula will prove valid only for motions with a velocity negligible in comparison with the velocity of light; it must represent only the limiting form of a general relativistic equation for the case of small velocities. Such a generalized relativistic formula was in fact derived by de Broglie. The initial development of the theory of spectra led one to suppose that precisely such a relativistic generalization of the wave equation would make it possible to include in the theory the explanation of the fine structure of the hydrogen spectrum. However, the most recent development of spectral theories has shown that the simple replacement of equation (20) by its relativistic generalization cannot be sufficient for this purpose; what appears necessary is the introduction into the theory, in one form or another, of the idea of an electron rotating about its own axis. Considerable progress has already been made recently in this direction; however, we cannot here undertake the consideration of this generalized wave equation.

The second shortcoming of formula (20) consists in its connection with equation (13). A characteristic feature of the latter equation is that in it the magnitude of the gradient of the function \(W\) is set equal to a certain function of the coordinates. This feature of the equation made it possible to derive from it the conception of “waves” fluctuating in space. Meanwhile, this relation could be obtained only because the system which we used as an example—namely, one particle in a central force field—possessed kinetic energy equal to the sum of the squares of the momenta (multiplied by a constant). But we can easily imagine systems not possessing this property. As a simple example one may indicate two particles of different masses moving in a force field, or a rigid rotating body of irregular shape. If we were to write, for the first of the systems mentioned, the kinetic energy and the momenta, we would obtain

\[ T=\frac{m_{1}}{2}\left(\dot{x}_{1}^{2}+\dot{y}_{1}^{2}+\dot{z}_{1}^{2}\right)+\frac{m_{2}}{2}\left(\dot{x}_{2}^{2}+\dot{y}_{2}^{2}+\dot{z}_{2}^{2}\right) \]

and \(p_{x_1}=m_1\dot{x}_1,\ p_{x_2}=m_2\dot{x}_2\), etc.; and only for \(m_1=m_2\) is \(T=\mathrm{const}\times \sum p_i^2\) obtained. Therefore, if we had taken as an example a mechanical system of a more general character, then instead of (13) we would have obtained another equation, which could not have been interpreted as an expression for a wave in three-dimensional Euclidean space. It is impossible, in the field connected with the motion of a system, to form a representation of a wave, and the path to equation (20) would be closed. The overcoming of this difficulty proves possible by means of a method that may be called “non-Euclidean geometry.” This mathematical theory gives formulas of a general character, which can be applied to any system, with any expression whatever for the kinetic energy. In the case of a single solitary particle in an empty field, these equations turn out to be identical with our equations (13) and (20). In the next-to-last geometry they continue to figure, concepts of wave, velocity of propagation \(c\), gradient, and Laplacian operator. I do not know, however, whether it makes sense to operate with these generalized concepts for those who are not sufficiently acquainted with this whole field. Therefore I shall confine myself to pointing out that non-Euclidean geometry gives a general wave equation, which contains (20) as a special case. This general equation has already succeeded in justifying itself when applied to certain models of atoms and molecules, as for example to the rigid rotator, which plays such an important role in the theory of band spectra.

However, the question of what these “waves” ultimately are becomes, in the transition from Euclidean space to the abstract configurational space, still darker and more obscure.

1. If the kinetic energy of the system is given in the form of a quadratic form of the velocities \(T=\sum_i\sum_j Q_{ij}\dot{q}_i\dot{q}_j\), and \(\Delta\) denotes the Laplace operator in the non-Euclidean space of configurations with metric \(ds^2=\sum\sum Q_{ij}\,dq_i\,dq_j\), then the general wave equation of de Broglie and Schrödinger has the form: \(h^2\Delta\psi+8\pi^2(E-V)\psi=0\).

It remains for us to take one more step in order to understand in what way wave mechanics can lead to the calculation of the energies of the stationary states of the atom.

It is widely known that in a body capable of serving as a medium for the propagation of waves and, at the same time, subject to some restriction or other in its motion, there arise so-called standing waves. Air in a closed vessel, a string clamped at its ends, a membrane fixed along its periphery may serve as examples of such bodies. Electricity behaves analogously in a circuit tuned to a definite frequency, and so on. In each of such systems, under appropriate conditions, standing waves arise, consisting of a characteristic alternation of nodes and antinodes. For their occurrence it is necessary that the frequency of the oscillation exciting the standing waves correspond to the “natural” or “resonant” frequency of the given system. To each such resonant frequency there corresponds a special pattern of the distribution of nodes and antinodes. As soon as a resonating system has been subjected to the action of an external wave of the appropriate frequency, the corresponding standing wave immediately arises in it; and, were it not for external and internal friction, once having arisen, the standing wave would have to preserve its existence forever. If an external wave acts on a resonator with a frequency that does not correspond to the resonator’s natural period, then a motion of a much more complicated character arises. The methods for calculating the natural frequencies of various systems and the standing waves corresponding to them constitute an essential field of theoretical acoustics.

The question arises: can the stationary states of natural atomic systems be regarded as standing waves, and the corresponding values of the energy of the atom as products of the frequencies of the atom’s natural oscillations by Planck’s constant \(h\)? Is it not possible that the problems of atomic theory can be solved by means of methods analogous to those applied in the investigation of macroscopic vibrators, such as acoustic instruments or oscillatory electrical systems? This idea was developed by E. Schrödinger in a series of articles beginning in 1926.

Examples of Stationary Oscillatory States

To illustrate the laws governing the formation of standing waves I shall give three examples: a stretched string, a membrane, and a sphere of liquid in a rigid shell. The first example is the simplest and the most widely known; all accounts of the theory of oscillations always begin with the study of a piano or violin string. From the physical point of view the problem of the string is characterized as a problem of one dimension (distance along the string), and from the mathematical point of view as a problem with two variables (the distance mentioned and time). The second example—a stretched membrane—is well known to anyone who has had occasion to deal with a telephone. However, the membranes used in practice for the most part do not fully correspond to the ideal case considered by us, since they are too thick. An ideally thin membrane is an example of an oscillatory system with two dimensions and three variable quantities. The example of the membrane will show us how important, in many cases, the proper choice of coordinates is, and we shall see what happens if one of the chosen coordinates turns out to be cyclic, i.e. an angle which practically returns to its initial value whenever it is increased by an integral multiple of \(2\pi\). In the problem of the membrane we shall encounter functions that are not so widely known as the simple sines and cosines that suffice in solving the problem of the stretched string. The third example we have chosen—a liquid sphere in a rigid vessel—is less frequently encountered in practice. It should help us transfer the results found in the study of the string and the membrane to problems of three dimensions and four variables. This problem will serve as the last step in the transition to the wave phenomena discovered by de Broglie and Schrödinger for illustrating the behavior of atoms. In order to pass to these latter, it will be enough to imagine strings and liquids not with constant density and elasticity, but with properties varying in a special way from point to point.

A stretched string. Let us imagine an infinitely long stretched string, whose direction coincides with the \(x\)-axis of the chosen coordinate system. Let us denote the tension of the string by the letter \(T\), and its linear density (i.e. the mass per \(1\ \mathrm{cm}\) of length) by the letter \(\rho\). In order to derive the differential equation of motion of the string, let us imagine it as consisting of a series of rectilinear segments (Fig. 1). When the string is displaced from its equilibrium position, the individual rectilinear segments no longer lie on one line, but, as Fig. 1 shows, form definite angles with one another. At the boundary of two segments the latter act on one another with a force determined by the tension \(T\). So long as the string is rectilinear, the forces to which each segment is subjected from its neighbors on one side and on the other are equal in magnitude and opposite in direction; their resultant is zero, and each segment is in equilibrium. But if the string is pulled aside, as in Fig. 1 (it being assumed that the string remains in the plane \(xy\)), then the mentioned forces, although still equal in magnitude, will no longer be opposite in direction. They have unequal components along the \(y\)-axis; these give a force different from zero, which draws the segment toward its normal position. More precisely, the following takes place: if the angle of the segment \(AB\) with the \(x\)-axis is denoted by \(\theta\) (Fig. 1), and the angle of the segment \(CD\) with the same axis by \(\theta+d\theta\), then the forces acting on the solid segment \(BC\) will have the components:

Fig. 1.

\[ \begin{aligned} X_1&=-T\cos\theta; & Y_1&=-T\sin\theta\\ X_2&=-T\cos(\theta+d\theta); & Y_2&=-T\sin(\theta+d\theta). \end{aligned} \]

Their resultants will be:

\[ \begin{aligned} X&=X_1+X_2=T\,[\cos(\theta+d\theta)-\cos\theta]\\ Y&=Y_1+Y_2=T\,[\sin(\theta+d\theta)-\sin\theta]. \end{aligned} \tag{21} \]

If, moreover, the angle \(\theta\) itself is small, i.e. if the deviation from the equilibrium position is insignificant, then, first, the component \(X\) of the resultant force is vanishingly small in comparison with the component \(Y\)—this is easily seen from considering the course of the curves \(\sin\) and \(\cos\) near zero—and, secondly, the component \(Y\) may be represented as:

\[ Y=T[\operatorname{tg}(\theta+d\theta)-\operatorname{tg}\theta]. \tag{21a} \]

Since for small \(\theta\), \(\sin\theta=\operatorname{tg}\theta\). Further, \(\operatorname{tg}\theta\) may be identified with the inclination of \(BC\) relative to the \(x\)-axis, i.e. with \(\dfrac{dy}{dx}\). We obtain:

\[ Y=T\frac{\operatorname{tg}(\theta+d\theta)-\operatorname{tg}\theta}{dx}\,dx =T\frac{d(\operatorname{tg}\theta)}{dx}\,dx =T\frac{d^2y}{dx^2}\,dx. \]

Equating the force to the product of the mass by the acceleration, we obtain:

\[ \rho\cdot\frac{d^2y}{dt^2}=T\frac{d^2y}{dx^2}. \tag{22} \]

Below we shall show that this equation, if one puts

\[ \frac{T}{\rho}=u^2, \]

is the equation (16) of the preceding paragraph, derived for the given particular case of a one-dimensional system. We see that this equation is strictly applicable to the limiting case of small deformations. But the elementary theory of oscillations deals precisely with these small oscillations.

Denoting differentiation with respect to time by dots, and differentiation with respect to spatial coordinates by primes, we may write (22) in the form:

\[ \ddot y=\frac{T}{\rho}y''. \tag{23} \]

Equation (23), which expresses the fact of a linear dependence of the second derivative of the displacement with respect to \(t\) on the second derivative of the same quantity with respect to \(x\), is the simplest of the wave equations.

We call this equation “wave” because it can (but need not) represent a wave. By a wave

we understand the deformation of the string—a curve or “something” else—which moves along an infinite string with constant velocity.

To illustrate this possible content of equation (23), let us suppose that at the moment \(t=0\) the string is displaced in such a way that it forms a sinusoidal curve with equation:

\[ y=A\sin mx \qquad (t=0). \tag{24} \]

Suppose further that all points of the string are moving at this moment with velocity

\[ \dot y=nA\cos mx \qquad (t=0) \tag{25} \]

parallel to the \(y\)-axis.

At some subsequent moment \(t\), the configuration and motion of the string are represented by the equations:

\[ y=A\sin (nt+mx);\qquad \dot y=nA\cos (nt+mx), \tag{26} \]

since these equations satisfy the general differential equation (23) and the initial conditions expressed in equations (24) and (25). In order that the initial conditions be satisfied, it is only necessary that between the constants \(n\) and \(m\), on the one hand, and the constants of the string \(T\) and \(\rho\), on the other, there exist a relation expressed by the formula:

\[ \frac{m}{n}=\sqrt{\frac{T}{\rho}}. \tag{27} \]

If condition (27) is satisfied, then the state of the string at all times remains expressed by equation (26).

Examining this equation more closely, we note that, according to it, those values of the displacement \(y\) and the velocity \(\dot y\) which at a given moment existed at some point of the string \(x_0\) can, after a time \(t\), be found at the point \(x_1\), separated from \(x_0\) by

\[ x_1-x_0=-\left(\frac{n}{m}\right)t. \]

In other words, these values move along the string with constant velocity. The entire configuration of the string, its sinusoidal form, and the distribution of velocities along it remain unchanged and merely slide along the string in the direction

decreasing \(x\). The form of the string is transmitted in this direction like a wave, and the ratio \(\frac{n}{m}\) measures the speed of motion of this wave:

\[ u=\frac{n}{m}=\sqrt{\frac{T}{\rho}}. \tag{28} \]

This conclusion justifies the name “wave” equation for equation (23), and of the coefficient \(\frac{T}{\rho}\) in that equation as the square of the wave speed.

The reader has probably noticed the circumstance that the result obtained by us could be achieved only with the aid of very narrow special conditions. We assumed the string to be infinitely long; we assumed the initial displacement to be produced along a sinusoidal curve. Likewise, we strictly prescribed the distribution of transverse velocities along the string at the initial moment. If we were to change these latter conditions, we would arrive at quite different results. If, for example, we were to suppose that at the moment \(t=0\) the string has a sinusoidal form and all its points are at rest, then the subsequent motion of the string would no longer be represented by equation (26). In this case we would have to resort to the general solution of the differential equation (23):

\[ y=C\sin(nt+mx)+D\sin(nt-mx) \tag{29} \]

and choose the constants \(C\) and \(D\) in such a way that they satisfy the initial conditions chosen by us:

\[ y=A\sin mx,\qquad y=0 \quad (\text{for } t=0). \tag{30} \]

For this purpose we set:

\[ C=D=\frac{1}{2}A \]

and obtain:

\[ y=A\sin nt \cos mx. \tag{31} \]

Equation (30) represents not a wave continuously moving along the string, but a standing oscillation like those,

which arise in a properly excited violin string or in Kundt’s tubes. At first sight it is hardly possible to guess that these standing waves are the result of the mutual superposition of two waves moving along the string toward one another with velocity \(u=\dfrac{n}{m}=\sqrt{\dfrac{T}{\rho}}\). However, investigation shows that a standing wave is always equivalent to two waves moving toward one another. In equation (31) the coefficients \(n\) and \(m\) are connected with one another by the wave-propagation velocity characteristic of the given string, and this equation can be rewritten in the form

\[ y=A\sin umt\cos mx,\qquad u=\sqrt{\frac{T}{\rho}}. \tag{32} \]

While the tension and density of the string, for a given \(m\), uniquely determine \(n\) (or conversely), nothing that has been said so far imposes any restrictions on the possible values of the coefficients \(n\) or \(m\). An infinitely long string can give vibrations with any wavelength. Such a string is also capable of simultaneously taking part in any number of vibrations of different wavelengths, with arbitrary relations among their amplitudes and phases. On this is based our right to prescribe any arbitrary initial conditions whatever concerning the form and velocity of motion of the string—of course, insofar as these conditions do not contradict the basic requirements of continuity and finiteness at all points of the string. In order to satisfy the requirement according to which the form of the string at the initial instant must be determined by any arbitrary function \(f(x)\), and the distribution of the transverse velocities of its motion by another, likewise arbitrary function \(g(x)\), it is sufficient to expand the functions \(f\) and \(g\) in Fourier series (or, if necessary, in a Fourier integral); each term in this expansion will correspond to a separate wave equation of type (29) with a special value of \(m\) and with values of \(C\) and \(D\) determined by the initial state of the string. The configuration of the string

at any later moment of time will be predetermined by the sum of all these wave equations. In this case the external appearance will correspond neither to an unchanging wave moving with constant velocity along the string, nor to a permanent distribution of nodes and antinodes. All the characteristic features of wave motion that ordinarily strike the eye may thereby be masked; nevertheless, mathematical analysis shows that the whole complex and changing picture of the string can be interpreted as the sum of sinusoidal waves, continuously running in both directions with one and the same constant velocity.

The situation changes, however, as soon as we introduce any boundary conditions; when they are introduced, the string proves capable of taking part only in oscillations of certain frequencies.

As the most usual and elementary boundary conditions, let us suppose that the string is fixed at the points \(x=0\) and \(x=L\), and let us concern ourselves only with the motion of the string on the interval lying between these two points.

In preparation for the subsequent conclusions it is necessary to return to the fundamental equation and solve it first. This equation is:

\[ \ddot{y}=u^2 y'', \tag{33} \]

where \(u\) denotes the velocity of propagation of a sinusoidal wave along an infinite string. We shall try to find a solution which would have the form of a product of a function depending only on the argument \(x\) and a function depending only on \(t\):

\[ y=g(t)\cdot f(x), \tag{34} \]

From the differential equation (33) it follows that the functions \(f\) and \(g\) must satisfy the following condition:

\[ \frac{f''}{f}=\frac{\ddot{g}}{u^2 g}. \tag{35} \]

Since the left-hand side of equation (35) does not depend on \(t\), and the right-hand side does not depend on \(x\), they can be equal to one another only on the condition that each separately is

constant. We shall denote the constant by the symbol \(-m^2\) (this notation has been chosen because—as we shall see below—defined in this way \(m\) is identical with the \(m\) encountered by us on p. 460). We may now write, instead of (35):

\[ \frac{f''}{f}=-m^2 \tag{35a} \]

and

\[ \frac{\ddot g}{u^2 g}=-m^2. \tag{35b} \]

Thus we have split our original equation (33) into two equations, each of which contains only one unknown. Solving them therefore presents no difficulty, and the general solutions have the form:

\[ \begin{aligned} f(x)&=A\cos mx+B\sin mx,\\ g(t)&=C\cos mut+D\sin mut. \end{aligned} \tag{36} \]

The quantity \(m\) has so far not yet been restricted in any way. We now turn to the boundary conditions. They are formulated as follows:

\[ f(0)=f(L)=0. \tag{37} \]

We have now arrived, by the simplest example, at the most characteristic problem of acoustics, which at the same time is decisive also for atomic theory in the form it assumes in wave mechanics—namely, the problem of “characteristic numbers.”

In order to make the function \(f\) satisfy the boundary conditions (37), we must obviously put \(A=0\) and \(\sin mL=0\) [in this case \(f(x)=0\) for \(x=0\) and \(x=L\)]. For this purpose we must choose for \(m\) the values:

\[ m=\frac{k\pi}{L}\qquad k=1,\ 2,\ 3,\ 4\ldots \tag{38} \]

Thus, the introduction of boundary conditions has led to the restriction of the possible values of the coefficient \(m\) to a definite series of numbers. Thereby the number of possible values of the wavelength \(\lambda\) is also restricted.

The values of \(m\) allowed under the given boundary conditions are called “characteristic numbers” (in German, Eigenwerte). To each “characteristic number” there corresponds a separate value of the wavelength and of the frequency of vibration,

\[ \nu=\frac{mu}{2\pi}, \]

the so-called “natural frequency” of the string (Eigenfrequenz).

Furthermore, to each characteristic number there corresponds a special solution of the differential equation—a special “fundamental function” (Eigenfunktion). In our example of a string fixed at the ends, the fundamental functions corresponding to the characteristic numbers \(m=\dfrac{k\pi}{L}\) are:

\[ y_k=\sin \frac{k\pi}{L}x\left(C_k\cos \frac{k\pi u}{L}t+D_k\sin \frac{k\pi u}{L}t\right). \tag{39} \]

Each fundamental function represents in this case a sinusoidal standing vibration, with nodes at the ends of the string and at \((k-1)\) equally spaced points between them. This kind of vibration can without difficulty be realized in practice with the aid of a violin string, provided \(k\) is not too large. The constants \(C\) and \(D\) determine the amplitude of the vibration and its phase at any given instant.

Of course, the motion of the string in practice need not necessarily correspond to one single fundamental function. On the contrary, the string may simultaneously take part in any number of different natural vibrations, each with its own value of the constants \(C_k\) and \(D_k\). Any number of fundamental functions (with different allowed values of \(m\), i.e. different integral multiples \(k\)) may coexist simultaneously. The actual displacement of the string at any given instant will be determined by the sum of the values of all these fundamental functions. In order for the motion of the string to be restricted to a single fundamental function, it is necessary to adjust, with infinite precision, the initial displacements and velocities of the string. On the other hand, for any arbitrary

with the choice of initial conditions, the subsequent motion of the string will be a superposition of various natural oscillations, with the values of the constants \(C_k\) and \(D_k\) characteristic of the given initial conditions; when these conditions are known, these constants can be calculated. This calculation is analogous to the calculation, in the mechanics of a material point, of the trajectory along which a particle will move when its position and velocity at the instant \(t=0\) are known. (The application of quantum conditions to the orbits of a particle corresponds to the determination of the natural frequencies; here lies the bridge between atomic models with electronic orbits and the atomic models of wave mechanics.) Both in acoustics and in wave mechanics, the process of determining amplitudes and phases is for the most part much more complicated than the process of calculating the natural frequencies; fortunately, it is often less essential, although in many cases it is necessary.

Example of a stretched membrane. The differential equation of a stretched membrane has the form:

\[ \Delta z=\frac{d^2 z}{dx^2}+\frac{d^2 z}{dy^2}=\frac{1}{u^2}\cdot\frac{d^2 z}{dt^2}. \tag{40} \]

Here it is assumed that the membrane is situated in the \(xy\)-plane and that \(z\) denotes the displacement of individual points of the membrane from their equilibrium positions in this plane in a direction perpendicular to it. The letter \(u\) denotes the velocity of propagation of a sinusoidal wave in an infinitely extended membrane, with tension \(T\), the same at all points of the membrane, and with constant surface density \(\rho\). This velocity is determined from the equation:

\[ u^2=\frac{T}{\rho}, \tag{41} \]

obtained by a natural generalization of equation (28). The actual motion of a bounded membrane may be very complicated, but it can always be decomposed into a certain number of sinusoidal waves traveling in opposite directions.

The symbol \(\Delta\) denotes the differential operator of Laplace; in rectangular coordinates it has the form

\(\dfrac{d^2}{dx^2}\) (for one dimension), \(\dfrac{d^2}{dx^2} + \dfrac{d^2}{dy^2}\) (for two) or \(\dfrac{d^2}{dx^2} + \dfrac{d^2}{dy^2} + \dfrac{d^2}{dz^2}\) (for three dimensions). When other coordinate systems are used, the Laplace operator, of course, assumes another form. In systems with two and with a larger number of dimensions, the problem of formulating boundary conditions is inseparable from the problem of choosing coordinates. If the membrane has a square form and is clamped along its edges, then we must choose rectangular coordinates; if the membrane is circular and is likewise fixed along its edge, then this condition can be formulated simply only in polar coordinates. The first of these two problems (the square membrane) differs by its extreme simplicity; the reader can easily solve it by a method analogous to that applied by us in the investigation of the string, and the results he will obtain will likewise be a simple generalization of the results obtained in the preceding paragraph. Of greater interest to us is the problem of the circular membrane, to which we shall now turn. In the investigation of such a membrane we must use polar coordinates, and the center of the membrane must, of course, serve as the origin of coordinates. The Laplace operator in polar coordinates in the plane has the following form:

\[ \Delta = \frac{d^2}{dr^2} + \frac{1}{r}\frac{d}{dr} + \frac{1}{r^2}\frac{d^2}{d\theta^2}. \tag{42} \]

We introduce this expression of the Laplace operator into the fundamental equation (40) and obtain:

\[ \frac{d^2 z}{dr^2} + \frac{1}{r}\cdot \frac{dz}{dr} + \frac{1}{r^2}\cdot \frac{d^2 z}{d\theta^2} = \frac{1}{u^2}\cdot \frac{d^2 z}{dt^2}. \tag{42a} \]

We seek the solution of this differential equation in the form of a product of a function \(f(r)\), depending only on the radius, by a function \(F(\theta)\), depending only on \(\theta\), and by a function \(g(t)\), depending only on \(t\). As in the preceding paragraph, we easily observe that each of these three functions is determined by an independent differential ...

equations, into which we can split the general equation (40). The path which leads to this conclusion is entirely analogous to the one we used in investigating the stretched string. First of all, substituting \(z=f(r)\cdot F(\theta)\cdot g(t)\) into equation (42), we have:

\[ \frac{1}{f}\cdot\frac{d^2 f}{dr^2} + \frac{1}{rf}\frac{df}{dr} + \frac{1}{r^2F}\frac{d^2F}{d\theta^2} = \frac{1}{u^2g}\cdot\frac{d^2g}{dt^2}. \tag{43} \]

Since the left-hand side of this equation does not depend on \(t\), and the right-hand side does not depend on \(r\) and \(\theta\), they can be equal to one another only on the condition that each of them is equal to a constant. We denote this constant, as in the preceding paragraph, by \(-m^2\), and thus obtain, instead of equation (42a), two independent differential equations:

\[ \frac{1}{f}\cdot\frac{d^2 f}{dr^2} + \frac{1}{rf}\cdot\frac{df}{dr} + \frac{1}{r^2F}\cdot\frac{d^2F}{d\theta^2} = -m^2 \tag{43a} \]

\[ \frac{1}{u^2g}\cdot\frac{d^2g}{dt^2} = -m^2. \tag{43} \]

The first equation contains only the spatial coordinates \(r\) and \(\theta\), the second—only the time \(t\). This latter equation has exactly the same form as the equation of the stretched string (35b), and we can now simply rewrite the solution (36b) of that equation:

\[ g(t)=A\cos mut+B\sin mut. \tag{44} \]

The result obtained by us in considering a string stretched between two points leads us to expect in advance that only certain values are possible for \(m\)—the characteristic numbers of the problem, which depend on the boundary conditions. This is indeed so. But before determining them, we must deal with the differential equation (43a) for the spatial coordinates \(r\) and \(\theta\).

The solution of equation (43a) is carried out by the same method of separation. If we multiply both sides of equation (43a) by \(r^2\), then we obtain a sum of three terms, which depend only on \(r\), and a term depending only on \(\theta\); this sum must be equal to zero. The latter is possible only in the

case, when the sum of three terms depending on \(r\) is equal to some constant, and the term depending on \(\theta\) is equal to the same constant. We shall denote this new constant by \(\lambda^2\). The two equations into which (43a) splits will then be:

\[ \frac{r^2}{f}\cdot \frac{d^2 f}{dr^2} + \frac{r}{f}\cdot \frac{df}{dr} + m^2 r^2 = \lambda^2. \tag{45a} \]

\[ -\frac{1}{F}\cdot \frac{d^2 F}{d\theta^2} = \lambda^2. \tag{45b} \]

We first solve (45b). This equation is entirely analogous to the equations already solved, (35a) and (43b), and its general solution will be:

\[ F(\theta)=C\cos \lambda\theta + D\sin \lambda\theta. \tag{46} \]

At first glance the coefficient \(\lambda\) appears to be entirely unrestricted. But in reality this coefficient carries certain restrictions within itself: for the coordinate \(\theta\) is cyclic in character, like geographical longitude. When \(\theta\) is changed by an integral multiple of \(2\pi\), we return to the former place, and the function \(F(\theta)\) must return to its initial value. To satisfy this condition it is necessary that \(\lambda\) have one of the values of the series:

\[ \lambda = 0,\ 1,\ 2,\ 3\ldots \tag{46a} \]

These are the characteristic numbers \(\lambda\), and the functions \(F(\theta)\) corresponding to each separate possible value of the coefficient \(\lambda\) are the fundamental functions of equation (45). In the case under consideration we have obtained the characteristic values of the coefficient and the fundamental functions not on the basis of any boundary conditions, but only on the basis of the fact that the coordinate is cyclic in character. Cases of this kind occur also in wave mechanics.

We pass to the third and last part of the problem—to the determination of the function \(f(r)\). This function is determined by the equation:

\[ \frac{d^2 f}{dr^2} + \frac{1}{r}\frac{df}{dr} + \left(m^2-\frac{\lambda^2}{r^2}\right)f = 0; \tag{47} \]

For each permitted value of \(\lambda\) we have a special equation (47). In the preceding section we had an analogous equation (35); the solution of that equation was a sine function of \(mx\). Equation (47) also leads to a function of the argument \(mr\); but this function is more complicated than the sine, namely the so-called Bessel function. To the values \(0, 1, 2, 3\ldots\) of the coefficient \(\lambda\) there correspond the so-called Bessel functions of order zero, 1, 2, 3, etc. We shall denote them by the symbols \(J_0(mr), J_1(mr), J_2(mr)\), etc.

Bessel functions are analogous to sines in that they too oscillate between positive and negative values as the argument \(r\) increases from 0 to infinity. For an innumerable set of discrete values of the argument, the Bessel function passes through 0. Unlike the zeros of the function \(\sin mr\), the zeros of the functions \(J(mr)\) do not lie at equal distances from one another. The corresponding values of the argument \(r\) may be found in tables of functions; we shall denote them by \(b^1, b^2, b^3\), in order of increasing magnitude. The superscript numbers denote, of course, not powers but ordinal numbers; this notation is adopted so as to leave free a place for the subscript distinguishing Bessel functions of different order.

Thus, the solution of equation (40) has the form:

\[ z = J_\lambda(mz)(C\cos \lambda\theta + D\sin \lambda\theta)(A\cos mut + B\sin mut). \tag{48} \]

This equation represents a standing oscillation of an infinitely extended membrane, in which \(\lambda\) lines passing through the origin remain stationary (nodal lines). Further, an innumerable number of concentric nodal circles, with center at the origin, remain stationary. The parts of the membrane situated between the nodal lines and the nodal circles are antinodes; they oscillate with frequency \(\frac{mu}{2\pi}\); the \(\lambda\) nodal lines are arranged at equal angles to one another; the radii of successive nodal circles \(r_1, r_2, r_3,\ldots\) are obtained by dividing by \(m\) the zeros \(b^1, b^2, b^3,\ldots\) of the Bessel function of order \(\lambda\).

What is the nature of the change in the character of the oscillation caused by fastening the membrane at the edges? Obviously, the oscillation represented by equation (48) can be realized in a membrane of radius \(R\), fixed immovably along its periphery, only in the case when \(R\) coincides with the radius of one of the nodal circles. Thus, \(R\) must be equal to \(\dfrac{b_i}{m}\). It will be more correct to express the converse: the standing oscillation (48) is possible in a membrane fixed along a circumference only in the case when the coefficient \(m\) satisfies one of the equations:

\[ m=\frac{b_i^1}{R}\ \text{or}\ \frac{b_i^2}{R},\ \text{or}\ \frac{b_i^3}{R}\ \text{and so on.} \tag{49} \]

These equations determine the characteristic numbers of the parameter \(m\) in the differential equation of the stretched membrane. There is a doubly infinite number of these characteristic numbers: to each of the infinite number of characteristic numbers \(\lambda\) there corresponds an infinite number of characteristic numbers \(m\). To each characteristic number \(m\) there corresponds a special natural frequency of the membrane and a special fundamental function—namely, the function (48) with the corresponding value of \(m\) taken from (49).

The constants \(A, B, C\), and \(D\) in the fundamental functions determine the amplitudes of the oscillations, their phase at each given instant, and also the directions of the nodal lines with respect to the chosen direction \(\theta=0\). An arbitrary number of natural oscillations may exist simultaneously. The actual motion of the membrane will be determined by the mutual superposition of these fundamental functions.

It hardly needs to be emphasized that all these results, just like the results obtained for a stretched string, are strictly valid only in the limiting case of infinitely small oscillations, and for practical purposes are valid with sufficient accuracy in the case of small oscillations.

Example of a liquid sphere. Among elementary oscillatory systems, a liquid sphere enclosed in a solid shell bears the greatest resemblance to the model of the atom

hydrogen in wave mechanics. The distribution of standing waves in both cases is almost completely identical.

The wave equation (16), written in polar coordinates for three-dimensional space, assumes a form that is somewhat frightening in its complexity:

\[ \ddot{\psi}=u^{2}\Delta\psi = u^{2}\frac{\operatorname{cosec}\theta}{r^{2}} \left[ \frac{d}{dr}\left(r^{2}\sin\theta\,\frac{d\psi}{dr}\right) + \frac{d}{d\varphi}\left(\operatorname{cosec}\theta\,\frac{d\psi}{d\varphi}\right) + \frac{d}{d\theta}\left(\sin\theta\,\frac{d\psi}{d\theta}\right) \right]. \tag{50} \]

The quantity \(\psi\) can no longer be considered in this case as a displacement from the position of equilibrium, since all three dimensions have already been used. The reader may, if he wishes, for the sake of vividness regard the quantity \(\psi\), for example, as the intensity of compression or rarefaction of the medium at a given point of the sphere, as in the case of sound waves. Perhaps the best preparation for wave mechanics will in general be not to associate any intuitive representations whatever with the quantity \(\psi\).

We attempt to find a solution of equation (50) in the usual way, i.e. we ask whether it can be solved by means of a function which is the product of four independent functions: one depending only on the time \(t\), another only on the radius \(r\), a third only on the longitude \(\varphi\), and a fourth only on the angle \(\theta\):

\[ \psi=g(t)\cdot f(r)\cdot \Phi(\varphi)\cdot \Theta(\theta). \]

As in the preceding examples, we find that the function of time \(g(t)\) must have the form

\[ g(t)=A\cos mut+B\sin mut. \tag{51} \]

The boundary conditions, as before, restrict the coefficient \(m\) and the frequency \(\dfrac{mu}{2\pi}\) to certain allowed values. The remaining functions of the spatial coordinates must satisfy the equations:

\[ \frac{1}{f} \left[ \frac{d}{dr}\left(r^{2}\frac{df}{dr}\right) + m^{2}r^{2}f \right] = -\frac{1}{X}\operatorname{cosec}\theta \left[ \frac{d}{d\varphi}\left(\operatorname{cosec}\theta\,\frac{dX}{d\varphi}\right) + \frac{d}{d\theta}\left(\sin\theta\,\frac{dX}{d\theta}\right) \right] = \lambda . \tag{52} \]

where \(Y\) denotes displacement along \(\varphi\), and \(\lambda\) is a constant which at first glance seems arbitrary, but in fact is restricted by certain values, for precisely the same reason as in the case of a flat membrane. For if the geographic longitude changes by an integer multiple of \(2\pi\), and the geographic latitude by an integer multiple of \(\pi\), then we return to the former place, and the function \(Y\) must acquire its former value. This is possible only if the condition is satisfied:

\[ \lambda = n(n+1) \qquad n = 0,\,1,\,2,\,3\ldots \tag{53} \]

These values of \(\lambda\) are, in essence, the characteristic numbers of the differential equation (52) for the variables \(\varphi\) and \(\theta\).¹

Corresponding to the characteristic numbers (53), the fundamental functions—the solutions of the second of the differential equations (52)—turn out to be functions of a completely new type, namely the so-called “spherical functions.” To each value of \(n\) in (53) there correspond “spherical functions of the \(n\)-th order.” Such a function consists of \(2n+1\) terms, each term containing an arbitrary coefficient. The values of the coefficients evidently correspond to different initial conditions, i.e. to different modes of vibration to which the same fundamental functions correspond.

What form does such a spherical function have? Each of its terms is the product of a function of the sine (or cosine) of \(\varphi\) separately, the so-called Legendre function, by a function of the variable \(\theta\).

¹ This and the following assertions concerning the functions \(Y_n\) can be proved by writing \(Y\), in the second of equations (52), as a product of a function of the variable \(\theta\) and a function of the variable \(\varphi\) separately, already employed five or six times in the course of the exposition, and leading to the splitting of it into two independent equations. The values of the parameter \(s\) in equation (54) are the characteristic numbers of the second of the latter equations. I have considered it desirable not to burden the exposition with the repeated repetition of the same procedures for solving equations. In the present case, moreover, there is the further circumstance that the separation of \(Y\) into two functions in the atomic model toward whose construction we are striving is not essential. The reader is therefore left to fill this gap himself.

If the Legendre functions are denoted by \(P_{n,s}\), then the fundamental function assumes the following form:

\[ Y_n(\theta,\varphi)=a_{n,0}P_{n,0}\cos\theta+ \sum_{s=1}^{n} a_{n,s}\cos(s\varphi)\cdot P_{n,s}(\cos\theta)+ \]

\[ +\sum_{s=1}^{n} b_{n,s}\sin(s\varphi)\cdot P_{n,s}(\cos\theta). \tag{54} \]

Each term of this equation represents a particular possible type of oscillation. The sum of all the terms represents the resultant motion arising from the superposition of these oscillations.

If we single out one definite oscillation, assigning to \(n\) some definite value \(n_1\), to \(s\) some definite value \(s_1\), and at the same time requiring that all the coefficients \(a\) and \(b\) in equation (54) vanish, with the exception of the coefficients \(a_{n_1,s_1}\) and \(b_{n_1,s_1}\), then we shall find that in this case the function \(Y=\Phi(\varphi)\cdot\Theta(\theta)\), and consequently also the function \(\psi=g(t)\cdot f(r)\cdot Y(\varphi,\theta)\), will assume the value \(0\) for \(s_1\) different values of the argument \(\varphi\) and for \(n_1-s_1\) different values of the argument \(\theta\). If we imagine a sphere described about the origin of coordinates, it is easy to see that the surface of this sphere will carry \(s_1\) meridional nodal lines and \(n_1-s_1\) nodal circles of latitude; along these lines the function will be once and for all equal to zero. If, from one spherical surface surrounding the origin of coordinates, we pass to the totality of all such surfaces, in other words to the whole volume of the sphere, we shall see that the oscillation of the fluid in the sphere is determined by the choice of two integers (I nearly said—quantum numbers!), and that the entire mass of fluid consists of compartments separated by \(s_1\) nodal planes intersecting along the axis \(\theta=0^\circ\), and by \(n_1-s_1\) double nodal cones with vertex at the origin of coordinates and with the axis \(\theta=0^\circ\) as their height.

Up to this point we have left out of consideration the dependence of the motion on the radius \(r\). The close analogy with the case of a stretched membrane makes the solution of this problem very easy. The differential equation (52) for \(f(r)\) resembles-

give. Consequently, equation (47) has several similar solutions:

\[ f(r)=\frac{1}{\sqrt[4]{r}}\,J_{n+\frac12}(mr). \tag{55} \]

The function (55) has the value \(f=0\) for an infinite series of discrete values of the variable \(r\), which we shall denote, in order of increasing magnitude, by the letters \(B_1,B_2,B_3\ldots\). At these places \(\psi\), and hence also the function \(\varphi\), vanish. In the geometric space of the ball of liquid, \(r\) can have any value; to this there will correspond an infinite series of concentric nodal spheres with radii

\[ \frac{B^1}{m},\quad \frac{B^2}{m},\quad \frac{B^3}{m}\ldots \]

If the liquid is bounded by a solid spherical surface of radius \(R\), then this surface must coincide with one of the nodal spheres just listed. This requirement can be satisfied only with the help of certain discrete proper values of the coefficient \(m\); these values are determined by the expression

\[ m=\frac{B^i}{R}. \]

The corresponding frequencies of the oscillations are the “proper frequencies” of the given ball; they are expressed by the formula

\[ \nu=\frac{B^i u}{2\pi R}. \]

The fundamental functions of the oscillation are expressed by equation (55), in which the parameter \(m\) is replaced successively by the permissible values

\[ \frac{B^i}{R}. \]

Thus the fundamental functions of oscillation of a liquid ball are products of a function of the radius \(r\), expressed by equation (55), with one of the permitted values of the constant \(m\), and a function (54) of the angles \(\varphi\) and \(\theta\), with permitted values of the constants \(n\) and \(s\), conditioned by the periodic character of these variables, and by a time function (51) with a frequency \(b\), determined by the boundary conditions of the ball. Each fundamental function, with definite values of \(m\), \(n\), and \(s\), corresponds to an oscillation in which the ball is divided into a definite number \(s\) of meridional planes, \((n-s)\) double cones, and a definite number of concentric spheres, along each of which the liquid remains permanently at rest. Within each region the liquid oscillates continuously with a definite frequency.

Atomic Models of Wave Mechanics.

The case of a “string” along which the wave velocity changes or even becomes imaginary.

Until now I have used the example of a stretched string, a membrane, and a liquid sphere to illustrate the properties of the differential equation:

\[ u^2 \Delta \psi = \frac{d^2 \psi}{dt^2}, \tag{56} \]

under the condition that \(u^2\) is a positive constant. In these cases \(u^2\) had the meaning of the quotient obtained by dividing one essentially positive quantity—the tension (or pressure)—by another, likewise essentially positive one—the density. This constant then turned out to have the physical meaning of the square of the velocity of propagation of sinusoidal waves in the given medium. In certain problems of wave mechanics we have to encounter a completely analogous equation. In a number of the most important applications of Schrödinger’s mechanics, however, one has to deal with equations of type (56) that differ from those considered above in that the coefficient \(u^2\) depends on the coordinates and sometimes even assumes negative values! From the point of view of ease of solution, equations of this kind may not differ essentially from those we have discussed; but the image of waves in an elastic medium, chosen by us for a visual explanation, proves unsuitable in this case. In the problem of one dimension, so long as \(u^2\) remains positive, we may imagine, as an example, a string whose density changes from point to point. When \(u^2\) passes through zero and becomes negative, the velocity of propagation of waves in our example becomes imaginary. Formally, nothing prevents one from speaking of a string with an imaginary velocity of propagation of waves; but in doing so the words are already deprived of almost any physical meaning. On the other hand, we know of no other words that could enliven, in this case, the monotonous sequence of equations.

Differential equations of type (55) with constant negative values of the coefficient \(n^2\) belong to the class of wave equations. Restricting ourselves to one dimension, we find for the solution of this equation in the case of a “string” with a constant speed of propagation a form of the following kind:

\[ z=(A\cos mUt+B\sin mUt)(Ce^{mx}+De^{-mx}), \tag{57} \]

where \(U\) denotes the real square root of \(-n^2\). With this function one obtains much less conveniently than with the usual sinusoidal function, obtained in the analogous case with a positive constant \(n^2\). Thus, for example, in this case it is impossible to find such characteristic numbers for the coefficient \(m\) for which the function would vanish identically at two specified points of the string, that is, the ordinary boundary conditions would be fulfilled. More precisely, the vanishing gives only one single value \(m=0\), which satisfies this condition, annihilating the entire function over the whole length of the string. Consequently, in this case \(\psi\) cannot be made to remain finite for any value of the variable except by equating to zero either the coefficient \(m\), or both constants \(A\) and \(B\), which means the complete annihilation of the function. (Conversion to infinity at certain points of the string, from the mathematical point of view, is not an obstacle to the formation of vibrations represented by sinusoidal functions of time.)

Let us now consider the more general equation

\[ \frac{d^2y}{dx^2}=(a-bx^2)\frac{d^2y}{dt^2}, \tag{58} \]

which may be interpreted as a wave equation for a string in which the velocity of wave propagation varies from point to point according to the condition \(u^2=(a-bx^2)\). In the middle part of the string, for values of \(x\) lying between \(-\sqrt{\frac{a}{b}}\) and \(+\sqrt{\frac{a}{b}}\), the velocity \(u\) will in this case have a real value; but on both sides of these two points it will be imaginary, all the way to \(x=\pm\infty\). By the usual methods we obtain from (58) the equations:

\[ y=f(x)g(t);\qquad g=A\cos \nu t+B\sin \nu t \]

\[ \frac{d^2 f}{d x^2}+\nu^2(a-bx^2)f=\frac{d^2 f}{d x^2}+(C-x^2)f=0 \tag{59} \]

[We take the constant \(\nu^2 b\) in (59) to be equal to 1, which does not harm the generality of the derivation.]

We must now deal with the solution of the equation for \(f(x)\). We try to find the solution for \(f(x)\) in the form of a power series multiplied by \(e^{-\frac{x^2}{2}}\) (a device often used in the theory of differential equations):

\[ f(x)=e^{-\frac{x^2}{2}}\sum_{n=0}^{\infty} a_n x^n . \tag{60} \]

We substitute this expression into the differential equation (59), cancel by \(e^{-\frac{x^2}{2}}\), and collect together all terms with the same exponent. Each such group of terms has the form:

\[ a_{n+2}(n+1)(n+2)x^n-a_n(2n+1-C)x^n . \tag{61} \]

Equation (59) will be satisfied if all these factors are equal to zero. Setting each group separately equal to \(0\), we obtain:

\[ \frac{a_{n+2}}{a_n}=\frac{(2n+1-C)}{(n+1)(n+2)} . \tag{62} \]

Let us now put \(a_0=0\), thereby causing all even coefficients to vanish; at the same time let us assign to the coefficient \(a_2\) some arbitrary value and compute, on the basis of this assumption, with the aid of (62), the values of all subsequent odd coefficients \(a_3, a_5, a_7\), etc. We may also proceed conversely—put \(a_1\) equal to 0, thereby eliminating all odd coefficients, assign some arbitrary value to \(a_0\), and compute all even coefficients \(a_2, a_4\), etc. By both methods we obtain solutions of equation (59) for any value of the parameter \(C\). However, it is easy to establish that for certain values of the parameter \(C\) the solutions will possess special properties.

Indeed, from (62) one may conclude that we shall obtain entirely different results depending on whether the parameter \(C\) is equal to one of the odd integers \(1, 2, 5\ldots\), or whether it has some other value. For if \(C\) is equal to an odd positive integer, then the series of coefficients terminates at the term with the corresponding value of \(n\); this and all subsequent coefficients turn out to be equal to zero. Thus the power series which we have chosen for solving our differential equation will consist of a finite number of terms. If \(C\) is not equal to an odd positive integer, then this power series will be infinite.

We have before us a new kind of “characteristic numbers.” If the parameter \(C\) in the differential equation for the eigen “string” which we are considering has one of the values

\[ C = 2n + 1,\quad n = 0, 1, 2, 3, \tag{63} \]

then the solutions have a special character.

Let us consider in what the peculiarity of these solutions consists. If the parameter \(C\) has some other value, different from (63), then the series \(\sum a_n x^n\) is infinite. When \(x\) increases to infinity, the sum of the series grows with such rapidity that it outweighs the simultaneous constant decrease of the factor \(e^{-\frac{x^2}{2}}\); thus the function \(f(x)\) at both ends of the interval \(-\infty < x < +\infty\) becomes infinitely large. If, on the contrary, \(C\) is equal to one of the “characteristic numbers” (63), then the series \(a_n x^n\) terminates: as \(x\) increases to infinity, the decay of the factor \(e^{-\frac{x^2}{2}}\) in this case outweighs the growth of the sum of the power series, and the function \(f(x)\) remains finite even for \(x = \infty\). Thus, the “characteristic numbers” of the coefficient \(C\), and only they alone, make it possible to give for the differential equation (59) solutions which remain finite for any value of the independent variable from plus to minus infinity. This finiteness condition replaces, in the present case, the usual boundary conditions of a stretched string.

The fundamental functions of equation (59), corresponding to these characteristic numbers, are:

\[ f_m(x)=e^{-\frac{x^2}{2}}H_m(x) \tag{64} \]

The symbol \(H_m(x)\) denotes a finite series \(\sum a_m x^m\), constructed by the methods described above and ending with the \(m\)-th term. These series are known under the name of Hermite polynomials.

Interpretation of the linear harmonic oscillator in wave mechanics.

The preceding paragraph contains everything necessary for Schrödinger’s theory of the linear harmonic oscillator. The simplest linear harmonic oscillator—i.e. a material point which is connected by an elastic force with the position of equilibrium and can oscillate about this position of equilibrium—is known to have played an extraordinarily important role in the history of quantum theory. It was the first system for which Planck proposed that it possesses the property of absorbing and emitting energy only in the form of quanta of a definite finite size. As is known, with the aid of this assumption Planck succeeded in solving the problem of black-body radiation, and it served as the foundation of the entire theory of quanta.

Let us imagine a particle of mass \(m\), capable of moving only along the \(x\)-axis and attracted toward the origin of coordinates by a force proportional to its distance from it (\(-k^2x\)). It is known that if a particle bound in this way is displaced from its position of equilibrium and then again left to itself, it will perform elastic oscillations about the center of equilibrium with frequency

\[ \nu_0=\frac{k}{2\pi\sqrt{m}}. \]

Its potential energy is determined by the following function of the coordinate \(x\):

\[ V=\frac{1}{2}k^2x^2=2\pi^2m\nu_0^2x^2. \tag{65} \]

The wave equation (20) takes the form:

\[ \frac{d^2\psi}{dx^2}+\frac{8\pi^2m}{h^2}\left(E-2\pi^2m\nu_0^2x^2\right)\psi=0. \tag{66} \]

Let us introduce the variable \(q = r \cdot 2\pi \sqrt{\dfrac{m\nu_0}{h}}\), which leads to an equation of the form

\[ \frac{E^2}{4q^2} + (C - q^2)^2 = 0, \qquad \text{i.e.} \qquad C = \frac{2E}{h\nu_0}. \tag{67} \]

i.e. of the form corresponding to the normal equation (53).

According to the theory of Hill’s determinant, stationary states of a linear oscillator are determined by those values of the parameter \(E\), i.e. of the oscillator energy, for which equation (67) has solutions that remain finite for any value of the independent variable \(x\), including \(x=\infty\). These energy values correspond to the values of the parameter \(C\) which, in the preceding paragraph, we called characteristic numbers and which are determined by equation (63).

The energy of a linear oscillator in its stationary states is thus determined by the equation

\[ E_n = \frac{h\nu_0}{2}(2n+1) \qquad n=0,1,2\ldots \]

Consequently,

\[ E_n=\frac{1}{2}h\nu_0,\quad \frac{3}{2}h\nu_0,\quad \frac{5}{2}h\nu_0\ldots \tag{68} \]

Thus, quantum mechanics leads to values of the energy of a harmonic oscillator with frequency \(\nu_0\) that are products of the fundamental factor \(h\nu_0\) by the successive “half-integer” numbers \(1/2\), \(3/2\), \(5/2\), etc.

Thus the harmonic oscillator is an example of a system with half-integer quantum numbers. In most earlier theories it was accepted (whether explicitly or tacitly) that the energy of Planck’s oscillator in stationary states is determined by integer multiples of the product \(h\nu_0\). However, already in the interpretation of certain band spectra it was noted that better agreement between theory and experiment is obtained by allowing half-integer quantum numbers for the vibrations of atoms in a diatomic molecule (which, in the first approximation, may be regarded as harmonic oscillators).

The fundamental functions corresponding to the individual stationary states of the harmonic vibrator are:

\[ \psi_n(x)=\mathrm{const}\cdot e^{-2\pi^2 m\nu_0\cdot \frac{x^2}{h}}\, H_n\!\left(2\pi x\sqrt{\frac{m\nu_0}{h}}\right). \tag{69} \]

The first five of these functions are shown graphically in Fig. 2.1 If he wishes, the reader may regard these curves as systems of “antinodes” and “nodes” that are formed on five vibrating strings, the velocity of propagation of vibrations in which varies according to five different

Fig. 2.

laws, obtained by substituting the first five values of the energy \(E\) [from the series (69)] into the equation:

\[ u=\frac{E}{\sqrt{2m\left(E-2\pi^2 m\nu_0^2 x^2\right)}}. \tag{70} \]

Thus, the various stationary states of the linear oscillator are represented, according to Schrödinger, not by the fundamental

oscillations and overtones of one string, but with the fundamental (and the only) oscillations of a number of different strings. [Perhaps it will be more convenient for some to imagine, as Dr. Fry proposes, instead of several strings, a single one, but with different propagation velocities for waves of different frequency.]

The atom of hydrogen is interpreted in wave mechanics. The atom of hydrogen is regarded in wave mechanics, as also in Bohr’s model, as potential energy \(V=-\frac{e^2}{r}\). I recall that this formula for the potential energy is obtained on the basis of representing the nucleus and the electron as point particles with charges \(+e\) and \(-e\), situated at a distance \(r\) from one another. Meanwhile the electron and the nucleus, as point particles, do not enter explicitly into the new theory, and nevertheless the potential energy, taken from the Bohr model, is laid at the foundation of Schrödinger’s model of the hydrogen atom.

The potential energy of such a form indicates the necessity of making use of polar coordinates in formulating the problem. The wave equation (16), by means of (15), will in this case be written in the form:

\[ -\frac{E^2}{2m\left(E+\frac{e^2}{r}\right)}\,\Delta\psi=\frac{d^2\psi}{dt^2}. \tag{71} \]

Substituting the frequency of the oscillations \(\nu\), expressed by \(\frac{E}{h}\), we obtain:

\[ \Delta\psi+\frac{8\pi^2 m}{h^2}\left(E+\frac{e^2}{r}\right)\psi=0. \tag{72} \]

The similarity of this equation with that derived above for the oscillations of a liquid sphere catches the eye; the analogy here is the same as that between a harmonic oscillator and a stretched string. In the hydrogen atom we have, as it were, a liquid sphere, the propagation velocity of the oscillations in which changes from point to point according to the formula:

\[ u^2=\frac{E^2}{2m\left(E+\frac{e^2}{r}\right)}. \tag{73} \]

We must now find the corresponding standing oscillations.

If \(E\) is a positive constant, then the velocity of propagation of waves throughout the entire mass of the sphere is real. In this case boundary conditions of the usual type can be prescribed (for example, the condition according to which the sphere must be bounded by a solid sphere of known radius). We can compute, for given boundary conditions, the “characteristic numbers” of the parameter \(E\), and the distributions of standing oscillations in the fluid corresponding to these characteristic numbers, as well as their frequencies. In the absence of any boundary conditions, equation (72) can be solved for any value of the parameter \(E\).

But if we assign negative values to \(E\), the situation changes. Now \(u\), the velocity of propagation of waves, is real only within the limits of a sphere of radius \(-\dfrac{e^2}{E}\); on the surface of this sphere it becomes equal to 0, while outside it becomes imaginary. This reminds us of the oscillator and the string that we used to illustrate it. In the latter case the velocity of propagation of oscillations was real in the middle part and imaginary at both ends. There are essential differences between the two cases: in the case of the hydrogen atom the variable \(r\) has only positive values, and at the point \(r=0\) the velocity, although infinitely large, is real.

Considering a string with variable and, in a certain region, imaginary velocity of propagation of oscillations, we found that the law of variation of the velocity can be chosen in such a way that standing oscillations with a constant distribution of nodes and antinodes and with definite proper periods of oscillation are formed in the string. For this purpose it was necessary to choose for the parameter one of the values of a certain series. The very same thing we shall have to do in the case of the hydrogen atom.

We try to find for equation (72), as before, a solution in the form of the product of a function of the angles \(\varphi\) and \(\theta\) by a function of the radius \(r\). Proceeding in this familiar way, we arrive at the equation:

\[ \operatorname{cosec}\theta\,\frac{d}{d\theta} \left(\sin\theta\,\frac{dY}{d\theta}\right) + \frac{d}{dz}\left(\operatorname{cosec}^{2}\theta\,\frac{dY}{dz}\right) = -\lambda Y . \tag{74} \]

and

\[ -\frac{d}{dr}\left(r^{2}\frac{dT}{dr}\right) + \frac{8\pi^{2}mr^{2}}{h^{2}}\left(E+\frac{e^{2}}{r}\right)T = \lambda T . \tag{75} \]

Equation (75) is identical with the one which we had occasion to encounter in discussing the hydrogen atom (52). Here, as there, it follows from the physical character of the variables \(z\) and \(\theta\) that the constant \(\lambda\) can assume only certain values—“characteristic numbers”:

\[ \lambda=l(l+1), \qquad l=0,1,2,3\ldots \tag{76} \]

Equation (75), therefore, differs only insignificantly from the corresponding equation (52) in the preceding section. Here there appears the difference between the properties of real liquids and the properties of that artificial “imaginary” liquid which serves as a material for constructing the model of the hydrogen atom in wave mechanics.

If in equation (75) we assign to the parameter \(E\) an arbitrary negative value, then in the general case an equation is obtained which has no solutions remaining finite both at the origin of coordinates and as \(r\) increases to infinity. In this case we have a position analogous to that which we saw in the example of the linear oscillator. And there, an arbitrary choice of the parameter, denoted by the letter \(G\), led to the impossibility of finding a solution in which the amplitudes would remain finite at both ends of the string. Schrödinger showed, however, that there is a series of values of the parameter \(E\)—characteristic numbers—which make it possible to give single-valued, continuous, and finite solutions for any value of the variable \(r\).

These characteristic numbers are as follows:

\[ E_n=-\frac{2\pi^{2}me^{4}}{h^{2}n^{2}}; \qquad n=1,2,3,4\ldots \tag{77} \]

The successive possible values of the energy in a system with potential energy \(-\dfrac{e^{2}}{r}\), i.e. the energies of stationary states

Introduction to Schrödinger’s Wave Mechanics

the hydrogen atom are determined by wave mechanics as quotients obtained by dividing the fundamental factor

\[ -\frac{2\pi^{2}me^{4}}{h^{2}} \]

by the squares of the consecutive integers 1, 2, 3, 4.

This conclusion agrees with the results of experiment. Formula (77) is nothing other than a repetition of Bohr’s formula, on which the whole modern theory of spectra is based. This formula has been so brilliantly justified in practice that hardly any new theory of the structure of atoms could hope for recognition if it, on its part, did not also lead to it.

If we wish to form a clear picture for ourselves of the Schrödinger hydrogen atom, then for each atom we must imagine a special fluid filling all infinite space, the velocity of propagation of oscillations in which, for each stationary state, depends differently on the radius. The law of variation of this velocity is obtained in each individual case by substituting into (73) the corresponding value of the parameter \(E\), chosen from the series (77). If in (73) we substitute some other, arbitrary chosen value of \(E\), then we can likewise picture the resulting equation as expressing the motion of an imaginary fluid; but this fluid will not be capable of maintaining within itself a standing and continuous oscillation with amplitudes finite everywhere. Only for the Bohr values of the energy do we obtain systems capable of resonating after the manner of a sphere made of a real fluid.

Our next task is to ascertain the character of those standing oscillations which correspond to the individual stationary states. This problem is far more complicated than in the case of the imaginary strings corresponding to the various stationary states of the harmonic oscillator. The difficulties are connected not only with the transition from one dimension to three, but also with the fact of mathematical “degeneracy” characteristic of the present problem. This degeneracy entails a non-uniqueness of the solution: to each allowed value of the energy \(E\) (apart from the first)

corresponds to one, while multiple different values. To clarify this fact, it is necessary to return to both equations (74) and (75).

Since equation (74) is identically satisfied with the equation we found for a pair from the real liquid, the distribution of the latter waves in the perturbed element III is not distinguished from that in the entire sphere, insofar as the matter concerns the dependence of the amplitudes on the nodal diameters \(z\) and \(b\). The perturbed liquid, just like the real one, is divided into sections bounded by nodal planes, nodal double cones and nodal spheres, and the distribution of the nodal planes and double cones is identical with that in the case of the real liquid for identical characteristics of the numbers. The only distribution of nodal spheres, which depends essentially on the function \(f(r)\) in the case II, is quite different, since, as equation (75) shows, for \(f(r)\) it is essentially determined by equation (52).

Thus, only one fundamental function of equation (75) corresponds to each value of the characteristic number for the parameter \(E\). In the second characteristic number there correspond to this number two fundamental functions, in the third—three, etc. This limitation of the multiplicity of fundamental functions is connected with the limitation to which the values of the parameter are subject. If, in the expression of the function \(\varphi\) in the form of a product of two functions—

\[ \varphi(r, b, z) = F(r)\cdot X_l(b, z) \tag{78} \]

we assign to the parameter \(E\) in the first set some one of the values permitted for it, then we still have a free choice among the various possible values of the parameter \(l\) in the second set. However, this choice is limited by one condition. We have the right to take for \(l\) a value no greater than the value \(n\) chosen by us; otherwise \(E_n\) will not be a characteristic number of equation (75) in the sense meant by us. Thus, for \(n=1\) we are restricted to the value \(l=0\); for \(n=2\), to the values \(l=0\) and \(1\), etc. To each characteristic

to the number \(E_n\) there correspond \((n-1)\) different spherical functions \(Y_1(\theta,\varphi),\,Y_2(\theta,\varphi)\ldots Y_{n-1}(\theta,\varphi)\) as possible solutions of equation (74). Each of these solutions gives a special fundamental function \(F_{n,l}(r)\) of equation (75). If we introduce a new variable

\[ \rho=\frac{2\pi\sqrt{-2mE_n}}{h}\,r =\frac{4\pi^2\eta e^2}{n h^2}\,r =\frac{1}{n a_0}\,r \]

instead of \(r\) \(\left(a_0=\dfrac{h^2}{4\pi^2 m e^2}\right.\) is the radius of the first “Bohr orbit”\()\), then the fundamental functions in question take the following comparatively simple form: \({}^{1}\)

\[ X_{n,l}(\rho)=\mathrm{const}\,\rho^{\,l} e^{-\rho} \sum_{k=0}^{n-l-1} \frac{(-2\rho)^k}{k!} \binom{n+l}{\,n-l-1-k\,}; \tag{79} \]

the function \(X_{n,l}\) has \((n-l-1)\) roots, so that the corresponding oscillation must be characterized by \((n-l-1)\) nodal spheres. Thus, to each permitted value \(E_n\) there correspond \(n\) different solutions of the general equation (72), which differ from one another in the number of nodal spheres:

\[ \psi_{n,l}(r,\theta,\varphi)=X_{n l}(\rho)Y_l(\theta,\varphi); \qquad l=0,1,2\ldots(n-1). \tag{80} \]

Each of these equations represents an allowed class of oscillations, each corresponding separately to one of the terms of which the spherical function is composed, according to equation (54).

Thanks to the subdivision of the spherical functions, for the \(n\)-th permitted value of the parameter \(E\) there are

\[ (1+2+3+\ldots n)=\frac{n(n+1)}{2} \]

different modes of oscillation.

Equation (80) represents the different oscillations in which the imaginary liquid formation that serves us as a model of the hydrogen atom may take part. We might try to describe in detail, and possibly visually, the character

\({}^{1}\) The multiplier in parentheses in equation (79) represents the “number of combinations of \((n+l)\) taken \((n-l-1-k)\) at a time,” i.e. the \((n-l-1-k)\)-th coefficient of the binomial \((a+b)^{n+l}\).

oscillations in each individual case. However, I doubt whether this undertaking makes sense. At one time much energy and art were expended on the description and depiction of various electronic orbits, whose shadows so quickly “became obsolete.” Who will dare assert that the same fate will not befall, after a few years, the picture that we can now form for ourselves with the aid of the imaginary oscillating fluid? Nevertheless it may be regarded as probable that, at least for the next few years, the image of an oscillating fluid will be the most adequate one in interpreting experimental data in the field of the theory of the structure of the atom. Therefore I shall nevertheless allow myself to indicate a few details of the character of the oscillatory processes corresponding to the three most “deep” (i.e. lowest in energy) states of the hydrogen atom.

The normal state, \(n=1\). One fundamental function; one exponential function of \(r\), decreasing from the origin of coordinates to infinity without nodal spheres. The corresponding angular function \(Y_0(\theta,\varphi)\) is a constant. The oscillation is represented by the equation:

\[ \psi(r)=\mathrm{const.}\ e^{-\frac{r}{a_0}};\qquad a_0=\frac{h^2}{4\pi^2 m e^2} \tag{81} \]

and possesses perfect spherical symmetry.

The first excited state, \(n=2\) (the final state after emission of a line of the Balmer series). Two fundamental functions, \(X_{2,0}\) and \(X_{2,1}\). The first represents an oscillation with one nodal sphere; the second has no nodal spheres at all, and the amplitude of the oscillation decreases exponentially on passing from the center to infinity. The first fundamental function must, in order to obtain the general formula of motion, be multiplied by \(Y_0(\theta,\varphi)\). Since \(Y_0\) is a constant, this oscillation is characterized by complete spherical symmetry. The second fundamental function is to be multiplied by \(Y_1\); this latter factor consists of the terms written out completely in equation (54); the various systems obtained as a result of multiplication

oscillations are characterized by the presence of nodal planes and cones, and therefore cannot possess spherical symmetry. The reader may clarify for himself, on the basis of equation (55), the arrangement of the nodal surfaces in these cases.

The second excited state, \(n=3\) (the initial state of the atom in the emission of the \(H\alpha\) line). Three fundamental functions \(X_{3,0}\), \(X_{3,1}\), and \(X_{3,2}\). The first corresponds to an oscillation with two nodal spheres and with perfect spherical symmetry. The second and third represent oscillations with one nodal sphere and with none at all. Being multiplied by the spherical functions \(Y_1\) and \(Y_2\), these oscillations acquire nodal planes and cones and lose spherical symmetry.

In general, a stationary state characterized by the value of the parameter \(E_n\) possesses \(n\) fundamental functions, corresponding to oscillations with \(0, 1, 2, 3,\ldots,(n-1)\) nodal spheres; to the fundamental function with the maximum number of nodal spheres there corresponds only one kind of oscillation, distinguished by complete spherical symmetry. The other fundamental functions each correspond to several kinds of oscillations, with different numbers of nodal planes and cones.

If the examples we have given are to serve as models of a language for describing atomic phenomena, then it is necessary to compile a dictionary with the help of which one could translate into this language the expressions that we are accustomed to use when explaining ourselves in the language of the present—that is, in the language of the atomic theory of Bohr and Sommerfeld. This dictionary will have to contain definitions of the following sort: the number \(n\) is the so-called principal quantum number of the electron orbit; the number \(l\), smaller by 1 than the so-called azimuthal or “subsidiary” quantum number \((k)\) in the Bohr model; the number \((n-l-1)\), determining the number of nodal spheres, corresponds to the radial quantum number of the electron orbit. To make the meaning of these definitions clearer, I shall recall that the model of the hydrogen atom proposed by Bohr and Sommerfeld assigned to the atom, in the \(n\)-th energy state, \(n\) different orbits, of which one is circular and the remaining \((n-1)\) are elliptical with different eccentricities. (The introduction into this model of the conception of rota-

…an electron, seem somewhat antiquated, so that from this point of view our vocabulary belongs not to the language of the present, but to the language of “hoary antiquity,” i.e. approximately 1925.) Elliptical orbits are selected in the Bohr–Sommerfeld model by means of a quantum condition according to which the integral $\int p_\varphi d\varphi$ of the angular momentum $p_\varphi$, taken along the whole closed orbit, must be equal to the product of $h$ by an integer $k$, less than or equal to the principal quantum number $n$. At the same time the integral $\int p_r dr$ of the radial momentum $p_r$ must be equal to the product of $h$ by the integer $(n-k)$, so that the sum of the integrals $\int p_\varphi d\varphi + \int p_r dr$ will be equal to $n$. The numbers $n$, $k$, and $(n-k)$ were called, in the Bohr model, the principal, azimuthal, and radial quantum numbers. (Definitions of the sort given above should permit the transition from the developed orbits in earlier atomic models to the various kinds of oscillations in models of the new type.)

Perturbations

According to what was set forth above, wave mechanics leads to the conclusion that to each allowed value of the energy $E_n$ there correspond $n$ different modes of oscillation, differing from one another in the number of nodal spheres (not to mention the different distributions of nodal planes and cones). The question naturally arises: is it possible in practice to distinguish these oscillations from one another and to establish to which of them, or to which combination of them, a given concrete state of the hydrogen atom corresponds?

Translating this question into the language of the Bohr–Sommerfeld model, we obtain the following formulation: can one in each concrete case say on which particular one of the permitted elliptical orbits the electron is located?

To this question Bohr’s theory replied that the electron is in fact not in a strictly Coulomb field, so that the force acting on it is not exactly inversely proportional to the square of its distance from the nucleus. To the Coulomb force of attraction of the electron by the nucleus there must, in an exact calculation, be added still some perturbing force. Calculation shows that in the presence…

the energy of the perturbing force, the energies of the different electronic orbits with the same quantum number \(n\) cease to be identical. Let us imagine, for example, an atom consisting of a nucleus with charge \(11e\) and 10 electrons grouped near it, while the eleventh electron is in an orbit with a considerably larger axis (the usual model of the sodium atom). In this case, to a first approximation, the same force will act on the outer electron as if it were in the field of a nucleus with charge \(1e\). However, in a more exact calculation it will be necessary to take into account that the inner electrons do not completely coincide with the nucleus; this fact can be allowed for by introducing into the calculation a special perturbing force. In the presence of such a force, those \(n\) electronic orbits which are possible for the outer electron at an energy equal, in the first approximation, to \(E_n\), will no longer be mutually identical: the \(n\)-fold stationary state of the atom will split into \(n\) different stationary states, somewhat different from one another. Even in the hydrogen atom, despite the absence of inner electrons, one cannot dispense with the introduction of a perturbing force: its source is the dependence, required by the theory of relativity, of the electron mass on its velocity. This circumstance causes the stationary states of hydrogen to split into separate levels lying close beside one another; in the spectrum this splitting is observed as the appearance of the so-called fine structure of the lines.

Something analogous is also observed in the more exact derivation of stationary states by the methods of wave mechanics. If one introduces a “perturbing term” into the expression representing the potential energy of the atom in the wave equation, then one may hope that this correction will make it possible to distinguish from one another the various fundamental functions which originally corresponded to one and the same characteristic number, and to determine which of the allowed oscillatory systems corresponds to reality. (In the language of mathematics one may say that the introduction of perturbing forces destroys the degeneration of the problem; this destruction may be complete or only partial.)

In this domain wave mechanics gives exactly the same results as the original atomic theory of Bohr and Sommerfeld. This result is not very encouraging, for several years ago it became clear that the Bohr–Sommerfeld theory needs to be extended in order that it may successfully explain the details of the fine structure of spectra. This improvement consisted in introducing the conception of the electron as rotating about its own axis. Something analogous must, evidently, be done in wave mechanics as well; otherwise it will reveal the same shortcomings as the original theory of electronic orbits, which did not take account of the electron’s own rotation—for example, this theory could not fully explain the analogy between the spectra of hydrogen and of alkali metals.

There is one case of perturbing forces in which the conclusions of the Bohr–Sommerfeld theory agree, in the first approximation, with the conclusions of wave mechanics and at the same time agree with the experimental data, without its being necessary to resort to the aid of a rotating electron. This is the domain of the so-called Stark effect, i.e. the case in which the perturbing force arises from an external electric field. Since this example may serve as a convenient transition to the question to which the last part of the present article will be devoted, I shall dwell on it in somewhat greater detail.

The Stark effect. Let us imagine a hydrogen atom which is in an external electric field of arbitrary direction. We shall take the \(z\)-axis of our system of coordinates along this direction. Owing to the presence of this field, an electron at the position \((x, y, z)\), in addition to the potential energy \(-\dfrac{e^2}{r}\) due to attraction to the nucleus (situated at the origin of coordinates), acquires the additional potential energy \(+eFz\). (Thus we continue to use the conception of a point nucleus and a point electron.)

The total potential energy of the system therefore consists of the ordinary term \(-\dfrac{e^2}{r}\) and the “perturbation term” \(+eFz\).

The wave equation takes the form:

\[ \Delta \psi+\frac{8\pi^2 m}{h^2}\left(E+\frac{e^2}{r}-eFz\right)=0. \tag{82} \]

In this case, for its successful solution the problem requires the use of parabolic coordinates. Instead of the planes, double cones, and spheres that we used earlier, it is now desirable to introduce planes and two families of paraboloids of revolution. The planes must intersect one another along a line parallel to the direction of the field, i.e. along the \(z\)-axis. Both families of paraboloids have a common focus situated at the center of coordinates, i.e. coinciding with the nucleus; their vertices are located on the \(z\)-axis, in two opposite directions from the center. The transition from rectangular coordinates to parabolic coordinates is accomplished by means of the equations:

\[ x=\sqrt{\xi\eta}\cos\varphi,\qquad y=\sqrt{\xi\eta}\sin\varphi,\qquad z=\frac{1}{2}(\xi-\eta). \tag{83} \]

The wave equation assumes, in the new coordinates, the form:

\[ \frac{d}{d\xi}\left(\xi\frac{d\psi}{d\xi}\right) + \frac{d}{d\eta}\left(\eta\frac{d\psi}{d\eta}\right) + \frac{1}{4}\left(\frac{1}{\xi}+\frac{1}{\eta}\right) \frac{d^2\psi}{d\varphi^2} + \frac{2\pi^2 m}{h^2} \left[ E(\xi+\eta)+2e^2-\frac{1}{2}eF(\xi^2-\eta^2) \right]\psi=0. \tag{84} \]

Trying to decompose this differential equation into equations each of which contains only one of the variables, by the method that we have already used many times, we obtain three equations containing, besides \(E\), two further parameters. For all three parameters only certain values are admissible—the characteristic numbers, whose selection is determined, on the one hand, by the cyclic character of the variable \(\varphi\), and, on the other hand, by the circumstance that only for certain values of the parameters do the solutions of the equations remain finite for any value of the independent variable.

Let \(E=0\); find the corresponding characteristic numbers and substitute them into the equations. We shall obtain determi-

...distribution of standing waves in our imaginary fluid; closer examination shows that in this case the fluid will be divided into compartments by nodal surfaces having the form of planes and paraboloids, with an axis parallel to the direction of the field, looking in two opposite directions. To each permissible value of the energy \(E\) there correspond \((1+2+3+4+\cdots n)\) different systems of standing waves, each of which is characterized by a special number \(k_1\) of nodal paraboloids of one direction, a number \(k_2\) of nodal paraboloids of the other direction, and, finally, a special number of nodal planes \(s\). The possible values of the numbers \(k_1\), \(k_2\), and \(s\) are restricted by the condition that they must be integers not less than 0 and not greater than \(n\), and that their sum must differ by \((n-1)\)—in other words, they must satisfy the equation:

\[ k_1+k_2+s+1=n. \tag{85} \]

(Translating these statements into the language of electron orbits, we call \(s\) the equatorial quantum number, determining the angular momentum of the electron about the direction of the field, measured in units \(\frac{h}{2\pi}\); \(k_1\) and \(k_2\) are, in essence, parabolic quantum numbers.)

Introducing now, into the calculation, the external field \(F\), we find that among the \((1+2+3+4\cdots n)\) different oscillations corresponding to one value \(E_n\), those characterized by the relation \(k_1=k_2\) retain their energy unchanged, while the others are shifted to varying degrees according to the well-known formula *) of Epstein:

\[ \Delta E=\frac{3}{8}\frac{Fh^2 n}{8\pi^2 me}(k_1-k_2). \tag{86} \]

Thus, the \(n\)-th stationary state splits up, or decomposes, into several separate stationary states; however, this decomposition is not complete: some of the total number \((1+2+3+4+\cdots n)\) of oscillations remain, even in the presence of the external electric field, energetically identical (a somewhat crude...

…iterations). The spectral line corresponding to the transition from the state \(E_i\) to the state \(E_j\) must thus split in an electric field into a number of separate lines lying close to one another. This so-called Stark effect clearly shows the independent existence of different oscillations which, in the absence of an external field, give the same energy and therefore cannot be distinguished from one another.

Before proceeding to the further exposition, I should like to touch on one small paradox which, perhaps, has already attracted the reader’s attention. It was said above that, in the absence of an external field, the oscillations of our imaginary fluid are characterized by the distribution of nodal planes and nodal paraboloids, whereas in the preceding paragraph we derived, for the unperturbed hydrogen atom, a distribution of standing waves determined by nodal planes, double cones, and spheres. A closer examination shows, however, that these two statements do not contradict one another. For an oscillation of one kind can be obtained as the sum of several oscillations of another kind. Let us take, for example, the first excited state of the hydrogen atom \((n=2)\). By means of the procedure described in the preceding paragraph, we find three systems of oscillations: 1) a system with only one nodal sphere, 2) a system with one double nodal cone, and 3) a system with one nodal plane. By the second method, adopted by us in studying the Stark effect, we likewise obtain three different systems of oscillations: 1) with one nodal paraboloid directed in one direction, 2) with one nodal paraboloid directed in the other direction, and 3) with one nodal plane. The last modes of oscillation in both groups are obviously identical. Oscillations 1) and 2) of the second group can be reproduced by mutual superposition—with appropriate intensity—of oscillations 1), 2), and 3) of the first group. If the field acting on the hydrogen atom is gradually weakened and in the end reduced to zero, then the atom remains oscillating according to modes 1), 2), or 3) of the second group; these kinds of oscillations, however, are accessible to analytical

images, as certain combinations of oscillations 1), 2), and 3) of the first type.

Let us now suppose that we have before us an unperturbed hydrogen atom (again in the first excited state). We apply to it a certain very weak field \(F\). Before the application of the field the atom can be characterized by any combination of oscillations 1), 2), and 3) of the first kind. As oscillations of the second kind, to which the atom must now pass, not every combination of oscillations of the first kind is suitable, but only definite ones, specially selected according to the amplitude of their combinations. If before the application of the field there was no such combination in the atom, the question arises: in what way can an extremely weak—in the limit, infinitely weak—external field \(F\) produce the corresponding regrouping of oscillations? (An analogous paradox also arises when the problem is considered from the point of view of the theory of electronic orbits.)

Interpretation of the rotator in wave mechanics. A rotator, i.e. a rigid body capable of rotating about a fixed or free axis, is one of the most important elements in the arsenal of physicists engaged in the construction of atomic and molecular models. This is the usual model used by physicists in explaining the phenomena of electric and magnetic polarization of gases, and also in interpreting the band spectra of diatomic and polyatomic molecules. Most of the models used in this latter case, however, combine with the idea of a rotator also the idea of an oscillator; in other words, they regard the rotator not as a rigid body, but as an elastic system capable of vibrating. In the present article we shall confine ourselves to the example of a rigid rotator of unchanging form.

The treatment of the rotator by the method of wave mechanics is unusually simple; however, in the general case, in order to carry it out it is necessary to use the wave equation in a generalized form, in the configuration space of many dimensions. This complication can be avoided if we restrict ourselves to the consideration of that simple idealized rotator which was introduced into science about fifty years ago for

explained: the specific heats of diatomic gases, such as hydrogen. This model consists of two spheres immovably joined to one another, like a gymnastic dumbbell. It is assumed, moreover, that the whole system can rotate only about axes perpendicular to the line connecting the centers of the two spheres, and not about an axis parallel to this line (the figure axis). The position of this model is determined by the angles $\theta$ and $\varphi$, which characterize (in a polar coordinate system) the direction of the figure axis in space. The energy of the system consists exclusively of the kinetic energy of rotation. Therefore the term $V$ disappears in this case from the wave equation. This circumstance greatly simplifies the problem. Denoting by $A$ the moment of inertia of the model with respect to the axis of rotation, we find for the wave equation the form:

\[ \Delta \psi + \frac{8\pi^2 E A}{h^2}\psi = 0. \tag{87} \]

In this equation the Laplace operator $\Delta$ must be expressed in polar coordinates, as was already done in equation (50). The terms containing the coordinate $r$, which is absent in the present case, may be omitted. Thus, for our problem we again obtain the second equation (52), with the special value of the constant denoted in that equation by the letter $\lambda$:

\[ -\operatorname{cosec}\theta\left[\frac{d}{d\varphi}\left(\operatorname{cosec}\theta\,\frac{d\psi}{d\varphi}\right)+\frac{d}{d\theta}\left(\sin\theta\,\frac{d\psi}{d\theta}\right)\right] = \frac{8\pi^2 E A}{h^2}\psi . \tag{88} \]

And in the present example the function $\psi$ must return to its initial value when $\varphi$ is increased by an integral multiple of $2\pi$, and $\theta$ by an integral multiple of $\pi$, for such a change of both coordinates leads to the return of the model to its initial position. We must again conclude that for the coefficient on the right-hand side of the equation only certain values are possible—characteristic numbers; the condition which this coefficient must satisfy is equivalent to the following condition for the energy of the rotator $E$:

\[ E = n(n+1)\frac{h^2}{8\pi^2 A} = \left(n+\frac{1}{2}\right)^2\frac{h^2}{8\pi^2 A} +\text{const}; \quad n=0,1,2,3. \tag{89} \]

Thus, owing to the cyclic character of the variables, the energy of the rotator proves to be quantized according to equation (89). This equation gives us the second (after the harmonic oscillator) example of the quantum numbers obtained.

Thus, the proper values of the energy of the rotator are determined in a particularly simple and clear manner. However, a complication leading to the need to pass to a non-Euclidean configuration space is already appearing on the horizon. Equation (87) differs from the wave equations which we have used up to now by the replacement of the mass \(m\) by the moment of inertia \(A\). This replacement is to a sufficient degree natural, and, making it, one may rely on “intuition.” But, strictly speaking, in order to justify this substitution it would be necessary to refer to the form which the expression for the kinetic energy takes in the general wave equation in this case. If we abandon the restriction introduced above and allow the rotator also to rotate about the figure axis, then the kinetic energy will take another form, and in this general case of a rigid rotator with a free axis we cannot do without writing the wave equation in its general, non-Euclidean form. This problem was solved by several investigators, and the application of the formula obtained in this way to certain problems of molecular spectra showed that the form of the general wave equation, derived by de Broglie and Schrödinger, is well justified in practice.

The polarization of gases in a magnetic or electric field may be considered by means of the representation of molecules as magnetic or electric dipoles. The calculation is simplest under the assumption that the direction of the magnetic (or electric) moment coincides with the direction of the figure axis of the molecule, which cannot rotate about this axis. Let \(M\) denote the (magnetic or electric) moment of such a molecule. Let, further, the field in which the molecule is situated be directed parallel to the \(z\)-axis (i.e. to the direction from which the angle \(\theta\) is counted). The field gives rise to a potential—

energy which must be added to the left-hand side of equation (88). This additional term has the form:

\[ - V\psi = (MH \cos \vartheta)\psi. \tag{90} \]

It is easy to infer that, in this case, the wave equation has characteristic numbers which restrict the possibilities of orientation of the molecule with respect to the field. This conclusion, which had already been drawn by the original atomic mechanics of Bohr and Sommerfeld, was confirmed in practice by the experiments of Gerlach and Stern. The actual polarization of a gas in an electric or magnetic field can be calculated only by making some additional assumption concerning the probability of the different orientations of the molecules with respect to the field at different temperatures. Having made such an assumption, we obtain a formula for the dielectric constant and for the magnetic susceptibility of the gas as functions of the applied field and temperature. Usually in these cases the assumption (equal probability of all allowed cases) leads to a formula which, at high temperatures, passes asymptotically into Langevin’s well-known empirical formula:

\[ \text{Susceptibility} = \frac{J}{H} = \frac{NM^2}{3kT}. \tag{91} \]

Interpretation of the free electron in wave mechanics. We shall now leave aside the calculation of characteristic numbers and stationary states and return to de Broglie’s original ideas.

The wave equation of an electron moving with velocity \(V\) in a space devoid of any electric or magnetic forces (or for any other particle flying with constant velocity along the \(x\)-axis), in its classical (i.e., non-relativistic) form, has the following form:

\[ \frac{d^2\psi}{dx^2} + \frac{8\pi^2 mE}{h^2}\psi = 0 \qquad \left(E = \frac{1}{2}mv^2\right) \tag{92} \]

This equation has a solution in the form of a sinusoidal function for any value of the parameter \(E\) and, consequently, does not require any energy scales restricted by definite discrete--

...values (the opposite result would be too paradoxical!) Assigning to the oscillation the frequency, determined by equation (19), \(\nu=\dfrac{E}{h}\), and the velocity of propagation, according to (15), expressed by the formula:

\[ u=\frac{E}{\sqrt{2mE}}, \]

we obtain for the wavelength associated with the motion of an electron (or of another free particle of mass \(m\)) the expression

\[ \lambda=\frac{E}{\sqrt{2mE}}\cdot\frac{h}{E} =\frac{h}{\sqrt{2mE}}=\frac{h}{mv}. \tag{93} \]

If we calculate the absolute value of \(\lambda\) for electrons flying with speeds of several hundreds or thousands of volts, i.e. for ordinary cathode rays, then we obtain waves whose length approximately corresponds to the length of X-rays. Thus, for example, an electron of 150 volts corresponds to a wavelength almost exactly of 1 Ångström.

This coincidence involuntarily suggests the possibility of diffraction of electron rays when they fall upon crystal lattices, which, as is well known, produce diffraction of X-rays of the same wavelength. Nothing that we have said so far concerning waves associated with material particles obliges us to accept this conclusion. On the contrary, the skeptic might with some justification assert that the hope of ever seeing in a tangible way the propagation of the waves described in our three-dimensional space is as unfounded as the hope of ever seeing with one’s own eyes the other mathematical fictions used in calculation—for example, \(x\) and \(y\) from some algebraic equation. In the theoretical discussion of this possibility, one cannot help thinking that only in certain very simple cases, for example in the case of a free electron or of a hydrogen atom, does wave mechanics lead to the representation of waves in three-dimensional Euclidean space. In other cases—an example may be the free rotator—the “waves” must be considered only in non-Euclidean

multidimensional space. Nevertheless, as mentioned earlier, in both cases we can represent the processes by means of “wave equations” constructed in a completely analogous way. To say that waves exist only in a non-Euclidean “configuration space” practically means almost the same thing as to say that, in the physical sense, they do not exist at all. Why, then, in the particular case that leads to waves in three-dimensional space, should these waves be more real than in the general case? Thus, a priori, the question remains open; however, experiment gives a completely unambiguous answer: diffraction of electron waves in crystal lattices does indeed exist. It was predicted by Elsasser and discovered by Davisson and Germer and, in another form, by G. P. Thomson.^1 On the basis of the diffraction pattern, one can, in the usual way, knowing the lattice constant, calculate the wavelength. In this way, from experiment, one obtains for de Broglie waves values that agree completely with those calculated on the basis of the relation \(\frac{1}{2}mv^{2}=h\nu\).

It must be noted once again that the velocity of propagation of matter waves by no means coincides with the velocity of motion of the material particle to which these waves are “attached.” The velocity of the wave is

\[ u=\frac{E}{\sqrt{2m(E-V)}}, \]

while the velocity of the particle is

\[ v=\sqrt{\frac{2T}{m}}=\sqrt{\frac{2(E-V)}{m}}. \]

What we measure in the study of diffraction is the wavelength, and not the velocity of its propagation and not the frequency of oscillation. This circumstance is very important, for the wavelength, as we shall now see, is a quantity that does not depend on the absolute value of the energy, which—at least in classical mechanics—is always known only up to an arbitrary additive constant. If we add to the kinetic energy of the particle, measured by us with respect to some arbitrarily chosen coordinate system,

^1 Cf. the articles by P. S. Tartakovsky, Uspekhi fizicheskikh nauk, 8, 338, 1928, and V. L. Granovsky, Uspekhi fizicheskikh nauk, 9, 308, 1929.

If we change the constant selected in some way and call the sum of the energies of the system \(E\), then by doing so we change the corresponding frequency of this particle. But since at the same time the velocity of wave propagation changes in the same respect, the wavelength remains unchanged. On the basis of the relations \(\lambda \nu = u\) and \(h\nu = E\), the wavelength is always determined by the equation

\[ \lambda=\frac{h}{\sqrt{2m(E-V)}}, \]

where \(V\) denotes the potential energy. Any increase of the energy, with the velocity of the particle unchanged, increases both terms under the radical sign—\(E\) and \(V\)—to the same degree, and thus the difference \(E-V\) remains unchanged. Returning to the preceding sections of our article, we see that the calculation carried out in § III of the stationary states was based on the preliminary known requirements for the wavelengths, and not for the frequencies; for the distribution of standing waves in space depends only on the wavelength, and not on the frequency of oscillation. The frequency of the oscillations accessible to direct measurement—namely, the frequency of oscillation of the light waves emitted by an atom in the transition from one stationary state to another—depends exclusively on the difference of the energies of these two states, and in no way fixes their absolute magnitude. In relativistic mechanics energy is determined absolutely, as the product of the mass of the particle and the square of the velocity of light. If one uses relativistic mechanics from the very beginning, then the question we have touched upon cannot arise at all. It should be noted, however, that the incompleteness of the definition of energy in classical mechanics in no way affects the practical conclusions of wave mechanics, so that its predictions in this domain cannot serve as arguments either for or against the relativistic formulae.

In relativistic mechanics the wave formula for an electron moving freely in space assumes the form:

\[ \frac{d^{2}\psi}{dx^{2}} = \frac{4\pi^{2}}{h^{2}c^{2}} \left(E-m_{0}^{2}c^{4}\right)=0; \qquad E= \frac{m_{0}c^{2}}{\sqrt{1-\dfrac{v^{2}}{c^{2}}}}. \tag{94} \]

The length of the corresponding wave is determined by the expression

\[ h \sqrt{1-\frac{v^2}{c^2}}\cdot \frac{1}{m_0 v}=\frac{h}{mv}, \]

the frequency of the oscillations by the expression

\[ m_0 c^2 \frac{1}{h\sqrt{1-\frac{v^2}{c^2}}}. \]

The waves propagate with velocity \(\frac{c^2}{v}\), exceeding the velocity of light.

I must confine myself only to mentioning the important fact that the velocity of the particle is connected in wave mechanics with the velocity of the wave by exactly the same relation as that which connects, in optics, the velocity of propagation of the phase with the so-called group velocity.

An attempt to give a physical explanation of the quantity \(\psi\).

Exactly thirty-three years have passed since the day when the outstanding English statesman Lord Salisbury, elected president of the British Association for the Advancement of Science, uttered in his inaugural address, rich in pearls of wit, the following memorable words, occasioned by the numerous attempts of the physicists of that time to give a visual explanation of the properties of the ether: “The chief, if not the sole, function of the ether seems to consist in being the subject of the verb ‘to oscillate’.” The very same thing we can say at the present time concerning the quantity \(\psi\). This quantity enables us to determine the energy of the stationary states of the atom. When this goal is attained, the function \(\psi\) disappears from our field of view. Just as a variable under the sign of a definite integral disappears after the integration has been completed, the quantity \(\psi\) loses its interest for us when the goal for the attainment of which it was introduced has been reached. One might even dispense altogether with giving this quantity a special notation: many mathematicians prefer simply to speak of the differential operator:

\[ \Delta - 8\pi^2 m (E - V)\cdot \frac{1}{h^2}. \]

7 Advances in Physical Sciences, Vol. IX, No. 4.

Schrödinger made an equally bold attempt—without a doubt no less complete and no less definitive, but all the more interesting—to ascribe to the quantity \(\psi\), in accordance with its meaning in the theory, a definite physical meaning. Schrödinger’s supposition consisted in his believing the square of the amplitude \(\psi\) to be proportional to the square of the density of electricity at a given point, thus “smearing” the electron over a comparatively large, practically macroscopic—though very small—space.

Let us look more closely at this theory and at its consequences.

In order to avoid, as far as possible, physical complications, I shall take the simplest possible example—namely, the harmonic linear oscillator. We shall replace this oscillator by an imaginary string, stretched along the \(x\)-axis. The velocity of propagation of waves in such a string, as we know, is determined by the equation

\[ \sqrt{1-\frac{x^{2}}{L^{2}}} \]

and is thus real along the whole string from the initial coordinate up to the point \(x=\pm L\) and has the form of this integral. I shall also use, in setting forth the matter here, an even simpler example, which will serve us as a preliminary stage in the study of the oscillator—namely, a real stretched string, if it is clamped at the points \(x=\pm L\), a string with a constant velocity of propagation of waves along it.

In both cases—in the study of the real and the imaginary string—the search for the characteristic numbers and fundamental functions leads us to the establishment of a definite system of natural or proper vibrations with definite frequencies \(\nu_0,\nu_1,\nu_2,\ldots\), each of which corresponds to a definite spatial distribution of standing waves with their nodes and antinodes, represented analytically by fundamental functions:

\[ y_i=f_i(x)\left(A_i\cos 2\pi\nu_i t+B_i\sin 2\pi\nu_i t\right);\qquad i=0,\ 1,\ 2,\ldots \tag{15} \]

For the real string the functions \(f_i(x)\) are ordinary sinusoidal functions; for the imaginary string, which symbolizes the linear oscillator, these functions are expressed by equation (60). I recall that in this latter case we

...we would be considering not one string with various oscillations, but as many strings, different in their properties, as there are different oscillations that we would have to assign to the oscillator.

If the real string performs the \(i\)-th oscillation, or if we pay attention to the oscillation of the \(i\)-th string from among the strings simulating the harmonic oscillator, then the function \(f_i(x)\) is proportional to the amplitude of this oscillation. The form of equation (95) shows that at each given point this amplitude does not depend on time.

If we regard the square of the amplitude of the oscillations as the density of electricity at a given point, then on the basis of the preceding considerations we must conclude that the distribution of electricity along the imaginary string, by which for clarity we replace the linear oscillator, is constant in time. For each given stationary state there exists a definite constant distribution of amplitudes, i.e., according to our interpretation, a constant distribution of electric density along the string. This means that the field of the string (and consequently the linear oscillator) is characterized by a single fundamental function; the string’s own oscillation is not accompanied by the motion of electric charges, and therefore there is no reason to expect radiation of electromagnetic energy into the surrounding space.

Let us now imagine that the real string oscillates at once in two different ways, corresponding to the numbers \(i\) and \(j\), or else that the \(i\)-th and \(j\)-th imaginary strings simultaneously perform their oscillations. In this case the oscillation is expressed by equation (96) (in this equation, for simplicity, I have put \(A_i = A_j = 1\), and \(B_i = B_j = 0\), which does not impede the generality of the conclusions):

\[ y = y_i + y_j = f_i(x)\cos 2\pi\nu_i t + f_j(x)\cos 2\pi\nu_j t \tag{96} \]

Equation (96) is easily reduced to the form:

\[ y = C\cos(2\pi\nu_i t - \alpha), \tag{97} \]

where

\[ C^2 = f_i^2 + f_j^2 + 2 f_i f_j \cos 2\pi(\nu_i-\nu_j)t, \tag{98} \]

К. К. Darrow

and \(a\) is a constant of no special significance for us.

In equation (97) we have before us an example of an oscillation with an amplitude changing, at each given point, with the passage of time. In this case the square of the amplitude consists of the constant term \(f_i^2+f_j^2\) plus a term varying with time according to a sinusoidal function. Here the frequency of the variation of the square of the amplitude (the period of the sinusoidal term) is determined by the difference of the frequencies of the two coexisting systems of oscillations.^1

Identifying once again the square of the amplitude with the density of electricity, we see that this density will no longer be constant in time, but at each point will vary periodically with frequency \((\nu_i-\nu_j)\). Thus, according to the laws of classical electrodynamics, one should expect the emission of energy of frequency \(\nu=\nu_i-\nu_j\).

We recall that the frequencies \(\nu_i\) and \(\nu_j\) are determined by the energies of the stationary states \(E_i\) and \(E_j\), according to the equation \(\nu h=E\).

If we had the right to make the somewhat vague but tempting supposition that an oscillator can simultaneously be in both stationary states, with energies \(E_i\) and \(E_j\), then a visual image of such an oscillator could be furnished by an imaginary string in which, at each point, the square of the amplitude \(\psi\) fluctuates with frequency \((\nu_i-\nu_j)\); and if we identify this square of the amplitude \(\psi\) with the density of electricity, then we may expect that such a system will give radiation with frequency

\[ \frac{E_i-E_j}{h}. \]

That which was formerly denoted as a transition from one stationary state to another, according to the hypothesis described, should be called a coexistence of these two

^1 The transformation of (96) into (97) reduces to the superposition—well known from the theory of oscillations—of two harmonic oscillations of different frequencies \(\nu_i\) and \(\nu_j\) into a certain resultant oscillation, the amplitude of which is no longer constant, but varies periodically with frequency \((\nu_i-\nu_j)\), i.e., beats arise.

states. (I recall once again that we are not speaking of the coexistence of two possible vibrations of one and the same string, but of the coexistence, as it were, of two independent strings, each with one single fundamental vibration.)

We now take one more step forward in the development of the hypothesis adopted, and to this end compute the integral:

\[ M \int_{-\infty}^{+\infty} x C^2 dx = \int_{-\infty}^{+\infty} x f_i^2 dx + \int_{-\infty}^{+\infty} x f_j^2 dx + \left\{ 2 \int_{-\infty}^{+\infty} x f_i f_j dx \right\} \cos 2\pi(\nu_i-\nu_j)t . \tag{99} \]

This integral measures the electric moment of the assumed distribution of electric charge along the string, with respect to its center. Indeed, the subintegral function is the product \(C^2 dx\)—the charge of an element of length of the imaginary string—by the distance \(x\) of this element from the center, i.e. the electric moment of the element with respect to the origin of coordinates; consequently, the integral represents the electric moment of the entire string with respect to the origin of coordinates. If this integral is equal to 0, then this means that on both halves of the string—the right and the left—there is one and the same quantity of electricity. If the integral has a positive or negative value, then we must conclude that the charge is situated on the string asymmetrically with respect to its middle. If the value of the integral turns out to be periodically varying—for example, if the coefficient of the cosine differs from 0—then this will be equivalent to an oscillation of the charge along the string.

The function \(f_i(x)\) was written out in full in equation (60). We showed at the proper time that \(f_i(x)\) is alternately even and odd in character. Thus, the functions \(f_0, f_2, f_4 \ldots\) are even, and the functions \(f_1, f_3, f_5 \ldots\) are odd functions of \(x\). The squares \(f(x)\) always have an even character, and the products of these squares by \(x\) are always odd functions of \(x\). Therefore the two

the first integrals in formula (99) vanish. As regards the integral

\[ \int_{-\infty}^{+\infty} x f_i f_j\,dx, \]

its integrand is an odd function in those cases in which \(i\) and \(j\) are both even or both odd numbers; in both these cases this integral also proves to be equal to 0. Thus, when two even or two odd stationary states coexist simultaneously, no oscillations of the electric charge occur, and the electric moment of the system remains constant. In the case of even \(i\) and odd \(j\) (and also in the converse case) the conclusion is not so simple. However, a mathematical investigation shows that the integral

\[ \int_{-\infty}^{+\infty} x f_i f_j\,dx \]

also proves to be equal to zero here, except in the case when \(i\) differs from \(j\) by 1. This leads us to the law:

When two oscillations of a harmonic vibrator coexist, only under the condition \(j=i\pm1\) does the electric moment of the depicted string, shown in our model as a vibrator, change with the passage of time according to the sinusoidal law, with frequency \((\nu_i-\nu_j)\); in all other cases the electric moment remains equal to zero at all times.

The physical conclusion from this theorem is that only the coexistence of two neighboring stationary states of the oscillator leads to the emission of energy. Translating into the language of the former theory, we obtain the familiar rule: spectral transitions are allowed only between two states of the oscillator whose quantum numbers differ from one another by \(\pm 1\). This “selection rule,” derived in the old quantum theory by means of the “correspondence principle,” is confirmed, as is known, by experiment on certain molecular spectra, namely in those cases in which we have the right to regard the oscillations of atoms in a molecule as harmonic.

Thus, in the case of the harmonic oscillator, the interpretation of the quantity \(\psi^2\) as an electric density,

leads to a double condition: 1) the distribution of electricity in each separate stationary state proves to be static; when two states coexist, the electric dipole has the form of a vibrating string with the same frequency which, according to Bohr’s theory (confirmed by experiment), is emitted in the transition from one of these states to the other—this is the first condition; 2) the second consists in the theoretical derivation of the rule according to which only a combination of two neighboring oscillator states leads to the emission of radiant energy. In this process the process occurring in the atom acquires a visual form: it consists in the oscillation of the electric charge about the center of equilibrium. (Schrödinger showed that if we consider a large number of stationary states, with high values of $i$ and arbitrarily chosen relative “amplitudes,” i.e. values of $A_i$ and $B_i$ in equation (95), and imagine all the corresponding intrinsic oscillations as coexisting simultaneously, then, as a result, the entire electric charge of the electron will prove to be concentrated in a small region of space. Thus the “spread-out” electron will once again gather into a single point, which will be found oscillating back and forth about the center of equilibrium, with frequency $\nu_0$. The amplitude of its oscillation will be approximately equal to the amplitude of oscillation of the material particle with which we began our reasoning at the time, i.e. of a particle of mass $m$ under the action of the elastic force $-4\pi^2 m\nu_0^2 x$; at the same time the energy of the oscillations is the same as for the stationary state which, in the summation, manifests itself with the greatest amplitude. This result thus shows that, when a large number of oscillatory processes with high values of $i$ are considered, one can continuously pass over from the picture given by wave mechanics to the usual picture of spatially bounded oscillating particles: for highly excited states there can indeed be inherent such a distribution of charges in which one may rightly speak of point charges describing definite orbits, whereas for states with small quantum num-

the charges themselves “spread out” until there remains only one continuous cloud of charge—fluctuating or at rest).

Still another advantage of identifying the quantity $\psi^2$ with the density of electricity is found in passing from one dimension to two and three. As an example I shall take the hydrogen atom in an electric field. We have represented this atom to ourselves in the form of a liquid formation performing standing oscillations. If two systems of oscillations of the perturbed atom exist simultaneously, then their coexistence leads to a real oscillation of the electric charge in the atom, with a frequency equal to the difference of the two proper frequencies. This beat frequency corresponds to the frequency in the transition from one of these stationary states to the other, according to the laws of the old quantum theory. If, in the particular case, both stationary states correspond to one and the same value of the quantum number $s$ [the azimuthal quantum number occurring in equation (85)], then the oscillations of the electric charge will occur, as can be shown, parallel to the direction of the external field. Thus the motion of the charge will not have a component in the direction perpendicular to the direction of the field. This result corresponds to the empirical rule according to which the light emitted by atoms in transitions corresponding to a change only of the quantum numbers $k_1$ and $k_2$, with the value of the equatorial quantum number $s$ remaining unchanged, is distinguished by being linearly polarized with its electric vector parallel to the direction of the external field. If, however, the numbers $s$ for the two coexisting stationary states differ by one, then the calculation leads to an oscillation of the electric charge perpendicular to the direction of the electric field. Experiment confirms that the light obtained in such transitions is polarized in the plane perpendicular to the direction of the field. If the numbers $s$ of both states differ from each other by more than 1, then the corresponding displacements of the charge turn out to be altogether insignificant, and, in accordance with this, lines due to transitions in which $s$

is greater than 1; in the spectrum they do not appear at all with any noticeable intensity.

We thus have three points at which the identification of the quantity \(|\psi|^2\) with the density of electricity leads to successful results. In a visual picture representing the atom by means of an imaginary elastic fluid, the electricity turns out to be distributed motionlessly in space so long as the atom is in a stationary state; thus the absence of radiation in a stationary state becomes intelligible. When two stationary states are present simultaneously, the charge oscillates with a frequency determined by the difference of their energies; this oscillation turns out to be appreciable only in the case in which, between the two stationary states, according to the laws of the old atomic mechanics (Bohr’s correspondence principle), spectral transitions are possible; the direction along which the charge oscillation occurs proves to correspond to the polarization observed in the rays emitted in the given transition; in cases corresponding to “forbidden” transitions there is, as a rule, no appreciable oscillation of the charge whatever. As an outline of a possible theory of the origin of the light emitted by the atom, Schrödinger’s hypothesis is distinguished by incomparable merits. In the earlier theories of atomic structure it had not been possible to satisfy even the most elementary requirement: the existence of a regular connection between the periods of motion of the particles composing the atom and the periods of the radiation emitted by it. In Schrödinger’s theory a relation of this kind was established for the first time, and if we now see, in a discharge tube, hydrogen emitting the red Balmer line with an oscillation frequency equal to \(4.57 \cdot 10^{14}\), then, according to this hypothesis, we are at least entitled to assert that in every hydrogen atom there actually takes place some oscillatory process characterized by the same frequency.

Even the relative intensities of the various spectral lines are accessible to theoretical calculation by means of the only recently developed conception of wave mechanics. We have seen that in the example of the harmonic oscillator the degree of attenuation

the value of the integral

\[ \int_{-\infty}^{+\infty} x f_i f_j\,dx \]

for all combinations of \(i\) and \(j\), with the exception of those in which \(i\) differed from \(j\) by 1, led to the disappearance from the spectrum of lines corresponding to transitions in which the quantum number \(n\) changes by more than 1. Is it not possible to suppose that, for any combination of two stationary states, the intensity of radiation polarized parallel to some direction \(x\) is determined by the magnitude of the integral

\[ \int_{-\infty}^{+\infty} x\psi_i\psi_j\,dx, \]

formed from the fundamental functions of both states, \(\psi_i\) and \(\psi_j\)? To develop this idea it is necessary to make some additional assumption, for the fundamental functions, in the form in which we have written them up to now, may be multiplied by any coefficient while remaining, nevertheless, the same fundamental functions. If the integral is to vanish (i.e. in deriving selection rules), these constant factors have no significance; but if the integral is not zero (i.e. in deriving intensity rules), then its absolute magnitude depends essentially on the choice of these constants, and therefore some definite assumption is needed concerning the values of these constants. Schrödinger made a simple and natural assumption concerning the above-mentioned coefficients when calculating the intensities of the components of the Stark effect for the Balmer lines. The experimental results confirmed his suppositions.

I cannot dwell here in greater detail on these questions; I shall mention only that it is precisely in this domain that the point of contact is found between Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics. The integrals that interest us enter into Heisenberg’s theory in the form of matrix elements. Heisenberg’s theory in practice turns out to be another way of obtaining the very same conclusions to which wave mechanics leads.

The elegant and visual representation of Schrödinger’s, which we have developed in this essay, nevertheless encounters certain...

Introduction to Schrödinger’s Wave Mechanics

many difficulties. Let us point out some of them. We can imagine that, with the simultaneous presence of two stationary oscillations, the atom must emit radiation into the surrounding space. But this flow of energy cannot continue without end; sooner or later one of the stationary states must “die out,” and the radiation must cease. Meanwhile, up to now we have not been able, with the help of wave mechanics, to form any idea of the mechanism of this dying out. Perhaps it would be possible to introduce into the theory some explanation of this phenomenon, in the form of an assumption of interaction between the waves \(\psi\) and the electromagnetic waves emerging from the atom. It is much more difficult, however, to imagine a way out of another dilemma before which we are placed by the identification of the square of \(\psi\) with the density of electricity. Our wave formula made it possible to derive the correct values of the energies of the stationary states for the hydrogen atom only because we adopted for the potential energy the expression \(V=-\dfrac{e^2}{r}\), which follows from the conception of a point electron. If we now pass to the hypothesis of a “blurred-out” electron occupying all the space around the nucleus, then how can we justify introducing into our formula an expression for the potential energy that is incompatible with this hypothesis? What right have we to determine the distribution of the electric charge by two different methods for two purposes, and to combine these two representations in a single formula?

Wave mechanics, while giving such seductively vivid explanations of many atomic processes, is still rich in such fundamental difficulties. Thus the admirers of Lessing, who says that the greater delight lies in drawing nearer to the truth than in possessing it, may feel full satisfaction from studying physics. Wave mechanics is still only an attempt, and not a final achievement. It is a plan of campaign rather than a conquest. It is impossible yet to foresee how this attempt will end. But in conclusion we must recall: twenty-five years ago no one suspected any kind of

other properties of light, apart from those connected with its wave nature. Since then, experiment has shown that in many respects light behaves like a stream of discrete particles.

At the present time, experiments, one after another, are showing that matter in many respects behaves like a wave phenomenon. The dual nature which troubled us so much in the case of light has, since the advent of quantum theory, proved to be characteristic of matter as well. The particle–wave dualism has thus become a universal principle. There are no particles without wave properties, and there are no waves without corpuscular properties. The generalization of this dualism may perhaps point the way to a higher unity.

1. In (69) and in all subsequent equations, \(\psi\) is presented only as a function of the coordinates \(f(x,y\ldots)\), and the dependence on \(t\) is omitted. But the time function is always a simple sinusoidal function; its greatest value is therefore equal to 1. The true “elongation” is expressed by the product \(f(x,y\ldots)\,g(t)\); consequently, \(f\) is the greatest value of the “elongation,” or the amplitude. All subsequent equations are thus limited to describing the spatial distribution of vibration amplitudes. These are “amplitude equations,” which only after multiplication by \(g(t)\) become “wave equations.” ↩↩↩↩↩↩