Schrödinger's original struggles with a complex wave function Available to Purchase

A quotation from the classic Methoden der mathematischen Physik, a most influential source for physicists at the time, including Schrödinger, illustrates this common practice: “We can obtain the solutions of the nonhomogeneous equation [$mq¨+xq̇+cq=k(t)$] in a very convenient form, if we first consider the special case of a periodic force with frequency ω expressed as $k(t)=keiωt$⁠, where the complex representation is used just to derive real results more easily, and where k is assumed to be real” (Ref. 4, p. 223, my translation).

All quotations from original papers written by Schrödinger are taken from Ref. 11.

In the paper, Schrödinger does not write $Re(ψ)$⁠, I chose to do so just to make the difference between Eqs. (3) and (4) explicit.

Note that one needs to square Eq. (5) to obtain Eq. (8).

The potential and limitations of the analogy between the Schrödinger equation and the one describing a vibrating plate are discussed in detail in Ref. 8.

It is important to stress that this relation is only valid for the conservative case, i.e., for time-independent potentials. Schrödinger was aware of this limitation and mentioned it explicitly in his letter to Lorentz.

The interested reader can find a technical presentation of the mathematical problem at hand, as well as the derivation of an equation for the non-conservative case, in Ref. 6.

This is not to be confused with the demand for ψ to be a function in real space. Schrödinger is very explicit about the fact that ψ is, in general, a function in configuration space.

Note that this is Schrödinger's complex conjugate equation.

It is possible to find different attempts to interpret the meaning of ψ in Schrödinger's original papers. In the one where he claims to prove the equivalence between his theory and the one developed by Heisenberg, Born, and Jordan (Ann. Phys.384(8), 734–756 (1926)), Schrödinger makes the assumption that “the space density of electricity is given by the real part of $ψ(∂ψ¯/∂t)$⁠.” In the end of his fourth communication on wave mechanics,¹³ he identifies the quantity $ψψ¯$ as the “weight function” of the distribution of charge so that $ρ=eψψ¯$ is the electric charge density. This is justified by the derivation of a continuity equation based on his time-dependent equation (see, e.g., Chap. 17.1 of M. Longair, Quantum Concepts in Physics— An Alternative Approach to the Understanding of Quantum Mechanics (Cambridge U. P., Cambridge, 2013).

In Schrödinger's publications, we find inconsistent assumptions about the nature of c_k, sometimes it is assumed to be real, at other times complex. In this particular point of the lecture where Eq. (17) is presented, it is not fully clear which one is the case. Nevertheless, the important thing is the conclusion that when one multiplies ψ by its conjugate, the resultant expression has a frequency which is the difference between two (eigen)frequencies, i.e., $νkk′=νk−νk′$⁠. It is well known that Schrödinger was troubled by the notion of quantum jumps and by the fact that the observed frequencies were unrelated to internal vibrations of the system. With his interpretation of $ψψ¯$⁠, he believes to have solved this impasse, which is clear from a later passage in the four lectures where he states: “Something exists in the atom which actually vibrates with the observed frequency, viz., a certain part of the electric density-distribution.”

In Ref. 7 (p. 183), Chen boldly claims that Schrödinger never fully accepted a complex ψ. The author refers to a correspondence that he had with Viktor Weisskopf who said that “Schrödinger persisted in his belief in the real wave function until his last days.” I was not able to find this correspondence or any further evidence to sustain this view.