ELEMENTS OF WAVE MECHANICS

N. N. Andreev. Leningrad.

In recent years the mechanics of the atom has clearly been going through a crisis: it has become apparent that the quantization of the motions of the constituent parts of the atom leads to results consistent with experiment only in the simplest cases of one electron; moreover, the process by which the atom emits electromagnetic energy has remained obscure, and all attempts to interpret it have ended unsuccessfully. No remedies helped against these two principal ailments of quantum theory, and the conviction gradually grew that, for the further development of the theory, it was necessary to put forward certain entirely new principles. At present we have two such attempts1: the first belongs to Heisenberg, Born, and Jordan—the so-called matrix mechanics of the atom; the author of the second is Schrödinger, and it is most properly given the name wave mechanics. It very soon became clear that, despite the sharp difference both in their physical premises and in their mathematical methods, the two theories are completely equivalent: their results and their possibilities are identical. But Schrödinger’s theory proved to be much broader in its content and was able to approach deeper questions than the theory of Heisenberg–Born–Jordan.

1. The life of the atom cannot be described in the language of the mechanics of a system of a finite number of points, even if one applies the generalization that the special principle of relativity gives and, from among the solutions permitted by Newtonian mechanics, artificially—by quantization—selects the most suitable ones—this is the fundamental thought that arises in everyone acquainted with the modern theory of the atom. It is therefore natural that there have repeatedly arisen attempts to describe the state of the atom by partial differential equations, in other words, instead of looking at the atom as a discrete system of points, to regard it as a continuous system. The first attempts at such a description of phenomena in the atom are rather old,

and an account of them is given in Kaiser’s Handbuch der Spectroskopie; the most remarkable of them belonged to the prematurely deceased Swiss physicist Ritz¹); in it he considers the atom as a two-dimensional continuous system, like a membrane, but described by a partial differential equation of the tenth order. The results obtained by Ritz are remarkable, but he was unable to give any physical interpretation to his equations, and soon himself abandoned them, adopting an entirely different point of view.

A generalization of an entirely different kind was proposed in 1924 by the young French physicist L. de Broglie²); his fundamental assertion was: with every particle, whether a quantum of light, an electron, or a proton, there is associated a certain “phase” wave. The consequences of this basic idea were derived by de Broglie with essential aid from the special principle of relativity and proved very promising; in particular, the quantization of the closed orbits of the hydrogen electron acquired a physical meaning: namely, it turned out that along each of these orbits there fits an integral number of phase waves. However, despite its dual essence, de Broglie’s theory received no further development, evidently owing to the narrowness of the path of generalization he had chosen.

Schrödinger, however, managed, making use of de Broglie’s ideas—as he himself mentions more than once—to take so great a step forward that it would not be unjust to attach his name to the theory he created. His basic thought was³): the generalization of modern, Bohr, quantum mechanics must be carried out along the same path along which geometrical optics is generalized into wave optics. The advisability of such a path is indicated by the well-known analogy, recognized since ancient times, between geometrical optics and Newtonian mechanics.

Since wave optics starts from a partial differential equation, the task of atomic theory is to find the corresponding partial differential equation for the atomic system as well.

¹) Zur Theorie der Serienspektren. Ann. d. Phys. 12, 264, 1903 and Phys. ZS. 4, 406, 1903.

²) L. de Broglie. A Tentative Theory of Light Quanta, Phil. Mag. 47, 446, 1924; Thèses, Ann. de Physique (see also Uspekhi Fiz. Nauk, 4, 329, 1924. Abstract by L. I. Frenkel).

³) It is difficult to assert with certainty that precisely this was Schrödinger’s first idea, especially since in his first article: Quantisierung als Eigenwertproblem, Ann. d. Phys. 79, 361, 1926, his starting point is different. But the main idea of his theory is undoubtedly as we state it; moreover, it is the most convincing didactically, although for reasons that will be discussed below, it is now abandoned.

2. Analogy between mechanics and geometrical optics.

This analogy, especially fully developed by Hamilton, despite its usefulness (Hamilton arrived by means of it both at the equations of mechanics bearing his name and at the so-called Jacobi–Hamilton equation), was firmly forgotten; nor was it revived by the advocacy of the famous Göttingen mathematician Felix Klein, who used it in his lectures on mechanics and pointed to it in his reports; therefore it will not be superfluous to set forth its content briefly1. Its essence is that the trajectory of a moving point may be regarded as the path of a light ray in an isotropic medium with a suitably chosen index of refraction; precisely by using this observation, Johann Bernoulli solved the problem of the brachistochrone2; it is also, evidently, the basis of Newton’s theory of light.

The most complete formulation of this analogy is best effected by comparing Fermat’s principle with the Hamilton–Jacobi principle. If we denote the index of refraction by \(n\), and the element of the path of a light ray by \(dl\), then Fermat’s principle asserts that the path of a light ray between two points \(A\) and \(B\) satisfies the condition of stationarity of the integral of the index of refraction taken along the path of the ray, i.e.

\[ \delta J=\delta\int_A^B n\,dl=0. \tag{1} \]

We shall not enter here into the proof of this proposition3, but shall illustrate the validity of Fermat’s principle for the most important special case—the law of refraction. Let us compare two paths: the actual path of the ray from \(A\) to \(B\) (Fig. 1) and the infinitely close path \(AO'B\). For the first we have:

\[ J=AO\cdot n_1+OB\cdot n_2, \]

Fig. 1.

for the second:

\[ J=AO'\cdot n_1+O'B\cdot n_2 \]

and, consequently,

\[ \delta J=(AO-AO')n_1+(OB-O'B)n_2. \]

Putting \(AM=AO,\ NB=O'B\), we find:

\[ \delta J=-MO'n_1+NO\,n_2. \]

But since the triangles \(OMO'\), \(ONO'\), for small angles \(A\) and \(B\), may be regarded as right triangles, and in the region \(OO'\) both paths \(AOB\) and \(AO'B\) form, with the perpendicular to the surface, angles satisfying the law of refraction (all this with accuracy up to infinitesimals of the second order), it is easy to notice that

\[ \frac{NO}{MO'}=\frac{n_1}{n_2}, \]

i.e. that \(\delta J=0\).

Fermat and Heron themselves, having formulated this principle for the reflection of light, set it forth somewhat differently, namely: that light chooses such a path from \(A\) to \(B\) for which its passage requires the shortest time. Indeed, if one takes the point of view of Huygens’ elementary principle, from which, as is known, it follows that the refractive index is the ratio of velocities, then instead of \(n\) we may write \(\dfrac{c}{v}\) (\(c\) is the speed of light in empty space, \(v\) in the medium under consideration), and therefore:

\[ \delta J=\delta c\int_A^B \frac{dl}{v} = c\delta \int_{t_A}^{t_B} dt=0. \tag{2} \]

In this form Fermat’s principle loses its purely geometrical meaning, since the concept of velocity now enters into its formulation. Therefore (2) already represents a principle not of geometrical optics, which contains no kinematic elements whatever, but a principle of primitive wave optics, in the spirit of Huygens.

1. The best known of the variational principles of mechanics is Hamilton’s principle:

\[ \delta \int_{t_A}^{t_B} (T-U)dt=0. \tag{3} \]

Here \(T\) is the kinetic energy, \(U\) the potential energy of the point—a function only of its coordinates, i.e. of position. Jacobi obtained from it

the principle of a purely geometrical character, excluding time by means of the relation

\[ T^{2}=(E-U)\frac{m}{2}\left(\frac{dl}{dt}\right)^{2} \]

\[ T=\sqrt{\frac{m}{2}}\sqrt{E-U}\cdot \frac{dl}{dt}. \]

Substituting in (3), instead of \(U\), the quantity \(E-T\) (\(E\) is the total energy of the point), we first find

\[ \delta\int_{t_A}^{t_B}2T\cdot dt=0, \tag{4} \]

the so-called Maupertuis principle (the variation is to be carried out while regarding \(E\) as constant!), and from this, by means of the above substitution, there follows:

\[ \delta\int_A^B \sqrt{2m(E-U)}\cdot dl=0 \tag{5} \]

the Hamilton–Jacobi principle, which contains no kinematic elements, and therefore has a purely geometrical character. Performing the variation with respect to the coordinates, we find from (5) the equations of the trajectory of the moving point, but, obviously, we cannot obtain any conclusions about its velocity.

The analogy between (1) and (5) is obvious: a material point whose total energy \(E\) is given moves in the force field \(U\) along the same trajectory as that possessed by a light ray moving in a medium with refractive index

\[ n=\sqrt{2m(E-U)}. \tag{6} \]

But here, strictly speaking, the analogy also ends; it cannot be extended to the domain of kinematics. Indeed, in primitive wave optics the refractive index is inversely proportional to the velocity of the ray, whereas (6) shows us that the corresponding quantity in mechanics is directly proportional to the velocity; it was precisely this circumstance that ruined Newton’s emission theory, which asserted that in a medium with a larger refractive index the velocity of the light particle is also greater.

1. De Broglie, however, succeeded in taking one more step along the path of developing our analogy1, by turning to the concept not of a wave,

but the group velocity. If, in a medium possessing dispersion, there propagates a group of waves of only slightly different frequencies, then, owing to their interference, beats are observed; the velocity of propagation of these beats turns out to be no longer \(\frac{c}{n}\), but

\[ \frac{c}{\dfrac{d(n\nu)}{d\nu}} \]

\({}^{1}\). Making use of this formula, we shall now show that the velocity of a moving point coincides with the group velocity of waves in a medium with refractive index (6); but for this a certain new hypothesis is necessary. In order that expression (6) define a dispersing medium (otherwise \(\frac{d(n\nu)}{d\nu}=n\), and a group of waves propagates with the same velocity \(\frac{c}{n}\) as an individual wave), we shall set

\[ E=h\nu . \tag{7} \]

This is, of course, an extremely essential hypothesis, but one already familiar to every physicist; without attempting for the moment to interpret it, we shall use it; let us note only one important feature of it: it determines the energy completely, and not merely up to an arbitrary constant, as is the case in Newtonian mechanics.

First of all let us observe that expression (6) has no dimension, as is required for the refractive index. Therefore instead of (6) we shall write:

\[ n=\frac{\sqrt{2mc^{2}(E-U)}}{E}. \tag{8} \]

This is the second hypothesis put forward by de Broglie. Of course, from the mechanical point of view the introduction into Jacobi’s principle (5), or into Maupertuis’ principle (4), of the constant factor \(\frac{c^{2}}{E}\) is quite possible, since in these principles the variation is understood precisely at constant energy; but in the presence of relation (7) we introduce

\({}^{1}\) Let us recall how this result is obtained in the presence of the simplest group of two waves having frequencies \(\nu\) and \(\nu+d\nu\). We have:

\[ \sin 2\pi\left(\nu t-\frac{x n\nu}{c}\right) + \sin 2\pi\left((\nu+d\nu)t-\frac{x n\nu}{c} -\frac{x}{c}\frac{d(n\nu)}{d\nu}\,d\nu\right) = \]

\[ =2\sin 2\pi\left[ \left(\nu+\frac{d\nu}{2}\right)t -\frac{x}{c}\left(n\nu+\frac{1}{2}\frac{d(n\nu)}{d\nu}\,d\nu\right) \right] \cos \pi d\nu\left(t-\frac{x}{c}\frac{d(n\nu)}{d\nu}\right). \]

The frequency of the second factor is very small in comparison with the frequency of the first; therefore the second factor may be regarded as an amplitude slowly varying with time; the velocity of propagation of this altered amplitude is, evidently,

\[ c\left/\frac{d(n\nu)}{d\nu}\right. . \]

thereby a quite definite dependence of the refractive index on the frequency, and this is essential.

From (8) we find:

\[ n'=\frac{d(\nu n)}{d\nu}=\frac{d(En)}{dE}=\frac{c\sqrt{2m}}{2\sqrt{E-U}}=\frac{c\sqrt{\frac{m}{2}}}{\sqrt{T}}=\frac{c}{v}. \tag{9} \]

Hence also:

\[ \frac{c}{\dfrac{d(\nu n)}{d\nu}}=\frac{c}{n'}=v, \tag{10} \]

i.e. in a medium with refractive index (8) the group velocity of the waves coincides with the velocity of a point moving according to the laws of Newtonian mechanics. Thus, the analogy between geometrical optics and mechanics is extended to complete coincidence1; but so long as no physical interpretation of (7) and (8) is given, this is a purely formal analogy. De Broglie puts forward the assertion that with the motion of a point there is always associated some oscillatory process governed by relation (7); the waves characterizing this process he called “phase” waves; however, he cannot give any physical picture explaining his assertions.

5. Quantization of the hydrogen atom. All the more remarkable is one of the consequences of the analogy established above, namely the quantization of the hydrogen atom. Substituting into (8)

\[ U=\frac{e^2}{r}, \]

where \(e\) is the elementary charge and \(r\) is the distance between the nucleus of hydrogen and its electron, we obtain a medium with a radially symmetric refractive index and a definite dispersion. Calculation shows that the rays of light in such a medium are ellipses having the nucleus as one of their foci2. There is no need to present this calculation, if only because we know this result in advance from the solution of the corresponding mechanical problem. The quantum conditions make it possible to single out from these ellipses certain definite ones, admissible by quantum theory. And, as De Broglie showed, the number of phase waves (their length is, evidently,

\[ \lambda=\frac{c}{n\nu}, \]

) arranged along the circumferences of these quantum ellipses is necessarily an integer. This remarkable result, as well as others obtained by De Broglie, involuntarily suggests the conclusion that De Broglie’s analogy really has some deep physical meaning, and is not purely

formal. Therefore it is natural to wish for a further development of this analogy, and it was precisely this that led Schrödinger to his theory.

1. As we have already noted, the optics by means of which de Broglie establishes the analogy is a primitive geometrical optics in the spirit of Huygens, incapable of explaining such phenomena as, for example, diffraction. This is an optics of such short wavelengths that diffraction and interference phenomena become imperceptible. But the success of de Broglie mechanics naturally suggests the thought that its further generalization is also useful, one which should be related to de Broglie mechanics as Fresnel wave optics is related to the optics of Huygens. This is precisely the fundamental idea of Schrödinger.

But all of Fresnel optics is contained in the equation:

\[ \frac{\partial^2 \varphi}{\partial x^2} +\frac{\partial^2 \varphi}{\partial y^2} +\frac{\partial^2 \varphi}{\partial z^2} =\Delta \varphi =\frac{1}{v^2}\frac{\partial^2\varphi}{\partial t^2}. \tag{11} \]

The connection of this equation with Huygens’ wave optics is easy to establish if we consider those of its solutions which correspond to a small wavelength, i.e. to a high frequency; therefore put

\[ \varphi=e^{2\pi i\left(t-\frac{\psi}{v}\right)} =e^{2\pi i\nu f} \tag{12} \]

\[ \psi=\psi(x,y,z), \]

i.e. we shall consider sinusoidal waves propagating with velocity \(v\), whose wavelength is

\[ \lambda=\frac{v}{\nu}, \tag{13} \]

and the front has the form:

\[ f(t,x,y,z)=\mathrm{const}\quad \text{or}\quad \psi(x,y,z)=\mathrm{const}+vt. \tag{14} \]

Substituting (12) into (11), we find:

\[ (2\pi i\nu)^2\varphi \left[ \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2 + \left(\frac{\partial f}{\partial z}\right)^2 \right] +2\pi i\nu\,\varphi\,\Delta f = \left(\frac{2\pi i\nu}{v}\right)^2 \left(\frac{\partial f}{\partial t}\right)^2 \varphi. \tag{15} \]

At high frequency the first term on the left considerably outweighs the second, and therefore we obtain:

\[ \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2 + \left(\frac{\partial f}{\partial z}\right)^2 =\operatorname{grad}^2 f =\frac{1}{v^2} \left(\frac{\partial f}{\partial t}\right)^2. \tag{16} \]

Substituting here \(\psi\), we find:

\[ \operatorname{grad}^2\left(\frac{\psi n}{c}\right) = \frac{n^2}{c^2}. \tag{17} \]

This equation is the fundamental equation of geometrical optics: if some surface \(\psi=\mathrm{const}\)—a wavefront surface—is given, then (17) shows how one should draw the lines having the direction \(\operatorname{grad}\psi\), i.e., the geometrical rays perpendicular to this surface; the fact that the direction of the gradient is determined only up to sign,

\[ \operatorname{grad}\left(\frac{n\psi}{c}\right)=\pm\frac{n}{c}, \tag{18} \]

corresponds to the well-known theorem of geometrical optics that reversing the direction of rays does not change their trajectories.

Equation (16), which contains (in appearance only) time, may reasonably be interpreted as the equation of primitive wave optics; as is also seen from equation (14), it contains the velocity of propagation \(v\), which determines how the surface

\[ \psi=\mathrm{const}+vt \]

moves in space.

It is not difficult to show further that from (17) there also follows Fermat’s principle1.

All our reasoning has assumed the constancy of \(n\), and consequently also of \(v\). But it can be shown that in the more general case as well, when the medium is inhomogeneous and equation (11) has to be replaced by a more complicated one, for short waves we obtain equations (16) and (17); however, for the sake of brevity we shall not present these calculations.

1. If in (17) we replace the refractive index \(n\) by its expression (8), we find:

\[ \operatorname{grad}^{2}\left[\psi\sqrt{2m(E-U)}\right]=2m(E-U), \tag{19} \]

and this equation, by its form, very strongly recalls the well-known equation of mechanics, the so-called Hamilton–Jacobi equation:

\[ \left(\frac{\partial S}{\partial x}\right)^{2} + \left(\frac{\partial S}{\partial y}\right)^{2} + \left(\frac{\partial S}{\partial z}\right)^{2} = 2m(E-U) \tag{20} \]

and the complete identity between the two equations is established by means of the relation:

\[ S=\psi\sqrt{2m(E-U)}. \tag{21} \]

From this it becomes clear how one must carry out the generalization of Newton–de Broglie mechanics: one must pass from equation (20), characteristic [in the presence of (7) and (8)!] of this “geometrical” mechanics, to a new equation by a procedure inverse to that by which we, from Fres—

…of wave optics to equations (16), (18), and (19). This is easiest of all to do by putting, in equation (11), the velocity \(v\) equal to \(\dfrac{c}{n}\), with \(n\) taken from (8). Then we obtain the first Schrödinger equation:

\[ \Delta \psi=\frac{2m(E-U)}{E^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}, \tag{22} \]

which is the starting point of his entire theory.

Before we proceed to applications of this remarkable equation, we must once more emphasize that our generalization is the simplest, but by no means the only possible one. True, in form it corresponds to Fresnel’s equation (11); but one must not forget that the latter contains, in the form of a coefficient, a constant velocity, whereas for us this coefficient is doubly variable: \(U\) is a function of the coordinates, and \(E\) is proportional to the frequency. Moreover, equation (11) does not correspond to any theory of light known to us if \(v\) is regarded as a function of position and frequency: both Maxwell’s and the elastic theory of light lead us, for an inhomogeneous medium, to equations of an entirely different kind. Finally, let us not forget that all our arguments and analogies related to the mechanics of a single point; but if we consider the mechanics of a system of \(N\) points, the situation will be more complicated: then Fresnel’s equation, referring to the space of three dimensions, can in no way be a generalization of equation (20), which in this case will refer to \(3N\) variables. Schrödinger therefore had, in order to continue the analogy and generalization in this case, to turn to the optics (geometrical and wave) of multidimensional spaces.

All these considerations show that it is impossible to ascribe a physical meaning to the analogy between mechanics and optics—for example, to identify a mechanical point with a group, or, as Schrödinger more aptly calls it, a “packet” of waves.

In his second paper Schrödinger¹) tried to do this, but at the present time this attempt must be regarded as unsuccessful, and Schrödinger himself has abandoned this point of view. It should be noted, however, that de Broglie is still inclined to adhere to it²).

Thus, before us is an analogy which, in all probability, is purely formal, of the kind we know rather well in theoretical physics; it is enough, for example, merely to recall the analogy between electrostatics and the steady flow of an incompressible fluid. But the method of formal

¹) Quantisierung als Eigenwertproblem. Zweite Mitteilung. Ann. d. Phys. 79, 489. 1926. See also Naturwissenschaften, 14, 664, 1926.

²) L. de Broglie. Les principes de la nouvelle mécanique ondulatoire. Jour. de Physique, 7, 321, 1926.

analogies is nevertheless very useful in areas where there is a need for generalization, and it has always been used and will be used.

1. A rotator with an axis fixed in space.
   The solution of equation (22) turns out, generally speaking, to be rather complicated and requires of the physicist familiarity with a mathematical apparatus still little accustomed to him, although historically not new. But in the case of a rotator with an axis fixed in space the solution can be carried through to the end by quite elementary means, and the method of quantization characteristic of Schrödinger’s theory appears very clearly. We shall therefore dwell especially on this example.

But before proceeding to the solution of our problem, let us make one general remark. The solution of (22) is found in the following way: first, particular solutions are sought for sinusoidal waves; then, bearing in mind that equation (22) is a linear equation, for which the principle of superposition is valid, the complete solution is found by summing the particular solutions found.

For sinusoidal waves we may put:

\[ \varphi=\psi(x,y,z)\cdot e^{2\pi i \nu t}, \]

and since for us the energy is proportional to the frequency, we may write instead of this:

\[ \varphi=\psi\cdot e^{\frac{2\pi iEt}{h}}. \tag{23} \]

Substitution in (22) gives us:

\[ \Delta\psi+\frac{8\pi^2 m}{h^2}(E-U)\psi=0. \tag{24} \]

This is the second Schrödinger equation; it is useful to note the simple recipe by which it is obtained from the Hamilton–Jacobi equation (20), which we shall rewrite as:

\[ p_x^2+p_y^2+p_z^2-2m(E-U)=0 \tag{25} \]

Here \(p\) are the components of the momentum vector, as is known, obtained by differentiating the Jacobi–Hamilton function \(S\) with respect to the coordinates. Obviously, in order to pass from (25) to (24), one must apply to the function \(\psi\) an operation of the following form:

\[ \left\{ \left(\frac{2\pi i}{h}\right)^2 \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) -2m(E-U) \right\}, \tag{26} \]

obtained from (25) by replacing \(p_x\) by the operator \(\dfrac{2\pi i}{h}\dfrac{\partial}{\partial x}\); at the same time, in carrying out the operation by means of the operator (26), it is necessary to per-

to perform its formal multiplication by \(\psi\). This recipe is useful in a very large number of cases in Schrödinger’s theory.

We now turn to the solution of our problem, and we shall proceed directly from Schrödinger’s second equation, where we put:

\[ U=0;\quad E=\frac{m r^2 \dot{\varphi}^{\,2}}{2}=\frac{M \dot{\varphi}^{\,2}}{2}=\mathrm{const}. \tag{27} \]

Here \(\dot{\varphi}\) is the angular velocity, \(M\) the moment of inertia, \(r\) the unchanging radius of rotation. Passing to polar coordinates \(r,\varphi,\vartheta\) and, taking into account the constancy of \(r,\vartheta\), we find in place of (24):

\[ \frac{1}{r^2}\frac{d^2\psi}{d\varphi^2}+\frac{8\pi^2 m}{h^2}E\psi=0 \]

or

\[ \ddot{\psi}+\frac{8\pi^2 M}{h^2}E\psi=0. \tag{28} \]

Owing to the constancy of the coefficient of \(\psi\), this equation is solved without difficulty; we find:

\[ \psi=\sin \omega\varphi \tag{29} \]

\[ \omega=\sqrt{\frac{8\pi^2 M E}{h^2}}. \]

After one revolution has elapsed, i.e. when \(\varphi\) increases to \(\varphi+2\pi\), our moving point returns to its former position; the phenomenon under consideration is periodic in Newtonian mechanics. But in Schrödinger’s mechanics we are dealing with the function \(\psi\), which must describe for us the physical aspect of the phenomenon. What the true physical meaning of this function is, we do not yet know; but if only it exists, then the function \(\psi\) must satisfy the fundamental requirements that a physicist imposes on every function of interest to him: it must be single-valued, continuous, and finite in all real space. In our particular case this means that \(\psi\) must have one and the same value for \(\varphi\) and for \(\varphi+2\pi\); indeed, in real space \(\varphi\) and \(\varphi+2\pi\) determine one and the same point.

Obviously, when applied to (29), this condition means that \(\omega\) is an integer \(n\); and to satisfy this one must correspondingly choose the value of the only parameter at our disposal, \(E\); we find:

\[ E=\frac{h^2 n^2}{8\pi^2 M}. \tag{30} \]

Thus, the energy of our motion turns out to be quantized, and the result of this quantization in the present particular case coincides with the result of the old quantum mechanics1.

In this example the meaning of quantization in Schrödinger’s theory is clearly seen: quantization consists in such a choice of the energy parameter for which \(\psi\) in (24) satisfies the conditions of continuity, single-valuedness, and finiteness in the region of admissible values of the coordinates. In such a method of quantization one cannot fail to see a great step forward in comparison with the old quantum theory: there it seemed to come, as it were, from heaven; here it appears as a condition long familiar to the physicist. If, for example, we solve the equation of oscillations of a string fixed at two points, then we obtain an entirely analogous quantization of the admissible periods of oscillation, which we call proper periods. Incidentally, the problem of finding the proper periods of a string also formally coincides with the problem just considered; indeed, the equation of the string is

\[ \frac{\partial^2 u}{\partial x^2}-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=0, \]

and we solve it by the assumption:

\[ u=U(x)e^{2\pi i t}, \]

where substitution of this assumption, corresponding to standing waves, leads us to the equation:

\[ \frac{\partial^2 U}{\partial x^2}+\frac{4\pi^2\nu^2}{v^2}U=0, \]

completely analogous to equation (28); the solution of this equation is

\[ U=\sin \omega x \]

for

\[ \omega=\sqrt{\frac{4\pi^2\nu^2}{v^2}}=\frac{2\pi\nu}{v}; \]

finally, if we impose the condition that the propagation of waves along the string have period \(x=l\), i.e. that the amplitudes of displacements and velocities repeat after a distance \(l\) (the so-called periodicity condition), then we shall see that for this the condition must be fulfilled:

\[ \omega l=\frac{2\pi\nu l}{v}=2\pi n, \]

analogous to (30). If from such a string we cut out a piece \(l\) and bend it, without changing the conditions of oscillation, into a circle, then its beginning and end

as a consequence of the periodicity condition, will turn out to perform one and the same motion—there will be no violation of continuity here.

The analogy is thus complete. But in other cases as well, of which we speak below, Schrödinger’s theory leads us to the solution of similar problems, with only the difference and complication that the coefficient of \(\psi\) in equation (24) is itself a function of the coordinates; in cases of one variable Schrödinger’s problem would become, for example, analogous to the problem of the vibrations of a string with non-uniformly distributed mass. But all this belongs to the old problems of mathematical physics, and methods for solving them have been sufficiently developed; always, as in the case of the string, solutions satisfying the conditions at the ends exist (in the above-mentioned physical sense) only for discrete (rarely for continuous) values of the parameter entering the coefficient of the unknown function; these values have received the name of characteristic numbers and are usually denoted by the letter \(\lambda_n\); their number is infinite. The integrals of the basic equation (24) corresponding to the characteristic numbers are called fundamental functions. Finding the characteristic numbers and the fundamental functions is, generally speaking, a difficult problem; but, of course, we cannot set forth here the techniques for its solution1.

One more remark concerning the meaning of the solutions being sought. Since we start from a particular solution of the form (23), then, obviously—as in the example of the string given above—we are seeking particular solutions corresponding to standing waves; this circumstance distinguishes Schrödinger’s mechanics from de Broglie’s mechanics, which considers traveling waves.

9. Other examples of quantization according to Schrödinger. The example considered above is the simplest with respect to the computational side of the matter. We shall not give detailed solutions for other cases of the mechanics of a point, since they contain no fundamental physical differences from the solution analyzed above, and the difference lies only in the purely mathematical side of the matter. We shall list only a few of the most interesting cases, whose solution could be carried through to the end.

Oscillator. In this case the energy of the point has the form:

\[ E=\frac{1}{2}m\dot{x}^{2}+\frac{1}{2}ax^{2} =\frac{1}{2}m\dot{x}^{2}+2\pi^{2}m\nu_{0}^{2}x^{2}. \]

Hence the coefficient of \(\psi\) in equation (24) is determined:

\[ \frac{8\pi^{2}m}{h^{2}}\left(E-2\pi^{2}m\nu_{0}^{2}x^{2}\right). \]

The solution of Schrödinger’s equation leads to characteristic numbers of the form:

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

i.e.,

\[ E_n=\left(n+\frac{1}{2}\right)h\nu_0. \]

Here \(\nu_0\) is the proper frequency of the oscillator in the mechanical sense of the word.

The result of the earlier quantum theory is well known:

\[ E_n=nh\nu_0. \]

As we see, the differences between the energy levels are the same in both theories; but in Schrödinger’s theory the levels themselves are shifted by \(\frac{1}{2}h\nu_0\); it is not superfluous to recall that this quantity occurs in the quantum theory of specific heats.

The hydrogen atom\(^1\). Schrödinger’s equation has the form:

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

The solution is carried out in polar coordinates \(r,\vartheta,\varphi\), and it is convenient to seek it in the form of a product of two functions, one of which depends only on \(r\), the other on \(\vartheta,\varphi\). Two separate equations are obtained: the one containing only the angular variables turns out to be a known equation, whose solutions are the so-called surface spherical functions; the other, containing only \(r\) and a certain parameter \(a=p(p+1)\) (\(p\) an integer), has characteristic numbers of two kinds: for \(E>0\) they form a continuous set, i.e. for every value \(E>0\) there exists a solution, single-valued, finite, and continuous throughout all space and vanishing at infinity; but for \(E<0\) the characteristic numbers, and consequently the energy levels, are discrete, namely:

\[ E_n=-\frac{2\pi^2me^4}{h^2(n+2)^2}\quad (n=0,1,2,\ldots), \]

where \(n\) must be no less than the above-mentioned \(p\). These results coincide with the earlier quantum theory, with the levels \(E_n<0\) corresponding to the elliptical paths of the electron, and \(E>0\) to hyperbolic ones. It is interesting to note that the earlier quantum mechanics admi-

\(^1\) Schrödinger. Quantisierung als Eigenwertproblem. Erste Mitteilung. Ann. d. Phys. 79, 361, 1926.

the energy level corresponding to rectilinear oscillation of the electron through the nucleus, and it was necessary simply to exclude this case by a prohibition, the meaning of which remained obscure; in Schrödinger’s mechanics this case is excluded automatically, by the absence of the characteristic number corresponding to it.

Schrödinger, and also Fock1, solved in the same way the problem corresponding to the Stark phenomenon—the splitting of spectral lines in an electric field—whereby a formula was found for the energy levels which in the first approximation coincides with the formula of the former quantum theory obtained by Epstein; in the second approximation the formulas diverge, especially for small quantum numbers2.

Diatomic molecules. Up to now we have spoken only of Schrödinger’s equation for a single point; but it can without difficulty be generalized to any number of degrees of freedom. There are several ways to this end; for example, one may proceed from the analogy with the optics of multidimensional media (which, of course, has no direct physical meaning), as Schrödinger does in his second paper, or else make use of the formal prescription indicated above; there are other ways as well. Such a more general Schrödinger equation has been applied to several cases in which it proved possible to carry the calculation through to the end. Thus, Fuess3 considered a diatomic molecule and arrived at results agreeing well with experiment, and here too the new theory showed a number of advantages over the old. Finally, let us mention that Alexandrow4 calculated the ionization potential of the hydrogen molecule, finding it equal to \(16.4\,V\), in complete agreement with the experimental results, whereas the former theory gave \(23.7V\)—a number that did not at all correspond to experiment; in general the old theory leads to incorrect results in all those cases where a system of more than two bodies is involved. Therefore the best test of the new theory would be a calculation of the spectrum of helium; however, this problem has not yet been solved, although approaches to its solution already exist.

1. The fundamental question of any theory of the atom is the calculation of the frequencies emitted by it and of their intensities, and quantization is only the first step toward solving this problem. In Bohr’s theory the second

step consisted in the assertion that the frequencies obtained are determined by the relation:

\[ h\nu = E' - E'', \tag{31} \]

whereas questions of intensity were decided with the aid of the correspondence principle; and both this latter and relation (31) were purely formal assertions, devoid of physical meaning. All attempts to find such meaning proved unsuccessful, although in order to find it recourse was had (Bohr, Kramers, Slater) to such desperate measures as abandoning the strict exactness of the law of conservation of energy and retaining for it only statistical validity. Therefore the questions of frequencies and intensities must be a touchstone for Schrödinger’s theory as well, and it is important to know how it resolves them; we shall now try to give as briefly as possible an account of what has so far been achieved by it in the solution of this problem.

Mathematically, Schrödinger’s equation may be regarded as definitively solved if the characteristic numbers have been found

\[ \lambda_1,\ \lambda_2,\ \lambda_3\ldots \]

and the corresponding (i.e. representing solutions of Schrödinger’s equation with the corresponding \(\lambda_n\) substituted into it) fundamental functions:

\[ \psi_1,\ \psi_2,\ \psi_3\ldots \]

Since Schrödinger’s equation is linear, the principle of superposition is valid for it, and in general its solution is

\[ \psi = c_1\psi_1 + c_2\psi_2 + \ldots = \sum c_n\psi_n, \tag{32} \]

where \(c_n\) are arbitrary constants. The meaning of the characteristic numbers, their connection with energy levels (which have such a vivid physical expression as excitation potentials by electron impact), has already been clarified. Therefore the next step in the development of Schrödinger’s theory must be the establishment of the physical meaning of the function \(\psi\), and it is natural to expect that it is precisely from this function that we must be able to calculate frequencies and intensities. But here one might expect that the problem can be divided into two parts. As formula (31) shows (a formula which has hitherto always justified itself), one must expect that the question of the frequency obtained is resolved only with the aid of the characteristic numbers \(\lambda_n\), or else with the aid of the functions \(\psi_n\) wholly determined by them, but that the coefficients \(c_n\) play no role here, since the latter are in no way connected with \(\lambda_n\) and may be chosen arbitrarily. As for the question of intensities, the values of \(c_n\) may be necessary for its solution. However, this very question may be posed in two ways. First, one may

N. N. ANDREEV

seek the intensity of the elementary act of radiation; for example, one may ask how the energy of electromagnetic waves is distributed around a given hydrogen atom, how the intensity is distributed along the lines of the Zeeman phenomenon, etc.; here too belongs the question of the polarization of radiation. But, on the other hand, in practically observed cases we are dealing not with the radiation of a single atom, but with a statistical phenomenon: a multitude of atoms, being in different states, radiate; and as a result we observe neither polarization nor any difference of radiation in different directions, and moreover we see not one frequency, but all possible frequencies.

Thus, we may seek the \(c_n\) corresponding to a given elementary act, or else, having decided to deal only with observable quantities, seek certain average statistical functions of \(\psi\), characterized by certain average values \(c_n\); we say functions of \(\psi\), and not the \(\psi\) themselves, because, as it turns out, no simple physical meaning can be found for the \(\psi\) themselves.

1. In accordance with all these considerations, let us first of all dwell on the adaptation of Schrödinger’s theory to Bohr’s frequency condition (31), and we shall operate only with \(\lambda_n\) and \(\psi_n\), forgetting about \(c_n\).

As (23) shows, we must proceed not from \(\psi_n\), but from

\[ \varphi_n=\psi_n e^{\frac{2\pi i E_n t}{h}}, \tag{33} \]

for they represent solutions of Schrödinger’s fundamental equation (22). It is easy to see that the \(\varphi_n\) themselves do not determine electromagnetic waves, since the radiation frequencies \(\nu_n=\frac{E_n}{h}\) are not present in the spectrum; but it is not difficult to notice that the products of fundamental functions by their conjugates have the desired form; for example, we have (the bar over a function is the sign of conjugation)

\[ \varphi_n\overline{\varphi_s} = \psi_n e^{\frac{2\pi i E_n t}{h}}\, \overline{\psi_s}\, e^{-\frac{2\pi i E_s t}{h}} = \psi_n\overline{\psi_s}\, e^{\frac{2\pi i t}{h}(E_n-E_s)}. \]

To the general solution, having the form:

\[ \varphi=\sum c_n\psi_n e^{\frac{2\pi i E_n t}{h}}, \tag{34} \]

the same remark is applicable:

\[ \varphi\overline{\varphi} = \sum\sum c_n\overline{c_s}\psi_n\overline{\psi_s}\, e^{\frac{2\pi i t}{h}(E_n-E_s)}; \]

for the sake of generality we assume that \(c_n\) may be complex. From this it is natural to draw two conclusions: 1) for the existence of radiation of an atom it is necessary that in it there should simultaneously exist two different pro-

processes, characterized by two different fundamental functions; 2) the quantities \(\psi_n \psi_s\) must be connected with the amplitude of radiation of the atom. If we recall that in electromagnetic theory radiation is produced, in the simplest case, by an oscillating electric dipole, then we shall wish \(\psi_n \psi_s\) to be some reflection of the amplitude of such a dipole. But the latter is a point image, whereas \(\psi_n \psi_s\) are spatial functions; they must therefore correspond to some spatial distribution of electricity, and not at all to a point one. All these rather indefinite considerations lead Schrödinger\(^{1}\) to the hypothesis that the quantity

\[ \varphi \overline{\varphi}. \]

may be regarded as the density of the distribution of electricity in the atom. Therefore the quantity of electricity in the volume element \(d\tau\) is

\[ \varphi \overline{\varphi}\, d\tau, \]

and the components of the dipole moment are this quantity multiplied by the coordinates of the element \(d\tau\); for example, the component of the dipole along the \(x\)-axis is

\[ x \varphi \overline{\varphi}\, d\tau. \]

The resultant dipole is the sum of all dipole elements, i.e.

\[ M_x=\int x\varphi \overline{\varphi}\, d\tau. \tag{35} \]

If the function \(\varphi\) consists of only one term

\[ \varphi_k=\psi_k e^{\frac{2\pi i E_k t}{h}}, \]

then \(\varphi_k \overline{\varphi}_k=\psi_k^2\), and there will be no dependence on time; it is clear that in this way it is explained why radiation frequencies \(\nu_k=\frac{E_k}{h}\) are not observed.

If, however, two terms, \(\varphi_k\) and \(\varphi_s\), are present in \(\varphi\), then we have:

\[ \varphi \overline{\varphi} = \left(\overline{c_k\varphi_k}+\overline{c_s\varphi_s}\right) \left(c_k\varphi_k+c_s\varphi_s\right) = \]

\[ = \overline{c_k}c_k\,\overline{\varphi_k}\varphi_k + \overline{c_s}c_s\,\overline{\varphi_s}\varphi_s + \overline{c_k}c_s\,\overline{\varphi_k}\varphi_s + \overline{c_s}c_k\,\overline{\varphi_s}\varphi_k; \]

the first two terms do not depend on time and therefore do not interest us; the second two reduce to the form:

\[ \psi_k\psi_s\gamma_k\gamma_s \left\{ e^{i(2\pi i \nu_{ks} t+\delta_{ks})} + e^{-i(2\pi i \nu_{ks} t+\delta_{ks})} \right\}, \]

\(^{1}\) Vierte Mitteilung. Ann. d. Phys. 81, 109, 1926.

where

\[ c_k=\gamma_k e^{i\delta_k};\qquad \lambda_{ks}=\lambda_k-\lambda_s; \]

\[ \nu_{ks}=\frac{E_k-E_s}{h}; \]

bearing in mind that \(x\) and \(\rho\) are real, we find

\[ M_x=2\gamma_k\gamma_s \cos(2\pi\nu_{ks}t+\delta_{ks})\int x\psi_k\psi_s\,d\tau . \tag{36} \]

Thus, the time-dependent part of the dipole moment has the required Bohr frequency

\[ \nu_{ks}=\frac{E_k-E_s}{h}. \tag{31} \]

This reasoning is not difficult to extend to any number of terms in (34), and in this way we obtain an idea of the physical meaning of Bohr’s condition (31).

12. If in (34) all terms are present \((c_n\ne 0)\), then the number of integrals entering into (36):

\[ x_{ks}=\int x\psi_k\psi_s\,d\tau \tag{36} \]

is, evidently, \(\infty^2\); writing them in the form of a table:

we obtain what is called a matrix; analogous tables we shall obtain also for any other function \(F\) (which, let us suppose, contains, as a factor, \(e^{2\pi i\nu_{ks}}\)) according to the rule:

\[ F_{ks}=\int F\psi_k\psi_s\,d\tau . \tag{37} \]

But precisely in the works of Heisenberg, Born and Jordan, subsequently developed by Dirac, Pauli and many others, every quantum quantity is represented in the form of a matrix. As an example, let us consider the radiation of an atom; the frequencies emitted by it are \(\nu_{ks}\); if we denote by \(x_{ks}\) the amplitudes of these radiations, then

\[ q_{ks}=x_{ks}e^{2\pi i\nu_{ks}t} \tag{38} \]

will give us a matrix element by which we can determine both the frequencies \((\nu_{k s})\) and the corresponding intensities \((x_{k s}^{2})\). The creators of matrix mechanics show that one can set up matrix equations for finding these quantities, and that these equations have a great formal analogy with Hamilton’s equations; every problem of the old quantum theory written in the form of Hamilton’s equations can be transformed into a problem of matrix mechanics by the simple formal replacement of ordinary quantities by matrix ones. We cannot enter here into an exposition of their arguments; let us say only that the elements of their matrices coincide with (37). This circumstance makes it possible to regard matrix mechanics as a part of Schrödinger’s mechanics, but it also reflects favorably on the latter. Namely, calculations of intensities by matrix mechanics gave favorable results; therefore one may also use, in Schrödinger’s mechanics, formula (36) for calculating intensities and polarizations. Schrödinger1 carried this out for the Stark phenomenon and found fairly good agreement with experiment, in any case better than was given by the old quantum theory with the aid of the correspondence principle.

1. From all that has been set forth above it is clear that Schrödinger’s theory, while often giving better numerical agreement with experiment than Bohr’s theory, surpasses the latter also in another respect: it gives an interpretation of quantization and of the frequency condition which apparently contains no internal contradictions and is, in any case, physically more acceptable than Bohr’s formal recipes. Of course, one may regret that it no longer deals with point electrons, but, so to speak, smears the electron over all space, and in addition does not give a clear model of the atom. But the first feature—the appeal to the equation of a continuous medium—is not so very new, as we pointed out at the beginning of our article, and one must simply become accustomed to it, just as we have become accustomed to Bohr’s physically absurd conditions; the second shortcoming—the absence of a model—may yet be remedied by the further development of the theory, although it must be noted that some authors (Born, Ya. I. Frenkel) hold the conviction that such a model is impossible in principle, since the very essence of Schrödinger’s mechanics is statistical. In our exposition we have tried not to adopt such a point of view, for it seems to us that it is not necessary for Schrödinger’s mechanics.

2. Having in view an exposition only of the foundations of Schrödinger’s mechanics, we have not touched on many questions connected with its further development. We have not spoken of its generalization to many degrees of free-

... freedom and in the domain of the mechanics of the special theory of relativity, which has already been done. Moreover, a very promising connection has also been established with the theory of relativity and Maxwell’s equations, so that there arises the hope of a very general and consistent exposition of the phenomena of radiation; finally, Bose–Einstein statistics1 also proves to be closely connected with Schrödinger’s theory; who knows whether we are not close to a very great synthesis of the principles of physics into one harmonious whole?

1. See about it It. Gamm, Advances in Physical Sciences, 2, 1926. ↩↩↩↩↩↩↩↩↩↩

2. G. Wentzel, Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik. Z. f. Phys. 38, 518, 1926. ↩↩↩

3. Fuess, Die Schwingungen zweiatomiger Molecüle in der Undulationsmechanik. Ann. d. Phys. 80, 367, 1926, and Zur Intensität der Bandenlinien u. des Affinitätsspectrums zweiatomiger Molecüle. Ann. d. Phys. 80, 281, 1926. ↩↩

4. W. Alexandrow, Das Wasserstoffmolecülion und die Undulationsmechanik. Ann. d. Phys. 81, 603, 1926. ↩