Schrodinger equation plausibility derivation

PROBLEM 3.1.1. In one dimension "derive," or give a plausibility argument, for the Schrodinger equation by combining the one-dimensional classical wave equation... 

This is the Schrodinger equation for a particle moving in one dimension. The development provided here is not a derivation of this central equation of quantum mechanics rather, it is a plausibility argument based on the idea that the motions of particles can be described by a wave function with the wavelength of the particle being given by the de Broglie relation. 

Having said that one cannot really derive the Schrodinger equation, we now want to make it plausible. There axe excellent biographies of Schrodinger and more complete histories of quantum mechanics than that provided here. These may provide insight into exactly how it really came about. The intent here is to show that it is in some sense reasonable, given what we know of classical physics. 

According to Ref. [65], an appropriate guiding criterion to define a physically plausible exterior orbital density consists in using the functional form for the wave function derived from exact solutions to the Schrodinger equation for the hydrogen atom confined by a soft spherical box [34], i.e. ... 

Even this procedure still contains an element of arbitrariness and reveals an unsolved problem of quantum chemistry. The representation of the exchange interaction by the local electron probability density (multiplied by a suitable numerical factor, of order 0.5) seems a plausible way to approximate a complex interaction that actually depends on the locations of pairs of electrons. However as yet, there has been no derivation from the many-electron Schrodinger equation of any systematic series of successive approximations to its solution, of which the local density approximation would be the first approximation. There is a belief and a hope that such a derivation will be achieved, but it remains a tantalizing challenge now. 

Schrodinger in 1926 first proposed an equation for de Broglie s matter waves. This equation cannot be derived from some other principle since it constitutes a fundamental law of nature. Its correctness can be judged only by its subsequent agreement with observed phenomena (a posteriori proof). Nonetheless, we will attempt a heuristic argument to make the result at least plausible. 

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