Wave Theory of the Mechanics of Atoms and Molecules1

E. Schrödinger, Zurich.

§ 1. Hamilton’s analogy between mechanics and optics.— § 2. The analogy can be generalized and leads to a real “physical” or “wave” mechanics instead of purely geometrical mechanics.— § 3. Significance of the wave type, macromechanical and micromechanical problems — § 4. The wave equation and its application to the hydrogen atom — § 5. The inner meaning of the appearance of discrete characteristic frequencies — § 6. Other problems. Intensities of the emitted light.— § 7. Derivation of the wave equation from Hamilton’s variational principle — § 8. The physical meaning of the wave function characterizes a continuous distribution of electricity in space, the oscillations of this distribution producing radiation on the basis of the laws of ordinary electrodynamics.— § 9. Non-conservative systems. Theory of dispersion and scattering. Theory of “transitions” between “stationary states” — § 10. The question of the theory of relativity and of the action of a magnetic field. Incompleteness of this part of the theory

§ 1. The theory set forth on the following pages is based on the extraordinarily interesting and profound investigations of L. de Broglie2 on the so-called “phase waves” (“ondes de phase”) and is applied to the motion of material particles, in particular to the motion of the electron or proton. The general point of view of the theory here expounded, published in a series of German articles3, consists in this: material points consist of systems of waves, or are even identical with them. Such an extreme conception may also be erroneous. In any case it gives no answer to the question of why in nature there occur precisely

of a given kind of waves, which must correspond to material particles with a definite mass and a definite charge. But, on the other hand, the opposite point of view, which leaves out of account the waves studied by L. de Broglie and considers only the motion of material particles, has led to such considerable difficulties in the theory of the mechanics of atoms—and this after a century of development and deepening of mechanics—that it seems not only not dangerous, but even desirable, to devote special attention, at least for a time, to the proposed point of view. In doing so, we must, of course, remember that full agreement in the study of the different aspects of physical phenomena can be attained only through a harmonious fusion of these two extreme points of view.

The principal advantages of the theory presented are the following:

a) The laws of motion and the quantum conditions are derived simultaneously from a simple Hamilton principle.

b) The disagreement which has existed up to now in quantum theory between the frequency of motion and the frequency of radiation is removed, since the latter frequencies coincide with differences of the former. A definite localization of electric charge in space and time may be associated with a system of waves, and the latter explains, on the basis of ordinary electrodynamics, the frequency, intensity, and polarization of the emitted light and makes all sorts of correspondence and selection principles superfluous.

c) It appears possible, with the aid of the new theory, to trace all the details of the so-called “transitions,” which until very recently had been quite mysterious.

d) There are several points of divergence between the new and the old theory concerning the individual values of energy or the levels of frequencies. In these cases the new theory is in better agreement with experiment.

To explain the line of thought, I shall take as an example of a mechanical system a material point of mass \(m\), moving in a conservative force field with potential energy \(V(x,y,z)\). All the following reasoning can be generalized to the motion of a “phase point,” representing the motion of any conservative system in “phase space” (the space \(q\), but not the space \(pq\))1.

We shall make this generalization in a somewhat different form in § 7. Using the usual notation, let us write the expression for the kinetic energy \(T\)

\[ T=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})=\frac{p_x^2+p_y^2+p_z^2}{2m}. \tag{1} \]

The well-known Hamiltonian action function \(W\)

\[ W=\int_{t_0}^{t}(T-V)\,dt, \tag{2} \]

regarded as a function of the upper limit of integration and of the final values of the coordinates \(x,y,z\), satisfies Hamilton’s equation\({}^{1}\) with partial derivatives:

\[ \frac{\partial W}{\partial t}+\frac{1}{2m}\left[\left(\frac{\partial W}{\partial x}\right)^2+\left(\frac{\partial W}{\partial y}\right)^2+\left(\frac{\partial W}{\partial z}\right)^2\right]+V(x,y,z)=0. \tag{3} \]

To solve this equation, we put, as usual,

\[ W=-Et+S(x,y,z), \tag{4} \]

where \(E\) is a constant of integration, i.e. the total energy, and \(S\) is a function only of the coordinates \(x,y,z\). Equation (3) may be written in the following form:

\[ |\operatorname{grad} W|=\sqrt{2m(E-V)}. \tag{5} \]

In this form it leads to a very simple geometrical interpretation. Suppose that \(t\) is constant. Any function \(W\) depending on the spatial coordinates can be described geometrically if one specifies a system of surfaces on which \(W\) is constant, and determines for each of these surfaces the constant value \(W_0\) which the function \(W\) assumes at the points of that surface. On the other hand, we can easily construct a solution of equation (5), starting from some arbitrary surface and an arbitrarily chosen value \(W_0\), which we assign to it. Let us choose the initial surface and the initial value, and let us take (also arbitrarily) one side of this surface as positive. We can easily construct

of the coordinate; in Cartesian coordinates the generalized momentum is the product of the mass of the point by the corresponding component of the velocity. To depict the state of motion of a system, statistical mechanics introduces the concept of “phase space \(pq\),” whose number of dimensions is twice as large as in “space \(q\).” Then the motion of the system at each moment is characterized by one point of “space \(pq\),” whose coordinates are the generalized coordinates \(q\) and the generalized momenta \(p\) of all the points of the given system.
Translator’s note.

\({}^{1}\) See, for example, M. Planck, Introduction to General Mechanics, Part II, Chs. III.—State Publishing House, Moscow–Leningrad, 1926.

Ed.

normal at each point of the chosen surface. The length of the normal is equal to¹):

\[ dn=\frac{dW_0}{\sqrt{2m(E-V)}}. \]

The aggregate of the points obtained in this way forms a surface to which we must assign the value \(W_0+dW_0\). Continuing this process, we shall obtain a system of surfaces and the corresponding values of the constants, i.e. the distribution of the function \(W\) in space, while \(t\) remains constant.

Let us now suppose that time varies. From equation (4) it follows that the system of surfaces does not change, but that the values of the constants pass along the normals from one surface to another with a certain velocity \(u\), which is expressed by²):

\[ u=\frac{E}{\sqrt{2m(E-V)}}. \tag{6} \]

The velocity \(u\) is a function of the constant energy \(E\) and, moreover, a function of the coordinates, since it contains \(V(x,y,z)\).

Instead of imagining the surfaces as fixed in space, and the values of the constant as passing from one surface to another, we may imagine that a definite numerical value \(W\) is associated with some individual surface, and that the surfaces themselves move continuously in such a way that each of them occupies exactly the position and shape of the next surface. Then the quantity \(u\), given by equation (6), determines the magnitude of the velocity of displacement of some surface along the normal at one of its points. Applying this point of view, we arrive at a visual representation which fully coincides with the picture of the propagation—

¹) From differential geometry it is known that the element of the normal to a surface whose equation is

\[ W(x,y,z)=W_0, \]

is equal to

\[ dn= \frac{\dfrac{\partial W}{\partial x}\,dx+\dfrac{\partial W}{\partial y}\,dy+\dfrac{\partial W}{\partial z}\,dz} {\sqrt{\left(\dfrac{\partial W}{\partial x}\right)^2+ \left(\dfrac{\partial W}{\partial y}\right)^2+ \left(\dfrac{\partial W}{\partial z}\right)^2}} = \frac{dW_0} {\sqrt{\left(\dfrac{\partial W}{\partial x}\right)^2+ \left(\dfrac{\partial W}{\partial y}\right)^2+ \left(\dfrac{\partial W}{\partial z}\right)^2}}. \]

Determining the value of the denominator from equation (3) and substituting \(\dfrac{\partial W}{\partial t}=-E\), on the basis of equation (4) we obtain the formula given in the text.

Translator’s note.

²)

\[ u=\frac{dn}{dt} = \frac{\dfrac{dW_0}{dt}} {\sqrt{2m(E-V)}} = [\text{on the basis of (4)}] = \frac{E}{\sqrt{2m(E-V)}}. \]

Translator’s note.

propagation of a system of stationary waves in an optically inhomogeneous (but isotropic) medium, where \(W\) is proportional to the phase, and \(u\) is the phase velocity (the refractive index is assumed proportional to \(\dfrac{1}{u}\)). It is obvious that the construction of normals \(dn\) considered is equivalent to Huygens’ principle. The orthogonal trajectories of our system of surfaces \(W\) form the system of rays of our optical picture. They represent the possible trajectories of material points in the mechanical problem. Indeed, it is known that

\[ p_x=m\dot{x}=\frac{\partial W}{\partial x} \tag{7} \]

(analogous equations for \(y\) and \(z\)^1). It should be noted that the phase velocity \(u\) is the velocity of a material point. The latter velocity is equal, on the basis of (7) and (5), to

\[ v=\sqrt{\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}} =\sqrt{\frac{2(E-V)}{m}} . \tag{8} \]

Comparing (6) and (8), we see that the quantities \(u\) and \(v\) vary inversely to one another.

It is easy to show the correspondence between the well-known mechanical principle of Hamilton and the equally well-known optical principle of Fermat.

§ 2. All that has been said above is by no means new. All this was known in its time to Hamilton much better than to the majority of physicists at the present time. Indeed, the theory of the propagation of light in an inhomogeneous medium, which Hamilton developed ten years before the mechanical theory, became, thanks to the striking analogy he discovered, the starting point for his famous theory of pure mechanics. Despite the great popularity which the latter acquired, the path that led to it was almost forgotten^2).

The following circumstance should be especially noted: although in the argument presented by us such concepts as “wave surfaces,” “Huygens’ principle,” and “Fermat’s principle” figure, nevertheless the entire analogy drawn relates rather to geometrical optics than to real physical or wave optics. Indeed, the chief and basic mechanical representation is the representation of the path of the trajectory of a material particle and corresponds to the representation of

^1) Equation (7) shows that the direction of the velocity coincides with the normal to the surface \(W(x,y,z)=W_0\).
Transl. note.

^2) See F. Klein, Jahresber. d. Deutsch. Math. Ver. 1, 1891; Zeits. f. Math. u. Phys. 46, 1901; (Ges. Abh. II, 601, 603); E. T. Whittaker, Analytical Dynamics, Chap. II; A. Sommerfeld, Atombau, p. 803. The analogy was again revealed in the domain of relativistic mechanics in the work of L. de Broglie mentioned above.

rays in the optical analogy. But the conception of rays is quite definite only in purely abstract geometrical optics. It loses almost all meaning in real physical optics as soon as the dimensions of the ray or of the material obstacles in its path become comparable with the wavelength. Even if this is not the case, the definition of rays in geometrical optics is nevertheless only approximate. It cannot at all be applied to the fine structure of real optical phenomena—to diffraction phenomena. Even if one somewhat generalizes geometrical optics by adding to it Huygens’ principle (in the simple form used above), one still cannot understand even the simplest diffraction phenomena unless one adds other rules, apparently strange, that determine the circumstances under which Huygens’ envelope surface has physical significance or else has none. (I have in mind the construction of “Fresnel zones.”) These rules would be completely incomprehensible to one who deals only with geometrical optics. Moreover, one may note that those concepts which are fundamental in real physical optics, i.e. the wave function itself (\(W\) is only the phase), the equation of wave propagation, the wavelength and the frequency of oscillations, do not enter at all into the analogy established above. The phase velocity \(u\) does enter, it is true, but, as we have seen, it is not very deeply connected with the mechanical velocity \(v\).

At first glance it does not seem interesting to work out Hamilton’s analogy in detail with respect to real wave optics. If one gives the concept of wavelength a definite physical meaning, then the concept of rays loses a definite physical meaning, at least in some cases—and therefore the analogy appears weakened or even entirely destroyed in those cases where the dimensions of the mechanical trajectories or their radii of curvature become comparable with the wavelength. To save the analogy, it would seem necessary to assign to the wavelength an extremely small value, so small that it would be small in comparison with all dimensions that can play any role in problems of mechanics. But then the development of wave images would seem superfluous, since geometrical optics is the limiting case of wave optics for vanishingly small wavelengths1.

Let us compare with these considerations the striking fact which we come to know every day with certainty: this fact consists in this, that ordinary mechanics is in reality not applicable to mechanical systems that are very small, namely of atomic dimensions. Taking into account this circumstance, which sets its seal on all modern physical thought, does it not seem

interesting to investigate the question: perhaps the inapplicability of ordinary mechanics to micromechanical problems proves to be of the same kind as the inapplicability of geometrical optics to the phenomena of diffraction and interference, and may it not be possible to overcome it in a similar way? In other words: Hamilton’s analogy must also be extended to wave optics, with a definite magnitude being assigned to the wavelength in each particular case. This quantity has real significance for the mechanical problem, namely: ordinary mechanics, with its conception of a moving point and its linear trajectory (or, in a more general sense, of an “image point” moving in coordinate space), is approximately correct only when it is applied to a trajectory that is large (or whose radii of curvature are large) in comparison with the wavelength. If this is not so, then one must study the phenomenon of wave propagation. In the simple case of a material point moving in an external force field, the wave phenomenon may be imagined as taking place in ordinary three-dimensional space. In the case of a more general mechanical system, this phenomenon must first be placed in coordinate space (the space of \(q\), and not the space of \(pq\))1, and then in some way projected into ordinary space. To a certain extent the equations of ordinary mechanics will prove no more suitable for the study of these micromechanical wave phenomena than the rules of geometrical optics are for the study of diffraction phenomena. In this investigation it is quite natural to make use of the generally known methods of wave theory, generalized in the appropriate way. The ideas we have sketched briefly may find their justification in the success which their applications have had.

§ 3. Let us turn to the system of surfaces \(W\) discussed in § 1. With them we shall associate the idea of stationary sinusoidal waves whose phase is determined by the quantity \(W\) in equation (4). The wave function \(\psi\) has the form:

\[ \psi = A(x,y,z)\sin\frac{W}{K} = A(x,y,z)\sin\left[-\frac{Et}{K}+\frac{S(x,y,z)}{K}\right], \tag{9} \]

where the function \(A\) is the “amplitude.” It is necessary to introduce a constant \(K\), which must have the physical dimension of action (energy \(\times\) time), since the argument standing under the sine sign must be a dimensionless number. Since the frequency of the wave (9) is, evidently,

\[ \nu=\frac{E}{2\pi K}, \tag{10} \]

there is a strong temptation to suppose that \(K\) is a universal constant, independent of \(E\) and independent of the nature

mechanical system: if we make such an assumption and put \(K\) equal to \(\dfrac{h}{2\pi}\), then the frequency \(\nu\) is expressed by the equation:

\[ \nu=\frac{E}{h}, \tag{11} \]

where \(h\) is Planck’s constant. Thus we arrive, by a simple route, at the known universal relation between energy and frequency.

In ordinary mechanics, what has a definite value is not the absolute magnitude of the energy, but differences between energy magnitudes. When this difficulty is encountered, the zero level of energy can be determined in a quite satisfactory way if one makes use of relativistic mechanics and the conception of the equivalence of mass and energy1. But here we have no need to dwell on this question. Although the frequency \(\nu\) of our waves in equations (10) or (11) also depends on the zero level of energy, the wavelength does not depend on it. And, in accordance with what was said above, it is precisely the wavelength that is of greatest interest to us. Comparing this quantity with the dimensions of the path or orbit of a material particle, calculated according to ordinary mechanics, will show us whether this calculation has physical meaning, and whether the methods of ordinary mechanics are approximately applicable to the solution of the particular problem. The wavelength is equal, according to equations (11) and (6), to

\[ \lambda=\frac{u}{\nu}=\frac{h}{\sqrt{2m(E-V)}}. \tag{12} \]

Here \(E-V\) is the kinetic energy \(\frac{1}{2}mv^2\), which indeed does not depend on the zero level of the total energy. Substituting its value, we obtain:

\[ \lambda=\frac{h}{mv}. \tag{13} \]

To answer the question whether ordinary mechanics may be applied to an electron moving in a Keplerian orbit of atomic dimensions, let us suppose that \(a\) is the length of the dimensions of the atom, and compare \(\lambda\) with the magnitude \(a\)

\[ \frac{\lambda}{a}=\frac{h}{mva}. \tag{14} \]

The denominator of the right-hand side has, undoubtedly, the same order of magnitude as the angular momentum of the electron, and the latter, as is known, is of the same order of magnitude as Planck’s constant

in the case of a Keplerian orbit of atomic dimensions. Thus, the order of magnitude of the ratio \(\frac{\lambda}{a}\) becomes equal to unity, and ordinary mechanics proves applicable to such an orbit no more than geometrical optics is applicable to the diffraction of light by a disk whose diameter is equal to the wavelength. If a physicist attempted to analyze the latter phenomenon with the aid of the concept of rays, to which he has become accustomed in macroscopic geometrical optics, he would encounter extremely serious difficulties and apparent contradictions. The “rays” (the lines determining the direction of the energy flux) would no longer be straight and would interact with one another in the strangest manner, in complete contradiction to the basic laws of geometrical optics. In the same way, the representation in terms of the trajectories of material points appears inapplicable to orbits of atomic dimensions. It is quite satisfactory that, by equating \(K\) (in essence) to Planck’s constant (equation 11), we obtain, for the limits of applicability of ordinary mechanics, a quantity of precisely the same order as had to be admitted in order, with the aid of the new representation, to overcome the difficulties of quantum theory. We may add that, on the basis of equation (13), for a Keplerian electronic orbit of an order of magnitude corresponding to a large quantum number, the ratio of the wavelength to the dimensions of the orbit has an order of magnitude equal to unity divided by the quantum number1.

Thus ordinary mechanics gives an ever better approximation to the limit as the quantum number (or the dimensions of the orbit) increases, as should be expected of any rational theory.

According to the fundamental equation \(\nu=\frac{E}{h}\) (eq. 11), the phase velocity \(u\), determined by equation (6), proves to depend on the frequency \(\nu\). Therefore equation (6) is a dispersion equation. This gives a very interesting illumination of the relation between the two velocities: the velocity \(v\) of the moving particle (equation 8) and the phase velocity \(u\) (equation 6). It is easy to prove that \(v\) is exactly equal to the so-called group velocity, derived from the dispersion formula (6)2. Using—

Having made use of this interesting result, one can obtain an idea of how ordinary mechanics can give an approximate description of our wave motion. By superposing waves with frequencies in a small interval between \(\nu\) and \(\nu+d\nu\), one can construct a “wave packet,” whose dimensions are comparatively small in all directions, although they may be quite large in comparison with the wavelength. It can be proved that the “center of gravity,” so to speak, of such a packet moves, according to the laws of wave propagation, along precisely the same trajectory as that along which a material point would move according to the laws of ordinary mechanics. This equivalence remains valid even in the case when the dimensions of the orbit are not large in comparison with the wavelength. But in the latter case it is of no significance, since the wave packet spreads out in all directions far beyond the limits of the trajectory. On the contrary, if the dimensions of the orbit are comparatively large, then the motion of the wave packet as a whole can give a sufficient idea of what is actually taking place, if we are not interested in its internal structure. As indicated above, this “motion as a whole” is governed by the laws of ordinary mechanics.

§ 4. We shall not dwell further on this question, but shall turn to applications of the theory to much more interesting micromechanical problems. As indicated above, in this case the wave phenomena must be studied in detail. This can be done only with the aid of the “equation of wave propagation.” What, then, is this equation? In the case of a single material particle moving in the field of an external force, the simplest procedure is to try to use the ordinary wave equation:

\[ \Delta\psi-\frac{\ddot{\psi}}{u^{2}}=0 \tag{15} \]

and to substitute for \(u\) the quantity which is given by equation (6) and depends on the spatial coordinates (through the potential energy \(V\)) and on the frequency \(\frac{E}{h}\). This latter dependence restricts the application of equation (15) to such functions \(\psi\) as depend on time only through the factor \(e^{-2\pi i \frac{E}{h}t}\). (Such a restriction is always imposed on the wave equation if we are dealing with dispersion.) Therefore we have:

\[ \ddot{\psi}=-\frac{4\pi^{2}E^{2}\psi}{h^{2}}. \]

\(^1\) In other notation:

\[ \frac{\partial^{2}\psi}{\partial x^{2}} + \frac{\partial^{2}\psi}{\partial y^{2}} + \frac{\partial^{2}\psi}{\partial z^{2}} - \frac{1}{u^{2}}\frac{\partial^{2}\psi}{\partial t^{2}} =0 \]

Translator’s note.

Substituting this value into equation (6) and into equation (15), we obtain:

\[ \Delta\psi+\frac{8\pi^{2}m(E-V)\psi}{h^{2}}=0, \tag{16} \]

where \(\psi\) may be regarded as depending only on \(x, y, z\). (We do not change the notation of the function, as, strictly speaking, we should.)

How are we now to deal with equation (16)? At first glance it seems of little use for solving atomic problems, for example, for determining the discrete energy levels in the hydrogen atom. Since this is a partial differential equation, it has a multitude of solutions—a multitude even of a higher transcendental order of magnitude than the system of ordinary differential equations of ordinary mechanics. But the inadequacy of the latter in atomic problems consists not at all in their giving too few possible orbits, but, on the contrary, too many. The task of the “quantum conditions” is precisely to select a discrete number of these orbits as “real” or “stationary,” according to the views accepted until recently. Our wave equation (16), which does not restrict but indefinitely increases the number of possibilities, would seem to make the situation even worse.

Fortunately, these fears prove to be mistaken, owing to an extremely interesting feature of equation (16) in atomic problems. If, for example, we take:

\[ V=-\frac{e^{2}}{r}, \tag{17} \]

where \(e\) is the charge of the electron, \(r=\sqrt{x^{2}+y^{2}+z^{2}}\), then for the simplified hydrogen atom, or the one-body problem, we obtain:

\[ \Delta\psi+\frac{8\pi^{2}m\left(E+\dfrac{e^{2}}{r}\right)\psi}{h^{2}}=0. \tag{18} \]

It turns out that for the greater part of the values of the energy, or of the constant frequency \(E\), this equation has no solutions at all that would be continuous, finite, and single-valued throughout all space. For these values of \(E\), every solution \(\psi\) satisfying two conditions (namely, continuity and single-valuedness) increases without bound both as one approaches infinity and as one approaches the origin of coordinates. The only values of \(E\) for which this does not occur, i.e. for which there exist solutions continuous, finite, and single-valued throughout all space, are the following:

\[ \begin{aligned} &1)\quad E>0\\ &2)\quad E=-\frac{2\pi^{2}me^{4}}{h^{2}n^{2}} \end{aligned} \tag{19} \]

\[ (n=1,2,3,4\ldots). \]

The first set of values corresponds to hyperbolic orbits in ordinary mechanics. According to the generally accepted view, in ordinary quantum theory hyperbolic orbits are not subject to quantization. In our interpretation this follows directly from the circumstance that all positive values of \(E\) give finite solutions. The second set of values corresponds exactly, according to Bohr, to the stationary energy levels of elliptical orbits.

Although I cannot dwell here on the exact and rather painstaking proof of the assertions just mentioned, it is nevertheless of interest to describe in general outline the solutions corresponding to the second series of levels \(E\). The solution can be expressed in three-dimensional\(^1\) polar coordinates, taking \(\psi\) to be equal to the product of functions of the polar angles by a function of the radius \(r\) alone. The first is a spherical function, whose order, increased by one, corresponds to the azimuthal quantum number. The functions of \(r\) entering into the solution are somewhat similar (in general outline) to Bessel functions, but with the difference that they have only a finite number of positive roots, and this number corresponds exactly to the radial quantum number. These roots lie within a region about the origin of coordinates whose dimensions are of the same order as the corresponding Bohr orbit.\(^2\) With further increase of \(r\) the function passes through a maximum or a minimum, and then decreases exponentially as \(r\) increases without bound. Thus, the entire wave phenomenon, though mathematically extending over all space, is in essence confined to a small sphere with a diameter of several ångströms, which may be called an “atom” according to wave mechanics. Some of the solutions mentioned (consisting of the product of a spherical function and a function of \(r\)) resemble the fundamental vibrations of an elastic sphere with a finite number of “nodal surfaces” in the form of spheres, cones, and planes. But one should certainly not think that the wave motion forming the atom is, in general, confined to one of these solutions, whose particular selection and separation depends to a considerable degree on the choice of coordinates. To each of the discrete values \(E\) there corresponds a finite number of particular solutions. By forming a linear expression from the latter and arbitrary constant factors, we obtain the most general solution of equation (18) for a particular value of \(E\). The number of arbitrary constants entering

\(^1\) In the present case, of course, one cannot restrict the problem to two dimensions, as in ordinary mechanics, since the wave phenomenon is essentially three-dimensional.

\(^2\) The dimensions of this region, in which the values of the function differ noticeably from zero, are approximately equal to \(\dfrac{a_n}{n}\), where \(a_n\) is the semi-axis of the elliptical orbit corresponding to the principal quantum number \(n\) (E. Schrödinger, Ann. d. Phys. 79, 371, 1926). — Transl. note

this expression, is exactly equal to the so-called “statistical weight” of this energy level, or, in other words, to the number of separate levels into which the latter splits, according to Bohr’s theory (and therefore also according to the theory here set forth), when perturbing forces appear which remove the so-called “degeneracy” of the problem1. In this connection one may recall that in ordinary mechanics the method mentioned does not give a completely exact number of states. Certain experimental data lead to the conclusion that one must, with the aid of an additional argument—more or less convincing from the theoretical point of view—exclude some of these states, namely those for which the equatorial quantum number is equal to zero2. We may note with satisfaction the fact that, according to the theory here set forth, the above-mentioned number of arbitrary constants, or, in other words, the number of separate levels or frequencies into which each degenerate level \(E\) is split when a perturbing potential appears, is obtained quite correctly from the very beginning. The theory needs no additions, since it excludes the vibrational state corresponding to the Bohr orbit with equatorial quantum number equal to zero.

To complete this description, we may add that to the lowest level \(E\), or, from the point of view of wave motion, to the “fundamental tone” corresponding to the normal state of the atom, there belongs only one kind of vibration, and moreover a very simple one: the function \(\psi\) exhibits complete spherical symmetry and has no nodal surfaces at all. Both the radial quantum number and the order of the spherical function vanish.

§ 5. Let us briefly consider the question of why equation (18) has finite solutions only for certain values of the constant \(E\). All the considerations set forth on the preceding pages would be quite ordinary for any physicist if the problem were the so-called “boundary-value problem,” i.e. if the function \(\psi\) were required only inside some given surface—say, a sphere of given radius—and had to satisfy certain conditions on the boundary of this sphere, for example, to vanish. But although this is not the case, the problem is in fact equivalent

problem of boundary conditions, where the boundary is a sphere of infinite radius. Thus, the quantities (19) are essentially what are called the “characteristic numbers,” and the corresponding solutions are the “fundamental functions” of the problem belonging to equation (18). From the mathematical point of view1, boundary conditions in the proper sense of the word are unnecessary and inapplicable for an infinite boundary, for the reason that we approach a singular point of equation (18) if we go off in any direction in space to infinity. This is readily seen if the equation is separated into two equations, as described above, using polar coordinates. The ordinary differential equation thereby obtained, with independent variable \(r\), has two singular points, at \(r=0\) and at \(r=\infty\). It has (for negative values of \(E\)) only one solution that remains finite at \(r=0\), and only one solution finite at \(r=\infty\). Although these solutions do not coincide, they occur for definite values of \(E\), expressed by formula (19).

Instead of dwelling on the purely mathematical side of the subject, I shall try to give an account of the special properties of equation (18) in such a way that the essence of the matter will also be clear to those accustomed to dealing only with the general principles of wave theory. If \(E\) is negative, then the brackets in equation (18) will be negative outside a sphere of some radius. If we recall the way in which equation (18) was derived from equation (15), we shall see that, when the value of the brackets in (18) is negative, the square of the wave velocity is negative, and the wave velocity is imaginary. What does this mean? As is well known, the Laplace operator is directly related to the mean excess of neighboring values of a function over the value of the function at the point under consideration. Therefore the ordinary wave equation (15), with a positive quantity \(u^{2}\), indicates an accelerated increase (or a retarded decrease) of the function at those points where its value is less than the mean value of neighboring values; and, conversely, a retarded increase (or an accelerated decrease) of the function at those points where its value is greater than at neighboring points2. Thus the ordinary wave equation indicates a definite tendency toward the smoothing out of all differences between the values of the function at different points, though not at the very moment of their appearance and not to an arbitrary degree, as in the case of the equation of heat conduction. Nevertheless, to a certain extent it prevents the function from increasing or decreasing without bound.

To be continued.

in itself to a high degree satisfactory. But, in addition, it proves possible to calculate the amplitudes of the harmonic components of the electric moment for some direction in space, for example, in the case of the Stark phenomenon, parallel to the electric field or perpendicular to the field. If the theory is correct, then the squares of these amplitudes must be proportional to the intensities of the separate components of the lines polarized in some direction. Rather difficult calculations were carried out, the result of which is shown in Fig. 11.

In comparing the theory with experiment it is necessary to bear in mind that the calculations were carried out only for the limiting case of a very weak external field, and that at the field strength used

Fig. 1.

in the experiment (about 100,000 volts/cm), both experiment and theory indicate a noticeable influence of the intensity. In particular, very weak or vanishingly small components are noticeably strengthened as the field increases. The sum of the intensities of all perpendicular components of one and the same Balmer line proves to be exactly equal to the sum of the intensities of the parallel components of the same line. This is in full agreement with the proposition established by Stark that the field does not produce polarization of the emitted light as a whole.

I must point out that I arrived at the above-mentioned almost classical calculation of the intensities by noting a posteriori (i.e. after the main features of wave mechanics had been developed) its complete mathematical agreement with the matrix theory proposed by Hei-

by Heisenberg, Born, and Jordan1. The results indicated in Fig. 1 may also be called results of the latter theory, although they have not yet been calculated by means of its direct application. The connection between the two theories is rather complicated and cannot in any way be seen at first glance.

§ 7. At the beginning of the article we pointed out that, in the theory being expounded, both the laws of motion and the quantum conditions can be derived from a single principle of Hamilton. To prove this, it is necessary to make sure that the wave equation (16) can be derived from a variational principle, since this equation is, in fact, the only fundamental equation of the theory (for the case of one material point moving in a conservative force field—the only one considered in detail on the preceding pages).

The connection between equation (16) and Hamilton’s principle is very simple—exactly the same as in ordinary problems on oscillations. Moreover, this connection makes it possible to generalize the theory for any conservative systems very simply and consistently.

Suppose that it is required to find the extreme values of the following integral, taken over all space:

\[ J_1=\iiint\left\{\frac{h^2}{8\pi^2 m}\left[\left(\frac{\partial \psi}{\partial x}\right)^2+ \left(\frac{\partial \psi}{\partial y}\right)^2+ \left(\frac{\partial \psi}{\partial z}\right)^2\right]+V\psi^2\right\}\,dx\,dy\,dz, \tag{20} \]

where “admissible” are all functions \(\psi\) that are single-valued, finite, have continuous derivatives, and give to the following “normalizing” integral a constant value, for example 1:

\[ J_2=\iiint \psi^2\,dx\,dy\,dz=1. \tag{21} \]

If the variation is carried out under the additional condition by the known methods, then we obtain equation (16) as the known necessary condition for the extreme value of the integral (20), where the constant \(-E\) serves as the Lagrange multiplier by which the variation of the second integral must be multiplied and added to the first in order to take into account the additional condition2. Thus,

normalized fundamental functions of equation (16) are the so-called extremals1 of the integral (20) under the normalizing condition (21), and the characteristic numbers, i.e. the admissible values of the constant \(E\), are nothing other than the extremal values of the integral (20). (This property of the Lagrange multiplier is well known. It is easy to verify it if one takes into account that the extremal value of \(J_1 - E J_3\) must be equal to zero, since any other value can be increased or decreased by simple multiplication by a constant factor.)

The integrand in (20) turns out to be connected by a very simple relation with the ordinary Hamiltonian function of our mechanical problem—in the sense of ordinary mechanics. The function mentioned is equal to (see § 1)

\[ \frac{p_x^{\,2}+p_y^{\,2}+p_z^{\,2}}{2m}+V(x,y,z). \tag{22} \]

We assume that this function is a homogeneous quadratic function of the momenta \(p_x\), etc., and of unity, and replace

\[ p_x,\ p_y,\ p_z,\ 1 \]

respectively by

\[ \frac{h}{2\pi}\frac{\partial\psi}{\partial x},\qquad \frac{h}{2\pi}\frac{\partial\psi}{\partial y},\qquad \frac{h}{2\pi}\frac{\partial\psi}{\partial z},\qquad \psi. \]

Then we obtain the integrand (20). Hence it is natural to generalize our variational problem, and at the same time equation (16), to any conservative mechanical system. The Hamiltonian function of such a system has the form:

\[ \frac{1}{2}\sum_{l=1}^{N}\sum_{k=1}^{N} a_{lk}p_l p_k+V, \tag{23} \]

where \(a_{lk}=a_{kl}\), and the quantities \(a_{lk}\) and \(V\) are certain functions of the \(N\) generalized coordinates \(q_1\ldots q_N\). We assume that (23) is a homogeneous quadra-

where

\[ F=\frac{h^2}{8\pi^2 m}\left[\left(\frac{\partial\psi}{\partial x}\right)^2+ \left(\frac{\partial\psi}{\partial y}\right)^2+ \left(\frac{\partial\psi}{\partial z}\right)^2\right]+V\psi^2-E\psi^2; \]

\[ \psi_t=\frac{\partial\psi}{\partial t},\qquad \psi_x=\frac{\partial\psi}{\partial x},\qquad \psi_y=\frac{\partial\psi}{\partial y},\qquad \psi_z=\frac{\partial\psi}{\partial z}. \]

The value of the Lagrange multiplier \(-E\) is determined from the boundary conditions. Substituting the value of \(F\) into the differential equation written by us, we obtain equation (16).

Translator’s note.

Translator’s note.

... function of \(p_1 \ldots p_N,\ 1\), and replace these quantities respectively by

\[ \frac{h}{2\pi}\,\frac{\partial\psi}{\partial q_1},\ldots,\frac{h}{2\pi}\,\frac{\partial\psi}{\partial q_n},\ \psi . \]

Denoting by \(\Delta_p\) the determinant

\[ \Delta_p=\left|\sum \pm a_{lk}\right|,^{1)} \]

we form the integral

\[ J_1=\int\cdots\int\left[ \frac{h^2}{8\pi^2}\sum_l\sum_k a_{lk} \left(\frac{\partial\psi}{\partial q_l}\right) \left(\frac{\partial\psi}{\partial q_k}\right) +V\psi^2 \right]\cdot \frac{1}{\sqrt{\Delta_p}}\,dq_1\cdots dq_N \tag{24} \]

taken over the whole coordinate space, and seek its extreme values under the additional condition:

\[ J_2=\int\cdots\int \psi^2\cdot \frac{1}{\sqrt{\Delta_p}}\,dq_1\cdots dq_N=1. \tag{25} \]

This leads to a generalization of equation (16), namely:

\[ \sqrt{\Delta_p}\sum_l \frac{\partial}{\partial q_l} \left( \frac{\sum_k a_{lk}\,\dfrac{\partial\psi}{\partial q_k}} {\sqrt{\Delta_p}} \right) +\frac{8\pi^2}{h^2}(E-V)\psi=0, \tag{26} \]

where \(-E\), as before, is the Lagrange multiplier for (25).

The double sum appearing in (26) is a kind of generalization of the Laplace operator in a non-Euclidean coordinate space of \(N\) dimensions. The necessity of the appearance of the factor

\[ \frac{1}{\sqrt{\Delta_p}} \]

in integrals of the type (24) or (25) is well known from Gibbs’s statistical mechanics;

\[ \frac{1}{\sqrt{\Delta_p}}\,dq_1\cdots dq_N \]

\(^{1)}\) In expanded form this determinant is equal to:

\[ \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1N}\\ a_{21} & a_{22} & \cdots & a_{2N}\\ a_{N1} & a_{N2} & \cdots & a_{NN} \end{vmatrix} \]

It is the discriminant of the quadratic form in (23) (without \(V\)), expressing the kinetic energy.

Translator’s note.

is simply the Euclidean element of volume, for example, \(r^2\sin\theta\,d\theta\,d\varphi\,dr\) in the case of a single material point with mass equal to unity, whose position is determined by the three polar coordinates \(r,\theta,\varphi\)¹).

(If the determinant is omitted, the integrals will not be invariant with respect to point transformations; they will depend on the choice of generalized coordinates.) Equation (26) was used to solve all the problems mentioned in § 6.

§ 8. We shall postpone discussion of the question of the real physical meaning of the wave function \(\psi\) (see § 6), in order to be able to consider it in the most general way for any arbitrary system. Equation (16), or, in the more general case, equation (26), expresses the dependence of the wave function \(\psi\) only on the coordinates, while the dependence on time is expressed, for each particular solution corresponding to a particular characteristic number \(E=E_l\), by the real part of the quantity

\[ e^{\left(\frac{2\pi E_l t}{h}+\theta_l\right)i};\quad i=\sqrt{-1}, \]

where \(\theta_l\) are phase constants. If \(u_l\) (where \(l=1,2,3\ldots\)) are the fundamental functions, then the most general solution of the wave problem is equal to (the real part of)

\[ \psi=\sum_{l=1}^{S} c_l u_l e^{\left(\frac{2\pi E_l t}{h}+\theta_l\right)i}. \tag{27} \]

(For simplicity we assume that all the characteristic numbers are simple and discrete.) The quantities \(c_l\) are real constants. Let us now form the square of the absolute value of the complex function \(\psi\). It is equal to

\[ \psi\bar{\psi} = 2\sum_{l,l'} c_l c_{l'} u_l u_{l'} \cos\left[ \frac{2\pi(E_l-E_{l'} )}{h}t+\theta_l-\theta_{l'} \right], \tag{28} \]

¹) In other words, the determinant \(\Delta p\) is equal to unity divided by the functional determinant (Jacobian) of the transformation of the Cartesian coordinates into generalized ones. This can be verified by substituting into (22) the expressions for the momenta \(p_x\), etc., in terms of the generalized coordinates, i.e.

\[ p_x=m\dot{x}=m\sum_i \frac{\partial x}{\partial q_i}\dot{q}_i \quad\text{etc.} \]

and calculating the coefficients \(a_{l,l'}\), taking into account the definition of the generalized momentum:

\[ p_l=\frac{\partial T}{\partial \dot{q}_l}, \]

where \(T\) is the kinetic energy; this is the first term in (23).

Translator’s note.

where the bar denotes the conjugate complex quantity. The quantity \(\psi \bar\psi\), like \(\psi\) itself, is in the general case a function of the generalized coordinates \(q_1,\ldots,q_N\) and of time, and not a function of ordinary space and time, as in ordinary wave problems. Hence a difficulty arises in investigating the physical significance of the wave function. In the case of the hydrogen atom (regarded as a one-body problem) this difficulty is absent. In this case one can calculate rather accurately the values of the intensities, for example, of the components in the Stark phenomenon (see § 6, Fig. 1), with the aid of the following hypothesis: the charge of the electron is not concentrated at one point, but is distributed over all space with a density proportional to the quantity \(\psi \bar\psi\).

It is necessary to bear in mind that even under this hypothesis the charge of the electron is in fact confined to a region of a few ångströms, since the wave function \(\psi\) practically vanishes at a large distance from the nucleus (see § 4). The fluctuation of the charge is determined by equation (28) as applied to the special case of the hydrogen atom. To find the radiation which, on the basis of ordinary electrodynamics, is produced by these charges, it is only necessary to compute the rectangular components of the total electric moment\(^1\)), multiplying (28), respectively, by \(x, y, z\) and integrating over the volume, for example\(^2\)):

\[ \iiint z\psi \bar\psi\, dx\,dy\,dz =2\sum_{l,l'} c_l c_{l'} \cos\left[\frac{2\pi(E_l-E_{l'})t}{h}+\theta_l-\theta_{l'}\right] \cdot \iiint z u_l\cdot u_{l'}\, dx\,dy\,dz . \tag{29} \]

Thus the total electric moment is regarded as the result of the superposition of dipoles which are associated with pairs of fundamental functions, oscillating harmonically with the frequencies

\[ \frac{E_l-E_{l'}}{h}, \]

well known from N. Bohr’s famous frequency conditions.

The intensity of the emitted radiation of a given frequency must apparently be proportional to the square of the quantity

\[ c_l c_{l'} \iiint z u_l u_{l'}\, dx\,dy\,dz . \]

\(^1\)) This operation is correct only for the reason, and only so long as, the charge is practically concentrated within a volume which is small in comparison with the optical wavelength corresponding to the frequencies

\[ \frac{E_l-E_{l'}}{h}. \]

\(^2\)) In the sum \(\sum_{l,l'}\) each pair of values \(l,l'\) must be taken only once, and the terms in which \(l=l'\) must be divided by 2.

In calculating the intensities of the components of the Stark effect (Fig. 1), the assumption is made that the quantities \(C_i\) are equal for each series of characteristic numbers obtained from one Balmer level (eq. 19–2) under the action of an electric field. Then the relative intensities of the fine-structure components must be proportional to the square of the triple integral. This assumption agrees well with experiment.

The triple integral may be regarded as equal to what in Heisenberg’s theory is called the “matrix element \(z(l,l')\).” In this lies an internal correspondence between the two theories. But an important advantage of the theory set forth here—however imperfect it may be in many respects—consists, in my opinion, in the fact that, starting from a definite localization of the charge in space and time, we are actually able to calculate, on the basis of ordinary electrodynamics, both the frequency and the intensity and the polarization of the emitted light. All the so-called selection principles follow automatically from the circumstance that the triple integral vanishes in particular cases.

How, then, are these ideas to be generalized for the case of more than one, for example, \(N\) electrons? Here Heisenberg’s formal theory proved very valuable. It showed, not so much with the aid of physical reasoning as thanks to its compact formal structure, that equation (29), representing the rectilinear component of the total electric moment, must be preserved, with only the following special features: 1) the integrals are not triple, but \(N\)-fold, taken over the whole coordinate space; 2) \(z\) is replaced by the sum \(\sum l_i z_i\), i.e. by the component \(z\) of the total electric moment, which is produced by a model of point charges in the configuration \((x_1, y_1, z_1; x_2, y_2, z_2; \ldots x_n, y_n, z_n)\), referring to the element of integration \(dx_1 \ldots dz_N\).

On this basis one has to make the following hypothesis concerning the physical meaning of \(\psi\)—a hypothesis which, of course, reduces to our previous hypothesis in the case of only one electron: the real continuous charge distribution is a kind of average over the continuous set of all possible configurations of the corresponding model of point charges, the average being taken in such a way that the quantity \(\psi \bar{\psi}\) is a kind of weight function in configuration space.

At the present time it is still impossible to put forward any quite definite experimental results in favor of the generalized hypothesis. But certain very general theoretical conclusions about the magnitude \(\psi \bar{\psi}\) lead me to the conviction that the theory is correct. For example, the value of the integral of \(\psi \bar{\psi}\), taken over all space, proves to be absolutely constant (as is to be expected, if \(\psi \bar{\psi}\) —

proper weight function) not only for a conservative system, but also for a nonconservative system. The investigation of the latter will be considered in general outline in the following paragraph.

§ 9. Equation (16), or, in a more general form, equation (26), which is fundamental for our whole discussion, was derived under the assumption that \(\psi\) depends on time only through the factor:

\[ e^{\pm \frac{2\pi iEt}{h}} \tag{30} \]

But from this it follows that

\[ \dot{\psi}=\pm \frac{2\pi iE}{h}\,\psi . \tag{31} \]

From this equation and equation (26) one can eliminate the quantity \(E\), and then there is obtained an equation which is valid in every case, whatever the dependence of the wave function \(\psi\) on time may be:

\[ \sqrt{\Delta_p}\sum_l \frac{\partial}{\partial q_l} \left( \frac{\sum_k a_{lk}\frac{\partial \psi}{\partial q_k}}{\sqrt{\Delta_p}} \right) -\frac{8\pi^2}{h^2}V\psi \mp \frac{4\pi i}{h}\frac{\partial \psi}{\partial t} =0 . \tag{32} \]

The double sign in the last term presents no important difficulty. Since physical meaning is assigned only to the product \(\psi\bar{\psi}\), we may take for \(\psi\) either of the two equations (32); then \(\bar{\psi}\) will satisfy the other equation, and their product will remain unchanged.

Equation (32) can be generalized for an arbitrary nonconservative system, if only one admits that the potential function \(V\) depends explicitly on time. Of greatest interest is the case in which a small term is added to the potential energy of a conservative system. The latter represents the nonconservative potential energy imparted to the system by a light wave incident upon it. We cannot here go into the details, but shall indicate only the main features of the solution. The action of the incident light consists in the fact that each free oscillation of the unperturbed system, of frequency

\[ \frac{E_l}{h} \]

is accompanied by two forced oscillations, with, in general, smaller amplitudes and with frequencies \(\frac{E_l}{h}\pm \nu\), where \(\nu\) is the frequency of the incident ray of light. Following the same principles as in the preceding paragraph, we obtain that each free oscillation, interacting with the accompanying it

by forced oscillations, gives rise to the forced emission of light with the difference frequency

\[ \frac{E_l}{h}-\left(\frac{E_l}{h}\pm \nu\right)=\mp \nu, \]

i.e. equal to the frequency of the incident ray. This forced emission must, of course, be identified with the secondary elementary waves that are necessary for the description of absorption, dispersion, and scattering. Indeed, calculation shows that their amplitudes increase very appreciably when the frequency \(\nu\) of the incident light approaches one of the emission frequencies \(\dfrac{E_l-E_{l'}}{h}\). The final formula almost coincides with the well-known dispersion formula of Helmholtz in the form given to it by Kramers1.

The case of resonance cannot yet be investigated quite satisfactorily, since the term expressing damping is absent from our fundamental equation, even in the case of a free conservative system. (The radiation which, according to the assumption of § 8, is emitted in the interaction of each pair of free oscillations must, of course, in some way change their amplitudes. But this does not follow from the assumptions made so far.) It is interesting to note, however, that even in the absence of a damping term we do not encounter the case of an infinitely large amplitude at resonance, which, as is known, is obtained in the classical treatment of the problem. Here one obtains only that the incident light wave, with arbitrarily small amplitude, amplifies the forced oscillation of the system up to a finite amplitude. Moreover, if at the beginning there exists only one free oscillation, for example the one corresponding to the energy \(E_l\), and if

\[ h\nu=E_{l'}-E_l, \]

then the forced oscillation, amplified up to a finite amplitude, is identical in structure and frequency with the free oscillation corresponding to \(E_{l'}\). At the same time the amplitude of the oscillation \(E_l\) appreciably decreases. The sum of the squares of all amplitudes remains constant under all circumstances. Such a course of the phenomenon sheds some light (though not complete) on the so-called transition from one stationary state to another, which until now has completely resisted calculation.

§ 10. In the foregoing communication the wave theory of mechanics was developed without reference to two very important subjects: 1) the changes in classical mechanics introduced by the theory of relativity,

2) the action of the magnetic field on the atom. But this may already be considered a detail, since L. de Broglie, whose fundamental investigations served as the basis for the theory expounded here, proceeded precisely from the relativistic theory of the motion of the electron and from the very beginning took into account both the magnetic and the electric field.

Of course, one may take the same starting point for the present theory as well and develop it sufficiently far, applying relativistic mechanics instead of classical mechanics and including the action of the magnetic field. In this way interesting results were obtained concerning the change in wavelength, intensity, and polarization of the fine-structure components and of the components of the Zeeman effect for the hydrogen atom.^1) I did not consider it necessary to develop this form of the theory here for two reasons. First, it has not yet been possible to generalize the relativistic theory to a system with several electrons. And it is precisely in this domain that one may expect the new theory to solve problems that were inaccessible to the old theory. Second, the relativistic theory of the hydrogen atom is apparently incomplete. The results prove to be in serious contradiction with experiment, since in Sommerfeld’s well-known formula for the displacement of the components of the natural fine structure the so-called azimuthal quantum number (and also the radial quantum number) comes out “half-integral,” i.e. equal to half an odd number, and not to an integer. Thus the fine structure turns out to be incorrect.

This difficulty is apparently closely connected with the Uhlenbeck and Goudsmit theory^2) of the rotating electron. But how the rotation of the electron can be taken into account in the present theory is not yet known.^3)

^1) V. Fock, ZS. f. Phys. 38, 242, 1926. Translator’s note.

^2) G. E. Uhlenbeck and S. Goudsmit, Physica, 1925; Nature, Feb. 20, 1926.

^3) An attempt to connect Schrödinger’s wave mechanics in relativistic form with the theory of the rotating electron was made by F. London (F. London, Naturwissenschaften, 15, 15, 1927). Translator’s note.

1. H. A. Kramers, Nature, May 10 (1924), Aug. 30 (1924); H. A. Kramers und W. Heisenberg, ZS. f. Phys. 31, 681, 1925. ↩↩↩↩↩↩↩↩↩↩↩↩

2. On the basis of the rules of the calculus of variations, finding the function \(\psi\) which brings the integral \(J_1\) to an extremum (maximum or minimum) under condition (21) reduces to the integration of a differential equation with partial derivatives:
   \[ \frac{\partial F}{\partial \psi} -\frac{\partial}{\partial t}\left(\frac{\partial F}{\partial \psi_t}\right) -\frac{\partial}{\partial x}\left(\frac{\partial F}{\partial \psi_x}\right) -\frac{\partial}{\partial y}\left(\frac{\partial F}{\partial \psi_y}\right) -\frac{\partial}{\partial z}\left(\frac{\partial F}{\partial \psi_z}\right)=0, \] ↩↩↩↩↩

3. E. Schrödinger, Ann. d. Physik 79, 361, 489, 734, 80, 437, 81, 109, 1926; Naturwissenschaften 14, 664, 1926. [These articles have now been issued as a separate book: E. Schrödinger, Abhandlungen zur Wellenmechanik — J. A. Barth, Leipzig, 1927. Ed.] ↩