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In 1926, Schrödinger published his wave equation, introducing a new way to describe quantum systems. Later that same year, Max Born proposed the probability interpretation, suggesting that the wave function’s squared amplitude, $|\psi|^2$ gives the probability of finding a particle in a specific location. Since my understanding of the wave function relies heavily on Born's rule, I'm unclear on how Schrödinger originally viewed it before Born’s interpretation came into play.

How did Schrödinger interpret the wave function when he first introduced his equation, given that the probabilistic view was only proposed later by Born?

Qmechanic
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David
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2 Answers2

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Erwin Schrödinger, starting from de Broglie's particle wave assumption, invented the Wave Mechanics form of Quantum Mechanics. He made the first calculation of the hydrogen atom showing that Bohr's artificial quantization conditions can be explained by the solution of an eigenvalue problem of a differential equation, of his Schrödinger wave equation. He was, of course, interested in a physical interpretation of the complex wave function. Thus he considered, only heuristically, an interpretation of $\psi \psi^*$ of the single electron wave function as the distributed charge density of an electron. Calculated intensities and polarizations of emitted light based on this picture agreed surprisingly well with experimental results. But this charge density interpretation ran into problems for multi-electron systems. Even though he also found a way for a charge interpretation in this case he eventually accepted Born's probability interpretation. See a detailed description in E. Schrödinger "Collected Papers on Wave Mechanics ", Blackie & Son Ltd., London and Glasgow 1928.

PS: It is interesting that this "wrong" charge density interpretation of the wave function is nowadays extensively used in chemistry. Even the 1998 Chemistry Nobel prize awarded to Walter Kohn for his famous "Density Functional Theory" Wikipedia uses this concept. So after all, the charge density interpretation is rather useful and not so far from the truth. See also PSE and ArXiv

freecharly
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Lets take a electron.Schrodinger believed that the charge and mass of the electron were spread in its own cloud, the solution of the wavefunction at one point(x,t) described how much mass and charge it had at that point.Of course for that to work out the integral of the "mass density" and the "charge density" of all points would be equal to the mass and the charge of the electron respectively.