Periodicity in Statistical Derivation of the Schrodinger Equation
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint |
| Terbitan: |
, 2018
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/1485120 |
Daftar Isi:
- U. Klein, in a paper entitled the Statistical Origins of Quantum mechanics, has derived the Schrodinger equation from the continuity equation for probability density (conservation of probability) and an averaged form of Newton’s second law. In this note, we examine how the periodicity of quantum mechanics exp(iEt-ikx) seems to follow as a possible solution of the probability continuity equation with no need for Newton’s second law. We examine this periodic type solution in the presence and absence of a potential and find that in the case of a potential, a solution involves a single exp(iEt), but a linear combination of different exp(-ikx) waves. We further suggest that this periodic type of solution may be applicable to areas other than quantum mechanics as it is simply a solution of the probability continuity equation which applies to general probabilistic problems. For example, it might be applied to the harmonic oscillator of statistical mechanics. We also examine what at first appears to be a problem as the quantum solution to the probability continuity equation appears to be based on energy conservation at each point x, while classical statistical mechanics only conserves energy overall, not at each point x. It is argued that in the case of the classical harmonic oscillator, the quantum type solution is related to a variational principle such that the energy term really represents overall energy in the system and the potential term is related to Shannon’s entropy as has been argued in another note.