Quantization as an Eigenvalue Problem

Fourth Communication

Schrödinger, E. Ann. Phys. 1926, 81, 109–139. Original German title: “Quantisierung als Eigenwertproblem (Vierte Mitteilung)” Source

NoteTo the Reader

This is the fourth and final paper in Schrödinger’s landmark 1926 series that established wave mechanics. The year 1926 was Schrödinger’s Annus Mirabilis: from January to June, the 38-year-old physicist published four papers that transformed quantum theory, deriving the time-independent equation, solving the hydrogen atom, proving equivalence with Heisenberg’s matrix mechanics, and here presenting the time-dependent Schrödinger equation that governs how quantum systems evolve.

For General Chemistry students, this paper explains why atoms absorb and emit light at specific frequencies. Schrödinger shows how an atom’s response to incoming light depends on its quantum state, deriving the dispersion formula that describes how light is scattered. The paper also addresses the physical meaning of the wave function ψ: the quantity |ψ|² represents a “weight function” giving the probability of finding the system in a particular configuration.

Section 7 is particularly important: Schrödinger argues that the wave function describes a “superposition” of all possible configurations of the system, with |ψ|² giving the weight of each configuration. This probabilistic interpretation, later refined by Born, became the standard understanding of quantum mechanics.

By E. Schrödinger

(Fourth Communication⁽¹⁾)

Contents: §1. Elimination of the energy parameter from the wave equation. The actual wave equation. Non-conservative systems.¹ — §2. Extension of perturbation theory to perturbations that explicitly contain time. Dispersion theory.² — §3. Supplements to §2: Excited atoms, degenerate systems, continuous spectrum. — §4. Discussion of the resonance case. — §5. Generalization for an arbitrary perturbation. — §6. Relativistic-magnetic generalization of the basic equations. — §7. On the physical meaning of the field scalar.³

¹ A “conservative system” conserves total energy. Nothing changes over time except the positions of particles. A “non-conservative system” has energy flowing in or out, like when light hits an atom. Schrödinger needs to handle these cases to explain how atoms absorb and emit light.

² “Dispersion” is how light interacts with matter. It explains why glass bends different colors differently, why the sky is blue. This section covers how atoms respond to incoming light waves.

³ Section 7 is the payoff for Gen Chem students: Schrödinger explains what the wave function ψ actually means. The quantity |ψ|² gives the probability of finding the electron in different locations.

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§1. Elimination of the Energy Parameter from the Wave Equation. The Actual Wave Equation. Non-conservative Systems

The wave equation (18) or (18′) from p. 510 of the second communication

\[(1) \qquad \Delta\psi - \frac{2(E-V)}{E^2}\frac{\partial^2\psi}{\partial t^2} = 0\]

or⁴

⁴ This is the time-independent Schrödinger equation you see in Gen Chem, written in slightly different notation. Δ (the Laplacian) measures how curved the wave function is. E is total energy, V is potential energy, h is Planck’s constant. The equation says: “curvature of ψ depends on the difference between total and potential energy.”

\[(1') \qquad \Delta\psi + \frac{8\pi^2}{h^2}(E-V)\psi = 0,\]

which forms the foundation of the new approach to mechanics attempted in this series of papers, suffers from the defect that it does not express the law of change for the “mechanical field scalar” ψ uniformly and generally. Equation (1) contains the energy or frequency parameter E and is, as expressly emphasized there, valid with a definite E-value for processes that depend on time exclusively through a definite periodic factor⁵

⁵ This exponential with i (the imaginary unit) describes oscillation. The frequency is E/h, which is Planck’s relation. Higher energy means faster oscillation. The wave function “vibrates” in time at a frequency set by the energy.

\[(2) \qquad \psi \sim P \cdot R \cdot \left(e^{\pm\frac{2\pi i E t}{h}}\right).\]

Equation (1) is therefore in reality no more general than equation (1′), which takes the just-mentioned circumstance into account and no longer contains time at all.

When we have occasionally referred to equation (1) or (1′) as the “wave equation,” this was actually improper; it would be more correctly designated as the “oscillation” or “amplitude” equation.⁶ We managed with it, however, because the Sturm-Liouville eigenvalue problem attaches to it—just as in the mathematically completely analogous problem of free vibrations of strings and membranes—and not to the actual wave equation.

⁶ Worth distinguishing: the time-independent equation (1′) tells you the shape of allowed standing waves. The time-dependent equation (coming up) tells you how those waves evolve. It’s like knowing what guitar string harmonics look like vs. knowing how the string actually vibrates when plucked.

In doing so, we had always presupposed that the potential energy V is a pure coordinate function and does not explicitly depend on time. But there is an urgent need to extend the theory to non-conservative systems, because only in this way can one study the behavior of a system under the influence of prescribed external forces, e.g., a light wave or a passing foreign atom.⁷ But as soon as V explicitly contains time, it is obviously impossible to satisfy equation (1) or (1′) by a function ψ that depends on time only according to (2). One therefore no longer manages with the amplitude equation, but must turn to the actual wave equation.

⁷ Here’s the motivation. Atoms don’t exist in isolation. Light shines on them, other atoms pass by. To understand absorption, emission, and scattering, Schrödinger needs equations that can handle time-varying conditions.

For conservative systems, this can easily be stated. (2) is equivalent to⁸

⁸ The second time derivative of ψ is proportional to ψ itself with a negative sign, the hallmark of oscillatory motion. Compare to a mass on a spring: a = −(k/m)x. The “spring constant” here involves energy and Planck’s constant.

\[(3) \qquad \frac{\partial^2\psi}{\partial t^2} = -\frac{4\pi^2 E^2}{h^2}\psi.\]

From (1′) and (3) one can eliminate E through differentiations and obtains in an easily understood symbolic notation

\[(4) \qquad \left(\Delta - \frac{8\pi^2}{h^2}V\right)^2\psi + \frac{16\pi^2}{h^2}\frac{\partial^2\psi}{\partial t^2} = 0.\]

Every ψ that depends on time according to (2), but with arbitrary E, must satisfy this equation; consequently also every ψ that can be developed in a Fourier series in time⁹ (naturally with coordinate functions as coefficients). Equation (4) is therefore evidently the uniform and general wave equation for the field scalar ψ.

⁹ A Fourier series breaks any function into a sum of sine and cosine waves. Since each individual wave satisfies the equation, any combination does too. This is the principle of superposition, which underlies all of wave mechanics.

As one sees, it is no longer of the quite simple type of the vibrating membrane, but rather is of the fourth order in the coordinates and of a very similar type to that occurring in many problems of elasticity theory.⁽¹⁾ One need not, however, fear excessive complication of the theory from this, or even the necessity of revising the methods given so far that attach to equation (1′). If V does not contain time, one can, starting from (4), make the ansatz (2) and then split the operator in (4) as follows:

\[(4') \qquad \left(\Delta - \frac{8\pi^2}{h^2}V + \frac{8\pi^2}{h^2}E\right)\left(\Delta - \frac{8\pi^2}{h^2}V - \frac{8\pi^2}{h^2}E\right)\psi = 0.\]

This equation can be tentatively split into two equations connected by “either—or,” namely equation (1′) and another that differs from (1′) only in that the eigenvalue parameter is minus E instead of plus E, which according to (2) does not lead to new solutions. The splitting of (4) is not mandatory, because for operators the theorem does not hold that “a product can only vanish if at least one factor vanishes.” This lack of inevitability does, however, attach to the methods for solving partial differential equations at every step. The procedure finds its subsequent justification through the proof of the completeness of the eigenfunctions found as functions of the coordinates.¹⁰ This permits, in connection with the fact that not only the real part but also the imaginary part of (2) satisfies equation (4), arbitrary initial conditions for ψ and ∂ψ/∂t to be fulfilled.

¹⁰ “Completeness” means the set of all eigenfunctions can represent any possible state. It’s like how any vector in 3D space can be written as a combination of x, y, z unit vectors. The orbitals (1s, 2s, 2p, …) form a complete basis for describing any electron state.

We see therefore that the wave equation (4), which already carries the dispersion law within it, can really serve as the basis of the theory of conservative systems developed so far. Its generalization for the case of a temporally variable potential function does, however, require some caution, because terms with time derivatives of V can appear, about which equation (4), by the manner of its derivation, can naturally give us no information. In fact, one encounters complications when attempting to transfer equation (4) as it stands to non-conservative systems, which seem to arise from a term with ∂V/∂t. I have therefore taken a somewhat different path in the following, which is computationally extraordinarily much simpler and which I consider to be correct in principle.

One need not push the order of the wave equation up to four in order to eliminate the energy parameter from it. The time dependence of ψ required for the validity of (1′) can be expressed instead of by (3) also by

\[(3') \qquad \frac{\partial\psi}{\partial t} = \pm\frac{2\pi i}{h}E\psi\]

One then arrives at one of the two equations¹¹

¹¹ The time-dependent Schrödinger equation, arguably the most important equation in all of quantum mechanics. Written in modern notation: iħ ∂ψ/∂t = Ĥψ. It governs how any quantum system evolves over time. The time-independent equation you see in Gen Chem is a special case, valid only when nothing external disturbs the system.

\[(4'') \qquad \Delta\psi - \frac{8\pi^2}{h^2}V\psi \mp \frac{4\pi i}{h}\frac{\partial\psi}{\partial t} = 0.\]

We shall require that the complex wave function ψ satisfy one of these two equations.¹²

¹² The wave function is complex, having both real and imaginary parts. This wasn’t obvious and troubled many physicists. Real physical quantities come from |ψ|², which is always real.

Since the complex conjugate function \(\bar{\psi}\) then satisfies the other equation, one may regard the real part of ψ as the real wave function (if one needs it).—In the case of a conservative system, (4″) is essentially equivalent to (4), since, when V does not contain time, the real operator can be decomposed into the product of the two conjugate complex ones.

§2. Extension of Perturbation Theory to Perturbations That Explicitly Contain Time. Dispersion Theory

The main interest¹³ is not in systems for which the temporal fluctuations of the potential energy V are of the same order of magnitude as the spatial ones, but rather in systems that, being conservative in themselves, are perturbed by the addition of small prescribed functions of time (and of the coordinates) to the potential energy.¹⁴ We therefore make the ansatz:

¹³ This section is about why atoms absorb light at specific frequencies. Schrödinger treats incoming light as a small “perturbation” that shakes the electron slightly. The math shows that the atom responds strongly only when the light frequency matches a transition frequency.

¹⁴ “Perturbation theory” works like this: start with a problem you can solve exactly (the hydrogen atom), then add a small disturbance (light wave) and calculate how the solution changes. Most real-world quantum problems are solved this way.

\[(5) \qquad V = V_0(x) + r(x,t),\]

where x, as often before, stands as representative of the totality of configuration coordinates. We regard the unperturbed eigenvalue problem (r = 0) as solved. The perturbation problem can then be solved by quadratures.

We shall not, however, treat the general problem immediately, but rather single out from the large number of important applications that fall under the above formulation, because of its outstanding importance which certainly justifies separate treatment, the problem of dispersion theory. Here the perturbing forces arise from an alternating electric field that is homogeneous and synchronously oscillating in the region of the atom, so that if we are dealing with linearly polarized monochromatic light of frequency ν, we have to make the following ansatz for the perturbation potential¹⁵:

¹⁵ The light wave oscillates at frequency ν. The cosine function captures this back-and-forth oscillation of the electric field that pushes the electron one way, then the other.

\[(6) \qquad r(x,t) = A(x)\cos 2\pi\nu t\]

thus

\[(5') \qquad V = V_0(x) + A(x)\cos 2\pi\nu t.\]

Here A(x) is the negative product of the light amplitude into that coordinate function which according to ordinary mechanics represents the component of the electric moment of the atom in the direction of the electric light vector¹⁶ (say −F∑e_iz_i, if F is the light amplitude, e_i, z_i are the charges and z-coordinates of the mass points, and the light is polarized in the z-direction; we take the temporally variable part of the potential function from ordinary mechanics with just as much or just as little right as we earlier took, e.g., in the Kepler problem, the constant).

¹⁶ Translation: A(x) measures how strongly the light’s electric field affects the electron at position x. It depends on the electron’s charge and position relative to the nucleus.

With the ansatz (5′), equation (4″) reads:

\[(7) \qquad \Delta\psi - \frac{8\pi^2}{h^2}(V_0 + A\cos 2\pi\nu t)\psi \mp \frac{4\pi i}{h}\frac{\partial\psi}{\partial t} = 0.\]

For A = 0, these equations transform through the ansatz:

\[(8) \qquad \psi = u(x)e^{\pm\frac{2\pi i E t}{h}},\]

(which is now meant not as “pars realis” but in the proper sense) into the amplitude equation (1′) of the unperturbed problem, and one knows (cf. §1) that in this way the totality of solutions of the unperturbed problem is found. Let¹⁷

¹⁷ E_k are the allowed energy levels (eigenvalues); u_k(x) are the corresponding wave functions (eigenfunctions). For hydrogen, E₁ is the ground state (−13.6 eV), E₂ is the first excited state, etc. The u_k are the 1s, 2s, 2p orbitals you learn in Gen Chem.

\[E_k \quad \text{and} \quad u_k(x); \qquad k = 1, 2, 3, \ldots\]

be the eigenvalues and normalized eigenfunctions of the unperturbed problem, which we regard as known and, in order not to lose ourselves in side questions that will have to be considered separately, which we shall assume to be discrete and mutually distinct (non-degenerate system without continuous spectrum).¹⁸

¹⁸ “Normalized” means the total probability of finding the electron somewhere is exactly 1. “Non-degenerate” means each energy level has only one wave function, which is a simplification. (In reality, the 2p level has three degenerate orbitals: 2p_x, 2p_y, 2p_z.)

Solutions of the perturbed problem we shall then have to seek, exactly as in the case of a time-independent perturbation potential, in the neighborhood of every possible solution of the unperturbed problem, hence in the neighborhood of an arbitrary linear combination with constant coefficients of the [according to (8) to be furnished with the appropriate time factors \(e^{\pm\frac{2\pi i E_k t}{h}}\)] u_k(x). The solution of the perturbed problem lying in the neighborhood of a particular linear combination will have the physical meaning that it is what initially establishes itself when, upon the arrival of the light wave, precisely this particular linear combination of free oscillations was present (perhaps with slight modifications during the process of “building up”).

But since the equation of the perturbed problem is also homogeneous—this lack of analogy with the “forced oscillations” of acoustics should be emphatically noted!¹⁹—it obviously suffices to seek the perturbed solution in the neighborhood of each individual

¹⁹ When you push a swing at the right frequency (forced oscillation), the amplitude grows and grows. But quantum systems don’t work that way. The response depends on the atom’s current state, not just the driving frequency. This is why absorption spectra depend on which states are populated.

\[(9) \qquad u_k(x)e^{\pm\frac{2\pi i E_k t}{h}}\]

which one can then combine linearly ad libitum, exactly as with the unperturbed solutions.

We therefore now make the following ansatz for the solution of the first equation (7)²⁰:

²⁰ The strategy: ψ = (unperturbed solution) + (small correction w). The atom starts in state u_k with energy E_k. When light hits it, the wave function changes slightly. We want to find that change w.

\[(10) \qquad \psi = u_k(x)e^{\frac{2\pi i E_k t}{h}} + w(x,t).\]

[The lower sign, i.e., the second equation (7), we leave aside from now on; it would yield nothing new.] The additional term w(x, t) may be regarded as small; its product with the perturbation potential may be neglected.²¹ If one takes this into account when substituting (10) into (7) and takes into account that u_k(x) and E_k are eigenfunction and eigenvalue of the unperturbed problem, one obtains:

²¹ This “neglect small × small” step is the heart of perturbation theory. Since both w and the light amplitude are small, their product is very small and can be ignored. This makes the math tractable.

\[(11) \quad \left\{\begin{array}{l} \Delta w - \dfrac{8\pi^2}{h^2}V_0 w - \dfrac{4\pi i}{h}\dfrac{\partial w}{\partial t} = \dfrac{8\pi^2}{h^2}A\cos 2\pi\nu t \cdot u_k e^{\frac{2\pi i E_k t}{h}} \\[12pt] = \dfrac{4\pi^2}{h^2}A u_k \cdot \left(e^{\frac{2\pi i t}{h}(E_k + h\nu)} + e^{\frac{2\pi i t}{h}(E_k - h\nu)}\right). \end{array}\right.\]

This equation is easily and essentially only satisfied by the ansatz²²:

²² Two frequencies appear: E_k + hν and E_k − hν. The atom’s response beats at frequencies that are the sum and difference of its natural frequency and the light frequency. Like AM radio, where you hear the difference frequency (audio) when you tune to a station.

\[(12) \qquad w = w_+(x)e^{\frac{2\pi i t}{h}(E_k + h\nu)} + w_-(x)e^{\frac{2\pi i t}{h}(E_k - h\nu)},\]

if one subjects the two functions w_± to the two equations²³

²³ The left side looks like the Schrödinger equation with modified energy. The right side is the “driving force” from the light. This is the equation we need to solve to find how light perturbs the atom.

\[(13) \qquad \Delta w_\pm + \frac{8\pi^2}{h^2}(E_k \pm h\nu - V_0)w_\pm = \frac{4\pi^2}{h^2}A u_k.\]

This step is essentially unique. It appears at first that one can add to (12) an arbitrary aggregate of unperturbed eigenfunctions. But this aggregate would have to be assumed small of first order (since this assumption is made about w) and then offers no interest for the time being, since it at most produces perturbations of the second order.

In equations (13) we now finally have before us those inhomogeneous equations that we could properly expect to encounter—despite the lack of analogy with actual forced oscillations emphasized above. This lack of analogy is extraordinarily important and manifests itself in equations (13) in the following two circumstances. First, as “second term” (“exciting force”) there appears not the perturbation function A(x) alone, but rather its product with the already present free oscillation amplitude.²⁴ This is indispensable for doing justice to the physical facts, for the reaction of the atom to an incident light wave depends eminently on the state in which the atom happens to be, whereas the forced oscillations of a membrane, plate, etc., are well known to be completely independent of the possibly superimposed eigenfunctions, and would thus furnish a completely useless picture. Second, on the left side of (13), in place of the eigenvalue, i.e., as “exciting frequency,” there appears not the frequency ν of the perturbing force alone, but rather one time its sum, the other time its difference with that of the already present free oscillation.²⁵ This is likewise an indispensable requirement, for otherwise the eigenfrequencies themselves, which after all correspond to the term frequencies, would function as resonance positions, and not, as must be required, and as equation (13) actually yields, the differences of the eigenfrequencies, and indeed, as one recognizes with satisfaction: only the differences of an eigenfrequency that is actually excited against all the others, not the differences of pairs of eigenfrequencies of which neither is excited.

²⁴ The A·u_k product means the atom’s response depends on which state it’s in. A ground-state hydrogen atom responds differently than an excited one. This explains why absorption spectra differ from emission spectra.

²⁵ The resonance frequency isn’t the light frequency ν alone, but ν ± (atom’s natural frequency). Resonance occurs when this combined frequency matches another energy level. This is Bohr’s frequency condition ΔE = hν.

To survey this more precisely, we carry out the solution procedure to the end. By a well-known method⁽²⁾ we find as unique solutions of (13)²⁶:

²⁶ Watch the denominator. When hν equals E_n − E_k (a transition energy), the denominator goes to zero and the response blows up. Resonance. This is why atoms absorb light only at specific frequencies matching their energy level differences.

\[(14) \qquad w_\pm(x) = \frac{1}{2}\sum_{n=1}^{\infty}\frac{a'_{kn}u_n(x)}{E_k - E_n \pm h\nu}\]

with²⁷

²⁷ This integral measures “overlap” between states k and n weighted by the perturbation A. If u_k and u_n have shapes that don’t interact with the light field, a′_kn = 0 and that transition is forbidden. This is the mathematical origin of selection rules.

\[(15) \qquad a'_{kn} = \int A(x)u_k(x)u_n(x)\varrho(x)\,dx.\]

ρ(x) is the “density function,” i.e., that function of the position coordinates by which equation (1′) must be multiplied in order to make it self-adjoint. The u_n(x) are assumed normalized. Furthermore, it is assumed that hν does not exactly coincide with any of the eigenvalue differences E_k − E_n. This “resonance case” will be discussed later (cf. §4).²⁸

²⁸ Schrödinger is being careful here. When the light frequency exactly matches a transition, special treatment is needed. That’s absorption, and it’s handled in Section 4.

If we now form from (14), according to (12) and (10), the total perturbed oscillation, we obtain²⁹:

²⁹ The complete perturbed wave function. The first term is the original state u_k. The sum represents “mixing in” of other states u_n, each weighted by how well it couples to the light (a′_kn) and how close the light frequency is to resonance (the denominators).

\[(16) \quad \left\{\begin{array}{l} \psi = u_k(x)e^{\frac{2\pi i E_k t}{h}} \\[8pt] \quad + \dfrac{1}{2}\displaystyle\sum_{n=1}^{\infty}a'_{kn}u_n(x)\left(\dfrac{e^{\frac{2\pi i t}{h}(E_k + h\nu)}}{E_k - E_n + h\nu} + \dfrac{e^{\frac{2\pi i t}{h}(E_k - h\nu)}}{E_k - E_n - h\nu}\right). \end{array}\right.\]

In the perturbation case, therefore, together with every free oscillation u_k(x), all those oscillations u_n(x) oscillate along in small amplitude for which a′_kn ≠ 0.³⁰ These are exactly those which, if they existed together with u_k as free oscillations, would give rise to an emission that is polarized (wholly or partly) in the polarization direction of the incident wave. For a′_kn is, apart from a factor, nothing other than the amplitude component falling in this polarization direction of the electric moment of the atom oscillating with frequency (E_k − E_n)/h according to wave mechanics, which occurs when u_k and u_n coexist.⁽³⁾—The co-oscillation does not, however, take place with the eigenfrequency E_n/h proper to these oscillations, nor with the frequency ν of the light wave, but rather with the sum and with the difference of E_k/h (i.e., the frequency of the one existing free oscillation) and ν.

³⁰ The light “mixes” other orbitals into the wave function. If you shine light on a hydrogen atom in the 1s state, you temporarily mix in some 2p, 3p, etc. character. The amount mixed depends on how close the light is to resonance.

As the real solution, the real part or the imaginary part of (16) can be considered.—We shall, however, operate in the following with the complex solution itself.

To recognize the significance of our result for dispersion theory, one must investigate the emission that arises from the coexistence of the excited forced oscillations with the originally already present free oscillation. For this purpose we form, according to the procedure used so far⁽⁴⁾—a critique follows in §7—the product of the complex wave function (16) with the conjugate complex value, i.e., the norm of the complex wave function ψ. In doing so, we note that the perturbation terms are small, so that their squares and mutual products are to be set aside. After easy reduction⁽⁵⁾ one obtains³¹:

³¹ |ψ|² = ψ\(\bar{\psi}\) gives probability density. The first term is the unperturbed electron distribution. The second term oscillates at the light frequency, representing the electron sloshing back and forth in response to the light wave. Note the denominator again: strongest response when hν matches a transition energy.

\[(17) \quad \psi\bar{\psi} = u_k(x)^2 + 2\cos 2\pi\nu t\sum_{n=1}^{\infty}\frac{(E_k - E_n)a'_{kn}u_k(x)u_n(x)}{(E_k - E_n)^2 - h^2\nu^2}.\]

According to the heuristic hypothesis³² about the electrodynamic meaning of the field scalar ψ, which led us in the Stark effect of hydrogen to the correct selection and polarization rules and to a quite satisfactory representation of the intensity ratios, the above quantity represents—apart from a multiplicative constant—the density of electricity as a function of the spatial coordinates and of time, if x represents only three spatial coordinates, i.e., if we are dealing with the one-electron problem. In a natural generalization of this hypothesis—about which more details in §7—we now regard, in the general case, as the density of electricity that is “connected with” one of the classical-mechanical mass points, or “arises from it” or “corresponds to it wave-mechanically,” the following: the integral of ψ\(\bar{\psi}\), multiplied by a certain constant, the classical “charge” of the mass point in question, over all those system coordinates which classically-mechanically determine the position of the other mass points. The total charge density at a spatial point is then represented by the sum, extended over all mass points, of the named integrals.

³² “Heuristic hypothesis” = educated guess that works. Schrödinger is being honest that the interpretation of ψ is not yet fully understood. Section 7 will develop this further.

Then, to find any spatial component of the total wave-mechanical dipole moment as a function of time according to this hypothesis, one has to multiply expression (17) by that coordinate function which classically-mechanically gives the dipole component in question as a function of the configuration of the point system, thus, e.g., by³³

³³ The electric dipole moment is charge times position, summed over all charges. An oscillating dipole radiates electromagnetic waves. This is how atoms emit and scatter light.

\[(18) \qquad M_y = \sum e_i y_i,\]

if one is dealing with the dipole moment in the y-direction. Then one has to integrate over all configuration coordinates.

Let us carry this out. For abbreviation, let us set

\[(19) \qquad b_{kn} = \int M_y(x)u_k(x)u_n(x)\varrho\,dx.\]

Let us further clarify the definition of a′_kn according to (15) by recalling that, if the incident electric light vector is given by

\[(20) \qquad \mathfrak{E}_z = F\cos 2\pi\nu t\]

A(x) has the meaning

\[(21) \qquad \left\{\begin{array}{l} A(x) = -F \cdot M_z(x), \\[4pt] \text{where } M_z(x) = \sum e_i z_i. \end{array}\right.\]

If one then sets, analogous to (19),

\[(22) \qquad a_{kn} = \int M_z(x)u_k(x)u_n(x)\varrho\,dx,\]

then a′_kn = −Fa_kn and one finds by carrying out the planned integration³⁴:

³⁴ The dispersion formula. This equation predicts how atoms scatter light. The sum runs over all possible transitions, weighted by transition strengths (a_kn, b_kn). When light frequency approaches a transition frequency, the denominator shrinks and scattering increases.

\[(23) \quad \left\{\begin{array}{l} \displaystyle\int M_y\psi\bar{\psi}\varrho\,dx = a_{kk} \\[12pt] \quad + 2F\cos 2\pi\nu t\displaystyle\sum_{n=1}^{\infty}\dfrac{(E_n - E_k)a_{kn}b_{kn}}{(E_k - E_n)^2 - h^2\nu^2} \end{array}\right.\]

for the resulting electric moment, to which the secondary radiation is to be attributed, that the incident wave (20) gives rise to.³⁵

³⁵ “Secondary radiation” is scattered light. When light hits an atom, it induces an oscillating electric dipole, which re-radiates. This is the quantum explanation for how materials scatter and refract light.

For the emission, only the second, temporally variable part matters, of course, while the first represents the temporally constant dipole moment that is possibly connected with the originally existing free oscillation. This variable part looks quite reasonable and should correspond to all the requirements one is accustomed to place on a “dispersion formula.” Note above all the appearance also of those so-called “negative” terms which—in the usual manner of speaking—correspond to the transition possibility to a lower level (E_n < E_k) and to which Kramers⁽⁶⁾ first drew attention on the basis of correspondence-principle considerations. In general, our formula—despite the very different notation and way of thinking—is probably to be called formally identical with the Kramers secondary radiation formula.³⁶ The important connection of the secondary radiation coefficients with the spontaneous radiation coefficients a_kn, b_kn is put in evidence, and indeed the secondary radiation is also precisely described as regards its polarization state.⁽⁷⁾

³⁶ Kramers had derived this formula using Bohr’s correspondence principle, a somewhat ad hoc method. Schrödinger derives the same result rigorously from his wave equation. This agreement was strong evidence that wave mechanics was correct.

As for the absolute magnitude of the scattered radiation or of the induced dipole moment, I believe that it too is correctly reproduced by formula (23), although an error in the numerical factor in setting up the heuristic hypothesis introduced above is of course within the realm of possibility. The physical dimension is in any case correct, for since the square integrals of the eigenfunctions are normalized to unity, the a_kn, b_kn are, according to (18), (19), (21), (22), electric moments. The ratio of the induced dipole moment to the spontaneous one is, when ν is far from the emission frequency in question, of the order of magnitude equal to the ratio of the additional potential energy Fa_kn to the “energy level” E_k − E_n.

§3. Supplements to §2: Excited Atoms, Degenerate Systems, Continuous Spectrum

For the sake of clarity,³⁷ some special assumptions were made in the preceding paragraph and some questions set aside that are now to be considered in retrospect.

³⁷ Section 3 extends the theory to real-world complications: atoms in excited states, orbitals at the same energy (like 2p_x, 2p_y, 2p_z), and ionization (the “continuous spectrum” where the electron escapes entirely).

First: what happens when the light wave finds the atom in a state in which not, as previously assumed, only the one free oscillation u_k is excited, but several, let us say two, u_k and u_l? As already noted above, one then simply has to link additively in the perturbation case the two perturbation solutions (16) corresponding to the index k and to the index l, after having furnished them with constant (possibly complex) coefficients that correspond to the strength assumed for the free oscillations and the phase relationship of their excitation. One surveys without actually carrying out the calculation that then in the expression for ψ\(\bar{\psi}\) and likewise in expression (23) for the resulting electric moment, not merely the corresponding linear aggregate of the previously obtained terms appears, i.e., of expressions (17) and (23) written once with k, the other time with l; rather, in addition “combination terms” appear, namely first, of highest order, a term with³⁸

³⁸ When two states are simultaneously excited, the atom emits at frequency (E_k − E_l)/h, the transition frequency between them. This is spontaneous emission arising from the “beating” of two wave functions.

\[(24) \qquad u_k(x)u_l(x)e^{\frac{2\pi i}{h}(E_k - E_l)t}\]

which reproduces the spontaneous emission that is connected with the coexistence of the two free oscillations; second, perturbation terms of first order that are proportional to the perturbing field amplitude and correspond to the interaction of the forced oscillations belonging to u_k with the free oscillation u_l—and of the forced oscillations belonging to u_l with u_k. The frequency of these newly appearing terms in (17) and (23) is, as one probably also surveys without carrying out the calculation, not ν, but rather³⁹

³⁹ This predicts light scattered at new frequencies. Not the incoming frequency ν, and not the atom’s natural emission frequencies, but combinations of both. This is Raman scattering, predicted theoretically here in 1926. C.V. Raman experimentally discovered it in 1928 (Nobel Prize 1930). Raman spectroscopy is now a standard analytical tool in chemistry.

\[(25) \qquad |\nu \pm (E_k - E_l)/h|.\]

(New “resonance denominators” do not, however, appear in these terms.) One therefore has here a secondary radiation whose frequency coincides neither with the exciting light frequency nor with a spontaneous frequency of the system, but is a combination frequency of both.

The existence of this remarkable kind of secondary radiation was first postulated by Kramers and Heisenberg (loc. cit.) on the basis of correspondence-principle considerations, then by Born, Heisenberg, and Jordan on the basis of Heisenberg’s quantum mechanics.⁽⁸⁾ Experimentally demonstrated it has not been, as far as I know, in any case yet. The present theory also very clearly shows that the occurrence of this scattering radiation is tied to special conditions, which probably require experiments to be set up specifically for this purpose. First, two eigenfunctions u_k and u_l must be strongly excited, so that all experiments performed on atoms in the normal state are excluded—and that is the overwhelming majority. Second, there must exist at least one third eigenfunction state u_n (i.e., possible; it need not be excited) that leads to strong spontaneous emission in combination with both u_k and u_l. For the extraordinary scattering radiation to be found is proportional to the product of the relevant spontaneous emission coefficients (a_knb_ln and a_lnb_kn). The combination (u_k, u_l) would not in itself need to emit strongly; it would not matter even if—in the language of the older theory—this “transition were forbidden.” Nevertheless, one will practically have to add this requirement as well, namely the requirement that the line (u_k, u_l) actually be strongly emitted during the experiment, because that is really the only means to make sure that really both eigenfunctions are strongly excited, and indeed in the same atom individuals and in a sufficient number of them. If one now considers that in the strong and most investigated term series, i.e., in the ordinary s-, p-, d-, f-series, the conditions are mostly such that two terms that combine strongly with a third do not do so with each other, then a special selection of the experimental object and of the experimental conditions really appears necessary in order to be able to expect with certainty the scattering radiation in question, especially since it is of a different frequency than the incident light and therefore cannot give rise to dispersion or rotatory polarization, but can only be noticed as all-around scattered light.

The quantum-mechanical dispersion theory of Born, Heisenberg, and Jordan cited above does not, as far as I can see, permit considerations of the kind just carried out, despite its great formal similarity with the present one.⁴⁰ For it speaks only of one mode of reaction of the atom to incident radiation. It grasps the atom as a timeless whole and is so far unable to say how in its language the undoubted fact can be expressed that the atom can be in different states at different times and then demonstrably reacts to incident radiation in different ways.⁽⁹⁾

⁴⁰ A jab at Heisenberg’s competing “matrix mechanics.” Schrödinger’s wave mechanics naturally handles time evolution, while Heisenberg’s matrices initially seemed more abstract. Later, Schrödinger proved the two approaches were mathematically equivalent.

We now turn to another question. In §2, all eigenvalues were assumed to be discrete and mutually distinct. We first drop the second assumption and ask: what changes if multiple eigenvalues occur, i.e., if degeneracy is present? Perhaps one expects that then similar complications will arise as we encountered in the case of a time-constant perturbation (third communication, §2), i.e., that only through solving a “secular equation” must a system of eigenfunctions of the unperturbed atom adapted to the special perturbation first be determined and used for carrying out the perturbation calculation. This is true in the case of an arbitrary perturbation r(x, t), as we had assumed it in Eq. (5) at the beginning of §2, but precisely in the case of perturbation by a light wave, Eq. (6), it is not true, at least in the first approximation followed so far and as long as the assumption is maintained that the light frequency ν does not coincide with any of the relevant spontaneous emission frequencies. For then the parameter value in the double equation (13) set up for the amplitudes of the perturbation oscillations is not an eigenvalue, and the equation pair always has the unique solution pair (14), in which no vanishing denominators appear, even when E_k is a multiple eigenvalue. Here, the sum terms for which E_n = E_k are not to be suppressed, any more than the sum term n = k itself. It is noteworthy that through these terms—if one of them actually appears with non-vanishing a_kn—the frequency ν = 0 also appears among the resonance frequencies. To the “ordinary” scattering radiation, these terms, as one recognizes from (23), deliver no contribution because of E_k − E_n = 0.

The simplification that one need not pay special attention, at least in first approximation, to a possibly present degeneracy occurs, as we shall consider further below (cf. §5), whenever, as is the case for the light wave, the time average of the perturbation function vanishes or, what is the same, when its temporal Fourier development contains no constant, i.e., time-independent term.

While our first assumption about the eigenvalues—that they should be simple—has thus actually proved to be superfluous caution, a departure from the second—that they should be wholly discrete—does indeed bring no changes in principle, but quite considerable changes in the external appearance of the calculation, insofar, namely, as integrals over the continuous spectrum of equation (1′) must be added to the discrete sums in (14), (16), (17), (23).⁴¹

⁴¹ The “continuous spectrum” refers to ionization. Bound electrons have discrete energy levels (−13.6 eV, −3.4 eV, etc. for hydrogen), but a free electron can have any positive energy. This continuum of states affects how atoms scatter high-energy light.

The theory of such integral representations has been developed by H. Weyl,⁽¹⁰⁾ although only for ordinary differential equations, but the transfer to partial ones should probably be permissible. The state of affairs is briefly this.⁽¹¹⁾ When the homogeneous equation belonging to the inhomogeneous equations (13), i.e., the vibration equation (1′) of the unperturbed system, possesses in addition to a point spectrum also a continuous spectrum extending from E = a to E = b, then an arbitrary function f(x) naturally can no longer be developed according to the normalized discrete eigenfunctions u_n(x) alone:

\[(26) \qquad f(x) = \sum_{n=1}^{\infty}\varphi_n \cdot u_n(x) \quad \text{with} \quad \varphi_n = \int f(x)u_n(x)\varrho(x)\,dx;\]

rather, an integral development according to the eigensolutions u(x, E) belonging to the eigenvalues a ≤ E ≤ b must be added:

\[(27) \qquad f(x) = \sum_{n=1}^{\infty}\varphi_n \cdot u_n(x) + \int_a^b u(x,E)\varphi(E)\,dE,\]

where we intentionally choose the same letter for the “coefficient function” φ(E) as for the discrete coefficients φ_n, in order to emphasize the analogy. If one has now once and for all normalized the eigensolution u(x, E) by furnishing it with an appropriate function of E in such a way that

\[(28) \qquad \int dx\,\varrho(x)\int_{E'}^{E'+\Delta}u(x,E)u(x,E')\,dE' = 1 \quad \text{or} \quad = 0,\]

according as E belongs to the interval E′, E′ + Δ or not, then in the development (27), under the integral sign, one is to set:

\[(29) \qquad \varphi(E) = \lim_{\Delta\to 0}\frac{1}{\Delta}\int\varrho(\xi)f(\xi)\cdot\int_E^{E+\Delta}u(\xi,E')\,dE'\cdot d\xi,\]

where the first integral sign refers as always to the basic domain of the variable group x.⁽¹²⁾ Assuming the fulfillability of (28) and the existence of the development (27)—both of which, as stated, have been proved by Weyl for ordinary differential equations—the determination of the “coefficient function” according to (29) is almost as immediately evident as the well-known determination of Fourier coefficients.

The most important and difficult task in the concrete individual case is thereby the carrying out of the normalization of u(x, E), i.e., the finding of that function of E by which the at first non-normalized eigensolution of the continuous spectrum lying before us is to be multiplied in order thereafter to satisfy condition (28). Also for this practical task, Mr. Weyl’s papers cited above contain very valuable guidance and some worked-out examples. An example from atomic dynamics is carried out in a paper by Mr. Fues on the intensities of band spectra appearing simultaneously in these Annalen.

We now apply this to our problem, i.e., to the solution of the equation pair (13) for the amplitudes w_± of the perturbation oscillations, whereby, however, we continue to assume that the one excited free oscillation u_k belongs to the discrete point spectrum. We develop the right-hand side of (13) according to the scheme (27):

\[(30) \quad \frac{4\pi^2}{h^2}A(x)u_k(x) = \frac{4\pi^2}{h^2}\sum_{n=1}^{\infty}a'_{kn}u_n(x) + \frac{4\pi^2}{h^2}\int_a^b u(x,E)\alpha'_k(E)\,dE,\]

where a′_kn is given by (15) and α′_k(E) by (29) through

\[(15') \qquad \alpha'_k(E) = \lim_{\Delta\to 0}\frac{1}{\Delta}\int\varrho(\xi)A(\xi)u_k(\xi)\cdot\int_E^{E+\Delta}u(\xi,E')\,dE'\cdot d\xi.\]

If one thinks of the development (30) substituted into (13), then also develops the sought solution w_±(x) in quite analogous fashion according to the eigensolutions u_n(x) and u(x, E), and takes into account that for the last-named functions the left side of (13) takes on the value

\[\frac{8\pi^2}{h^2}(E_k \pm h\nu - E_n)u_n(x) \qquad \text{and} \qquad \frac{8\pi^2}{h^2}(E_k \pm h\nu - E)u(x,E)\]

respectively, then one obtains by “coefficient comparison” as generalization of (14)

\[(14') \quad w_\pm(x) = \frac{1}{2}\sum_{n=1}^{\infty}\frac{a'_{kn}u_n(x)}{E_k - E_n \pm h\nu} + \frac{1}{2}\int_a^b\frac{\alpha'_k(E)u(x,E)}{E_k - E \pm h\nu}\,dE.\]

The further execution is entirely analogous to that in §2. One finally obtains as additional term to (23)

\[(23') \qquad +2F\cos 2\pi\nu t\int_a^b\frac{(E-E_k)\alpha_k(E)\beta_k(E)}{(E_k - E)^2 - h^2\nu^2}\,dE\]

with

\[(22') \qquad \alpha_k(E) = \lim_{\Delta\to 0}\frac{1}{\Delta}\int\varrho(\xi)M_z(\xi)u_k(\xi)\cdot\int_E^{E+\Delta}u(\xi,E')\,dE'\cdot d\xi\]

\[(19') \qquad \beta_k(E) = \lim_{\Delta\to 0}\frac{1}{\Delta}\int\varrho(\xi)M_y(\xi)u_k(\xi)\cdot\int_E^{E+\Delta}u(\xi,E')\,dE'\cdot d\xi\]

(please note the full analogy with the formulas of §2 marked with the same numbers without primes).

The preceding sketch of the calculation can naturally be no more than a general framework; it is only meant to show that the much-discussed influence of the continuous spectrum on dispersion, which appears to exist experimentally,⁽¹³⁾ is demanded by the present theory exactly in the expected form, and it was meant to outline the path along which the problem will have to be attacked computationally.

§4. Discussion of the Resonance Case

We have so far always assumed⁴² that the frequency ν of the impinging light wave does not coincide with any of the emission frequencies coming into consideration.

⁴² This section explains absorption. When light frequency exactly matches a transition energy, something qualitatively different happens. The atom absorbs the photon and jumps to a higher energy state.

⁴³ The Bohr frequency condition: hν = ΔE. When the photon energy exactly matches the energy difference between states k and n, we have resonance. Absorption or stimulated emission.

We now assume that, say,⁴³

\[(31) \qquad h\nu = E_n - E_k > 0,\]

where we return for the rest, for the sake of easier speaking, to the restrictive assumptions of §2 (simple, discrete eigenvalues, a single free oscillation u_k excited). In the equation pair (13) the eigenvalue parameter then takes on the values

\[(32) \qquad E_k \pm E_n \mp E_k = \begin{cases} E_n \\ 2E_k - E_n. \end{cases}\]

I.e., for the upper sign an eigenvalue, namely E_n, is present.—Then two cases are possible. Either the right-hand side of this equation, multiplied by ρ(x), stands perpendicular to the corresponding eigenfunction u_n(x), i.e.,⁴⁴

⁴⁴ This integral equals zero when the transition is “forbidden” by selection rules. For example, s → s transitions are forbidden because the integral of an s orbital times an s orbital times x (the position operator) vanishes by symmetry.

\[(33) \qquad \int A(x)u_k(x)u_n(x)\varrho(x)\,dx = a'_{kn} = 0\]

or physically: u_k and u_n would, if they existed side by side as free oscillations, give rise either to no spontaneous emission at all or to one that is polarized perpendicular to the polarization direction of the incident light. In this case the critical equation (13) still possesses a solution, which is still given by (14), in which the catastrophe term vanishes. Physically this means—in the old manner of speaking—that a “forbidden transition” cannot be excited by resonance, or that a “transition,” even if it is not forbidden, cannot be excited by light that oscillates perpendicular to the polarization direction of that light which would be emitted in the “spontaneous transition.”⁴⁵

⁴⁵ Selection rules explained mathematically! A transition can only occur if light of the right polarization shines on the atom. The wave function overlap integral determines whether the transition is allowed.

Or second, (33) is not satisfied. Then the critical equation possesses no solution. The ansatz (10), which assumes an oscillation that differs only slightly—by quantities of the order of the light amplitude F—from the originally existing free oscillation, and is the most general under this assumption, therefore does not lead to the goal. Thus there exists no solution which differs only by quantities of the order of F from the originally existing free oscillation; the incident light therefore has an altering influence on the state of the system that bears no proportion to the magnitude of the light amplitude.⁴⁶ What influence? This too can be judged without new calculation by starting from the case where the resonance condition (31) is not exactly but only approximately satisfied. Then one sees from (16) that u_n(x), because of the small denominator, is excited to unusually strong forced oscillations and that—what is no less important—the frequency of these forced oscillations approaches the natural eigenfrequency E_n/h of the eigenfunction u_n. (All this is quite similar, but yet in a distinctive way different from otherwise known resonance phenomena; otherwise I would not discuss it so extensively.)

⁴⁶ When resonance conditions are met and the transition is allowed, something dramatic happens. The atom absorbs the photon and changes state. The perturbation approach breaks down because the change isn’t “small” anymore.

With gradual approach to the critical frequency, therefore, the previously not excited eigenfunction u_n, whose possibility is responsible for the crisis, is excited more and more strongly and at the same time more and more closely with the eigenfrequency proper to it. In contrast to ordinary resonance phenomena, however, and indeed even before the critical frequency is reached, there comes a moment when our solution no longer correctly captures the state of affairs, even under the assumption that our obviously “damping-free” wave ansatz is exactly correct. For we had indeed regarded the forced oscillation w as small compared to the existing free oscillation and [in equation (11)] neglected a quadratic term.

I believe the preceding considerations already allow one to see with sufficient clarity that the theory will in the resonance case really yield that result which it must yield in order to be in agreement with the Wood resonance phenomenon:⁴⁷ a building up of the eigenfunction u_n giving rise to the crisis to finite magnitude, comparable to that of the originally present u_k, from which then naturally “spontaneous emission” of the spectral line (u_k, u_n) follows.

⁴⁷ R.W. Wood discovered in 1904 that sodium vapor, when illuminated with its resonance wavelength (the D lines), re-emits that same light. This “resonance fluorescence” is what Schrödinger’s theory must explain: absorption followed by re-emission.

But I would prefer not yet to attempt at this point to carry through the calculation for the resonance case, because the result would be of only slight value as long as the feedback of the emitted radiation on the emitting system has not been taken into account. Such a feedback must exist, not only because there is no reason to make a fundamental distinction between the light wave incident from outside and the light wave emitted by the system itself, but also because otherwise, in a system left to itself, when several eigenfunctions are simultaneously excited, the spontaneous emission would continue without limit. The required feedback must bring about that in this case, hand in hand with the light emission, the higher eigenfunctions gradually die away and finally the ground oscillation alone remains, which corresponds to the normal state of the system. The feedback is obviously the exact analogue of the reaction force of radiation (\(\frac{2e^2}{3mc^3}\ddot{v}\)) in the classical electron. This analogy also allays the rising concern about the previous non-consideration of the feedback. The influence of the relevant (probably no longer linear) term in the wave equation will generally be small, exactly as in the electron the reaction force of radiation is generally very small compared to the inertial force and to the external field force. In the resonance case, however—exactly as in electron theory—the coupling with the eigenlight wave will become of the same order of magnitude as that with the incident wave and will have to be taken into account if one wants to correctly calculate the “equilibrium” between the various eigenfunctions that establishes itself for a given irradiation.

But let it be expressly noted: to avoid a resonance catastrophe, the feedback term would not be required!⁴⁸ Such a catastrophe can under no circumstances occur, because according to the theorem on the persistence of normalization proved below in §7, the configuration-space integral of ψ\(\bar{\psi}\) always remains normalized to the same value even under the action of arbitrary external forces—and indeed quite automatically, as a consequence of the wave equations (4″). The amplitudes of the ψ-oscillations therefore cannot grow without limit; they have “on average” always the same value. If one eigenfunction is built up, another must decrease for it.

⁴⁸ In classical physics, driving a system at resonance can make amplitudes blow up to infinity. Here, conservation of probability prevents that. If probability “flows into” one state, it must “flow out of” another. Total probability stays at 100%.

§5. Generalization for an Arbitrary Perturbation

If an arbitrary perturbation is present,⁴⁹ as initially assumed in Eq. (5) at the beginning of §2, one will develop the perturbation energy r(x, t) in a Fourier series or a Fourier integral in time.

⁴⁹ Section 5 extends the light-wave treatment to any time-dependent perturbation. Any disturbance can be broken into sine waves (Fourier analysis), and each component treated as we did the light wave. It handles electric fields, magnetic fields, collisions, anything.

The terms of this development then have the form (6) of the perturbation potential of a light wave. One sees without further ado that one then simply obtains in (11), on the right-hand side, two series (or possibly integrals) of imaginary e-powers instead of just two pieces. If none of the exciting frequencies coincides with a critical frequency, one obtains the solution exactly by the path indicated in §2, namely as Fourier series (or possibly Fourier integrals) in time. There is probably no point in writing down the formal developments here, and a more precise pursuit of individual problems lies outside the scope of the present communication. But one important circumstance, which was already touched upon in §3, must be mentioned.

Among the critical frequencies of equation (13) there generally also figures the frequency ν = E_k − E_k = 0. For here too an eigenvalue appears as eigenvalue parameter on the left-hand side, namely E_k. If therefore in the Fourier development of the perturbation function r(x, t) the frequency 0, i.e., a time-independent term, occurs, one does not reach the goal exactly by the earlier path. But one easily recognizes how it is to be modified, for the case of a temporally constant perturbation is familiar to us from before (cf. third communication). One then has to take into consideration a small displacement and possibly splitting of the eigenvalue or the eigenvalues of the excited free oscillations, i.e., one has to write in the exponent of the e-power of the first term on the right-hand side of Eq. (10), instead of E_k: E_k plus a small constant, the eigenvalue perturbation. This eigenvalue perturbation is determined, exactly as described in the third communication §§1 and 2, from the requirement that the right-hand side of the critical Fourier component of the present Eq. (13) should stand perpendicular to u_k (or possibly: to all eigenfunctions belonging to E_k).

The number of special problems that fall under the formulation of the present paragraph is extraordinarily large.⁵⁰ Through superposition of the perturbation by a constant electric or magnetic field and by a light wave, one arrives at magnetic and electric birefringence and at magnetic rotatory polarization. Also resonance radiation in a magnetic field belongs here, but for this purpose the resonance case discussed in §4 must first be subjected to an exact solution. Furthermore, one will be able to treat in the indicated way the effect of an α-particle or electron flying past an atom,⁽¹⁴⁾ if the encounter is not too close to be able to calculate the perturbation of each of the two systems from the unperturbed motion of the other. All these questions are a mere matter of calculation, once the eigenvalues and eigenfunctions of the unperturbed systems are known. It is therefore very much to be hoped that one will succeed in determining these functions at least approximately also for higher atoms, in analogy to the approximate determination of the Bohr electron orbits that belong to the various term types.

⁵⁰ Schrödinger lists phenomena his theory can now explain: Faraday rotation (how magnetic fields rotate light polarization), the Kerr effect (electric-field-induced birefringence), and even atomic collisions. Wave mechanics was proving to be a universal tool for atomic physics.

§6. Relativistic-Magnetic Generalization of the Basic Equations

In connection with the physical problems just mentioned,⁵¹ in which the magnetic field, hitherto entirely left aside in this series of communications, plays an important role, I would now like to communicate quite briefly the presumed relativistic-magnetic generalization of the basic equations (4″), even if I can do so for the time being only for the one-electron problem and only with the greatest reserve.

⁵¹ This section tackles magnetic fields and Einstein’s relativity. Schrödinger derives a more complete equation that explains the Zeeman effect (how magnetic fields split spectral lines) and hydrogen’s fine structure. But as he admits, it’s incomplete without electron spin.

⁵² Electron spin was just being proposed (Goudsmit and Uhlenbeck, 1925). Schrödinger’s equation doesn’t include it naturally. The fully correct treatment requires the Dirac equation (1928), which incorporates both relativity and spin, and predicts antimatter as a bonus. Dirac received the 1933 Nobel Prize.

The latter for two reasons. First, the generalization rests for the time being on purely formal analogy. Second, it leads, as already mentioned in the first communication,⁽¹⁵⁾ in the case of the Kepler problem formally to the Sommerfeld fine-structure formula, and indeed with “half-integral” azimuthal and radial quanta, which is today generally regarded as correct; but there is still missing the supplement, necessary for producing numerically correct splitting patterns of the hydrogen lines, which in Bohr’s picture is supplied by the Goudsmit-Uhlenbeck electron spin.⁵²

The Hamilton partial differential equation for the Lorentz electron can easily be put in the following form:

\[(34) \quad \left\{\begin{array}{l} \left(\dfrac{1}{c}\dfrac{\partial W}{\partial t} + \dfrac{e}{c}V\right)^2 - \left(\dfrac{\partial W}{\partial x} - \dfrac{e}{c}\mathfrak{A}_x\right)^2 - \left(\dfrac{\partial W}{\partial y} - \dfrac{e}{c}\mathfrak{A}_y\right)^2 \\[12pt] \qquad - \left(\dfrac{\partial W}{\partial z} - \dfrac{e}{c}\mathfrak{A}_z\right)^2 - m^2c^2 = 0. \end{array}\right.\]

Here e, m, c are charge, mass of the electron, and speed of light; V, \(\mathfrak{A}\) are the electromagnetic potentials of the external electromagnetic field at the electron’s location. W is the action function.

From the classical (relativistic) equation (34) I now seek to derive the wave equation for the electron through the following purely formal procedure, which, as one easily considers, would lead to equations (4″) if it were applied to the Hamilton equation of a mass point moving in an arbitrary force field of ordinary (non-relativistic) mechanics.—I replace in (34) after expanding the squares the quantities

\[(35) \quad \left\{\begin{array}{l} \dfrac{\partial W}{\partial t}, \quad \dfrac{\partial W}{\partial x}, \quad \dfrac{\partial W}{\partial y}, \quad \dfrac{\partial W}{\partial z}, \\[12pt] \text{respectively by the } \textit{operators} \\[8pt] \pm\dfrac{h}{2\pi i}\dfrac{\partial}{\partial t}, \quad \pm\dfrac{h}{2\pi i}\dfrac{\partial}{\partial x}, \quad \pm\dfrac{h}{2\pi i}\dfrac{\partial}{\partial y}, \quad \pm\dfrac{h}{2\pi i}\dfrac{\partial}{\partial z}. \end{array}\right.\]

The linear double operator so obtained, applied to a wave function ψ, I set equal to zero:

\[(36) \quad \left\{\begin{array}{l} \Delta\psi - \dfrac{1}{c^2}\dfrac{\partial^2\psi}{\partial t^2} \mp \dfrac{4\pi ie}{hc}\left(\dfrac{V}{c}\dfrac{\partial\psi}{\partial t} + \mathfrak{A}\,\text{grad}\,\psi\right) \\[12pt] \qquad + \dfrac{4\pi^2e^2}{h^2c^2}\left(V^2 - \mathfrak{A}^2 - \dfrac{m^2c^4}{e^2}\right)\psi = 0. \end{array}\right.\]

(The symbols Δ and grad here have the elementary three-dimensional Euclidean meaning.) The equation pair (36) would be the presumed relativistic-magnetic generalization of (4″) for the case of a single electron, and indeed it too would be to be understood in the sense that the complex wave function should satisfy either the one or the other equation.

For the hydrogen atom, one can derive from (36) the Sommerfeld fine-structure formula exactly by the method described in the first communication, and likewise (neglecting the term with \(\mathfrak{A}^2\)) the normal Zeeman effect can be derived, as well as the well-known selection and polarization rules together with intensity formulas; they follow from the integral relations between the spherical harmonics cited at the end of the third communication.

For the reasons named in the first paragraph of this section, I refrain for the time being from the detailed reproduction of these calculations and refer also in the following final paragraph to the “classical” and not to the still imperfect relativistic-magnetic version of the theory.

§7. On the Physical Meaning of the Field Scalar

In §2 the heuristic hypothesis⁵³ about the electrodynamic meaning of the field scalar ψ, previously used for the one-electron problem, was summarily generalized to an arbitrary system of charged mass points, and a more detailed discussion of this procedure was promised.

⁵³ The most important section for understanding quantum mechanics. What does ψ actually mean? Schrödinger grapples with this question and arrives at the probability interpretation that defines quantum mechanics today.

We had there calculated the density of electricity at an arbitrary spatial point as follows: one picks out one mass point, holds fixed the coordinate triple which according to ordinary mechanics describes its position, integrates ψ\(\bar{\psi}\) over all the other system coordinates, and multiplies the result by a certain constant, the “charge” of the mass point picked out; in the same way one proceeds with every mass point (coordinate triple), where the mass point picked out in each case is assigned the same position each time, namely the position of that spatial point at which one wishes to know the electricity density. The latter is equal to the algebraic sum of the partial results.

This prescription is now equivalent to the following conception, which better brings out the actual meaning of ψ. ψ\(\bar{\psi}\) is a kind of weight function in the configuration space of the system.⁵⁴ The wave-mechanical configuration of the system is a superposition of many, strictly speaking of all, kinematically possible point-mechanical configurations. In this, each point-mechanical configuration contributes to the true wave-mechanical configuration with a certain weight, which weight is given precisely by ψ\(\bar{\psi}\). If one likes paradoxes, one can say the system finds itself, as it were, in all kinematically conceivable positions simultaneously, but not in all “equally strongly.”⁵⁵ In macroscopic motions, the weight function practically contracts to a small region of practically indistinguishable positions, whose center of mass in configuration space traverses macroscopically perceptible distances. In microscopic motion problems, at any rate, the changing distribution over the region is also of interest, and for certain questions even of primary interest.⁵⁶

⁵⁴ The probability interpretation. |ψ|² tells you how likely you are to find the electron at each position. This is the “electron cloud” picture from Gen Chem: dense where the electron is likely to be, sparse where it’s unlikely. Schrödinger calls this a “weight function.” Max Born (Nobel Prize 1954) later formalized this as probability, giving |ψ|² its standard interpretation.

⁵⁵ This sentence captures quantum strangeness. Before measurement, the electron doesn’t have a definite position. It exists in a superposition of all possible positions, with |ψ|² giving the weight of each.

⁵⁶ For atoms, the “spreading out” of ψ matters. For baseballs, ψ is so concentrated that quantum uncertainty is undetectable. This is why quantum mechanics only becomes important at atomic scales.

This reinterpretation may at first seem shocking, after we have often spoken so concretely of the “ψ-oscillations” as of something quite real.⁵⁷ Something tangibly real does indeed underlie them, however, also according to the present conception, namely the highly real, electrodynamically effective fluctuations of the electric spatial density. The ψ-function is not to be and is not meant to do more and not less than to allow the totality of these fluctuations to be mathematically controlled and surveyed through a single partial differential equation. That the ψ-function itself cannot and may not in general be directly interpreted three-dimensionally spatially, as much as the one-electron problem invites this, because it is in general a function in configuration space, not in real space, has been repeatedly emphasized.^{(16) 58}

⁵⁷ Schrödinger initially hoped ψ was a real physical wave, like a water wave. The probability interpretation was harder to accept. It meant giving up on visualizing what electrons “really do.”

⁵⁸ For one electron, ψ(x,y,z) is a function of 3D space, easy to visualize as orbitals. For two electrons, ψ(x₁,y₁,z₁,x₂,y₂,z₂) is a function of 6D “configuration space.” You can’t draw it, but the math still works.

Of a weight function in the sense set forth above, one will wish that its integral over the entire configuration space constantly remain normalized to one and the same unchangeable value, preferably to unity.⁵⁹ In fact, one easily convinces oneself that this is necessary in order that, according to the above definitions, the total charge of the system remain constant. And indeed, this requirement is of course also to be made for non-conservative systems. For naturally the charge of a system must not change when, e.g., a light wave arrives, continues for a time, then ceases again.

⁵⁹ Normalization: ∫|ψ|²dτ = 1. The total probability of finding the electron somewhere must always equal 100%. Schrödinger is about to prove this is automatically preserved by his equation, which would be a consistency check.

(N.B.: This also holds for ionization processes. A detached particle is initially still to be reckoned with the system, until the detachment is also logically—through splitting of the configuration space—carried out.)

The question now arises whether this required persistence of the normalization is also really guaranteed by the change equations (4″) of p. 112, to which ψ is subject. If it were not the case, that would be quite catastrophic for our entire conception.⁶⁰ Fortunately, it is the case. We form

⁶⁰ Schrödinger is checking internal consistency. If his equation caused total probability to change over time, the whole theory would be nonsense. Electrons can’t appear from or vanish into nothing.

\[(37) \qquad \frac{d}{dt}\int\psi\bar{\psi}\varrho\,dx = \int\left(\psi\frac{\partial\bar{\psi}}{\partial t} + \bar{\psi}\frac{\partial\psi}{\partial t}\right)\varrho\,dx.\]

Now ψ satisfies one of the two equations (4″), \(\bar{\psi}\) therefore the other. Hence the above integral becomes, apart from a multiplicative constant:

\[(38) \qquad \int(\psi\Delta\bar{\psi} - \bar{\psi}\Delta\psi)\varrho\,dx = 2i\int(J\Delta R - R\Delta J)\varrho\,dx,\]

where for the moment

\[\psi = R + iJ\]

is set. The integral (38) vanishes identically according to Green’s theorem;⁶¹ the only condition which the functions R and J must satisfy for this—to vanish sufficiently strongly at infinity—means physically nothing other than that the system considered be practically restricted to a finite region.

⁶¹ Green’s theorem converts a volume integral to a surface integral. If ψ vanishes at infinity (the electron stays in some finite region), the surface integral is zero, so the total probability stays constant.

One can also turn the foregoing somewhat differently, by not integrating right away over the entire configuration space but merely transforming the time differential quotient of the weight function into a divergence by Green’s transformation. One thereby obtains insight into the flow conditions, first of the weight function and through it: of the electricity. The two equations

\[(4'') \quad \left\{\begin{array}{l} \dfrac{\partial\psi}{\partial t} = \dfrac{h}{4\pi i}\left(\Delta - \dfrac{8\pi^2}{h^2}V\right)\psi \\[12pt] \dfrac{\partial\bar{\psi}}{\partial t} = -\dfrac{h}{4\pi i}\left(\Delta - \dfrac{8\pi^2}{h^2}V\right)\bar{\psi} \end{array}\right.\]

one multiplies by ρ\(\bar{\psi}\) and ρψ respectively and adds them:

\[(39) \qquad \frac{\partial}{\partial t}(\varrho\psi\bar{\psi}) = \frac{h}{4\pi i}\varrho\cdot(\bar{\psi}\Delta\psi - \psi\Delta\bar{\psi}).\]

To carry out the transformation of the right-hand side in extenso, one must recall the explicit form of our multi-dimensional non-Euclidean Laplace operator⁽¹⁷⁾:

\[(40) \qquad \varrho\Delta = \sum_k\frac{\partial}{\partial q_k}\left[\varrho T_{p_k}\left(q_l, \frac{\partial\psi}{\partial q_l}\right)\right].\]

One then easily finds through a small transformation:

\[(41) \quad \left\{\begin{array}{l} \dfrac{\partial}{\partial t}(\varrho\psi\bar{\psi}) = \dfrac{h}{4\pi i}\displaystyle\sum_k\dfrac{\partial}{\partial q_k}\left[\varrho\bar{\psi}T_{p_k}\left(q_l, \dfrac{\partial\psi}{\partial q_l}\right)\right. \\[12pt] \qquad\qquad\qquad\qquad\qquad \left.- \varrho\psi T_{p_k}\left(q_l, \dfrac{\partial\bar{\psi}}{\partial q_l}\right)\right]. \end{array}\right.\]

The right-hand side appears as the divergence of a multi-dimensional real vector, which is obviously to be interpreted as the current density of the weight function in configuration space. Eq. (41) is the continuity equation of the weight function.⁶²

⁶² A continuity equation says “what flows in must equal what accumulates.” If probability increases somewhere, it must have flowed there from somewhere else. Probability isn’t created or destroyed, it just moves around. Same principle as conservation of mass in fluid mechanics.

From it one can obtain the continuity equation of electricity, and indeed one such holds individually for the charge density “arising from each individual mass point.” Let us consider, say, the α-th mass point; let its “charge” be e_α, its mass m_α; for simplicity, let its coordinate space be described by Cartesian coordinates x_α, y_α, z_α. We denote the product of the differentials of the other coordinates abbreviatingly by dx′. Over them we integrate Eq. (41), with fixed x_α, y_α, z_α. In this integration all terms on the right-hand side drop out except for three, and one obtains:

\[(42) \quad \left\{\begin{array}{l} \dfrac{\partial}{\partial t}\left[e_\alpha\displaystyle\int\psi\bar{\psi}\,dx'\right] = \dfrac{he_\alpha}{4\pi im_\alpha}\dfrac{\partial}{\partial x_\alpha}\left[\displaystyle\int\left(\bar{\psi}\dfrac{\partial\psi}{\partial x_\alpha} - \psi\dfrac{\partial\bar{\psi}}{\partial x_\alpha}\right)dx'\right] + \\[12pt] \quad + \dfrac{\partial}{\partial y_\alpha}\left[\displaystyle\int\left(\bar{\psi}\dfrac{\partial\psi}{\partial y_\alpha} - \psi\dfrac{\partial\bar{\psi}}{\partial y_\alpha}\right)dx'\right] + \cdot \} \\[12pt] = \dfrac{he_\alpha}{4\pi im_\alpha}\text{div}_\alpha\left[\displaystyle\int\left(\bar{\psi}\,\text{grad}_\alpha\psi - \psi\,\text{grad}_\alpha\bar{\psi}\right)dx'\right]. \end{array}\right.\]

In this equation div and grad have the ordinary three-dimensional Euclidean meaning, and x_α, y_α, z_α are to be conceived as Cartesian coordinates of real space. The equation is the continuity equation for the charge density “arising from the α-th mass point.” If one forms the others analogously and adds all, one obtains the overall continuity equation. It is of course to be emphasized that, as always in such cases, the interpretation of the integrals on the right-hand side as components of the current density is not absolutely mandatory, because a divergence-free vector could be added.

To give an example, one obtains for the conservative one-electron problem, when ψ is given by

\[(43) \qquad \psi = \sum_k c_k u_k e^{2\pi i\nu_k t + i\vartheta_k} \qquad (c_k, \vartheta_k \text{ real constants})\]

as current density J

\[(44) \quad \left\{\begin{array}{l} J = \dfrac{he_1}{2\pi m_1}\displaystyle\sum_{(k,l)}c_k c_l(u_l\,\text{grad}\,u_k - u_k\,\text{grad}\,u_l) \\[12pt] \qquad\qquad\qquad \cdot\sin[2\pi(\nu_k - \nu_l)t + \vartheta_k - \vartheta_l]. \end{array}\right.\]

One sees, and this holds generally for conservative systems, that when only a single eigenfunction is excited, the current components vanish and the distribution of electricity becomes temporally constant;⁶³ the latter one indeed also immediately sees, since ψ\(\bar{\psi}\) becomes temporally constant. This also still holds when several eigenfunctions are excited, but all belong to the same eigenvalue. On the other hand, the current density then need no longer vanish, but rather a stationary current distribution can and will in general be present. Since in the unperturbed normal state one or the other certainly holds, one can in a certain sense speak of a return to electrostatic and magnetostatic atomic models. With this, the radiation-free character of the normal state does indeed find a surprisingly simple solution.⁶⁴

⁶³ Why ground-state atoms don’t radiate. In classical physics, orbiting electrons should radiate and spiral into the nucleus. But Schrödinger shows that in a stationary state (single eigenfunction), nothing oscillates. The charge distribution is static. No oscillation means no radiation.

⁶⁴ This solved a 15-year-old puzzle. Since Rutherford’s 1911 nuclear model, classical electromagnetism predicted atoms should be unstable: orbiting electrons should radiate energy and spiral into the nucleus in about 10⁻¹¹ seconds. Bohr’s 1913 model forbade this by fiat. Schrödinger’s wave mechanics finally explains why: the ground state has no time-varying charge distribution, so nothing radiates.

I hope and believe that the foregoing approaches will prove useful for explaining the magnetic properties of atoms and molecules and further also for explaining the conduction of electricity in solid bodies.⁶⁵

⁶⁵ Wave mechanics did indeed go on to explain magnetism in materials and electrical conduction. The “band theory” of solids, developed in the following years, applies Schrödinger’s equation to electrons in crystals and explains conductors, insulators, and semiconductors.

A certain hardness lies without doubt at present still in the use of a complex wave function.⁶⁶ Were it fundamentally unavoidable and not a mere computational convenience, this would mean that fundamentally two wave functions exist, which only together give information about the state of the system. This somewhat unsympathetic conclusion admits, I believe, the much more sympathetic interpretation that the state of the system is given by a real function and its derivative with respect to time. That we can give no more precise information about this yet is connected with the fact that in the equation pair (4″) we have before us only the—for calculation indeed extraordinarily convenient—surrogate of a real wave equation of probably the fourth order, whose formulation, however, would not succeed for me in the non-conservative case.

⁶⁶ Schrödinger was uncomfortable with complex numbers being fundamental to physics. History proved he was wrong to worry: the wave function really is irreducibly complex. The phase (related to the imaginary part) is physically meaningful and produces interference. Without complex ψ, there would be no quantum interference, no atomic orbitals, no chemistry as we know it.

Zurich, Physical Institute of the University.

(Received June 21, 1926)

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Notes

1. Cf. Ann. d. Phys. 79, pp. 361, 489; 80, p. 437. 1926; further on the connection with Heisenberg’s theory: ibid. 79, p. 734.

2. E.g., for the vibrating plate: \(\Delta\Delta u + \frac{\partial^2 u}{\partial t^2} = 0\). Cf. Courant-Hilbert, Ch. V, §5, p. 256.

3. Cf. third communication §§1 and 2, text at equations (8) and (24).

4. Cf. the following and §7.

5. Cf. Ann. d. Phys. 79, p. 755. 1926; further the calculation of the Stark effect intensities in the third communication. At the first-named place, instead of ψ\(\bar{\psi}\), the real part of ψ² was proposed. That was an error, which was already corrected in the third communication.

6. We take the eigenfunctions u_n(x) as real for simplicity, as previously always, but note that under certain circumstances it is much more convenient, indeed absolutely necessary, to work with complex aggregates of the real eigenfunctions, e.g., for the eigenfunctions of the Kepler problem with \(e^{\pm m\varphi i}\) instead of \(\frac{\cos}{\sin}m\varphi\).

7. H. A. Kramers, Nature 10 May 1924; ibid. 30 August 1924; H. A. Kramers and W. Heisenberg, Ztschr. f. Phys. 31, p. 681. 1925. The correspondence-principle description of the polarization of scattered light given at the latter place (Eq. 27) is formally almost identical with ours.

8. It is hardly necessary to say that the two directions we have for simplicity called “z-direction” and “y-direction” need not be perpendicular to each other. One is the polarization direction of the incident wave, the other is that polarization component of the secondary wave in which one happens to be interested.

9. Born, Heisenberg, and Jordan, Ztschr. f. Phys. 35, p. 572. 1926.

10. On this difficulty of grasping the temporal course of an event, compare especially the closing words in Heisenberg’s most recent presentation of his theory, Math. Ann. 95, p. 683. 1926.

11. H. Weyl, Math. Ann. 68, p. 220. 1910; Gött. Nachr. 1910. Cf. also E. Hilb, Sitz.-Ber. d. Physik. Mediz. Soc. Erlangen 43, p. 68. 1911; Math. Ann. 71, p. 76. 1911.—I owe not only these literature references to Mr. H. Weyl, but also very valuable oral instruction in these not quite simple matters.

12. The presentation given here I owe to Mr. E. Fues.

13. As Mr. E. Fues informs me, in practice one may very often suppress the limiting process and write \(\int\varrho(\xi)f(\xi)u(\xi,E)\,d\xi\) for the inner integral; namely, always when \(\int\varrho(\xi)f(\xi)u(\xi,E)\,d\xi\) exists.

14. K. F. Herzfeld and K. L. Wolf, Ann. d. Phys. 76, pp. 71, 567. 1925; H. Kollmann and H. Mark, Die NW. 14, p. 648. 1926.

15. A very interesting and successful attempt to compare the effect of charged particles flying past with the effect of light waves through Fourier resolution of their field is found in E. Fermi, Ztschr. f. Phys. 29, p. 315. 1924.

16. Ann. d. Phys. 79, p. 372. 1926.

17. Ann. d. Phys. 79, pp. 526, 754. 1926.

18. Ann. d. Phys. 79, p. 748. 1926, equation (31). The quantity designated there by \(\Delta_p^{-\frac{1}{2}}\) is our “density function” ρ(x) (e.g., r² sinθ for a polar coordinate triple). T is the kinetic energy as a function of position coordinates and momenta; the index on T means the derivative with respect to a momentum coordinate.—In equations (31) and (32) loc. cit. the index k was unfortunately used twice by mistake, once as summation index, then also as representative index in the argument of the functions.