Does the Quantum Wavefunction Follow from a Fourier Series Treatment of the Classical Potential?
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint Journal |
| Terbitan: |
, 2019
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/3547067 |
Daftar Isi:
- In classical physics, there is a potential V(r) for a conservative force and force is linked to changes in momentum through Newton’s second law. For the sake of argument, imagine one may write V(r) as a Fourier series i.e. Sum over k Vk exp(ikx) and attribute physical relevance to this series. In other words, V(r) is a kind of average of microscopic granular system in which each exp(ikx) can impart momentum on a quantum particle. In addition, the Fourier series seems to have the appearance of a vector with exp(ikx) as a basis. It may also be noted that exp(ikx)exp(ipx) = exp(i (k+p)x). Thus, the potential seems to consist of ‘particles’ represented by exp(ikx) which may merge with quantum particles represented by exp(ipx) to form a boosted quantum particle exp(i (p+k)x). At first, it may seem exp(i (p+k)x) represents increased kinetic energy, but actually this merge should create potential energy. This perhaps gives a microscopic picture of potential energy. Thus, a merge of a Fourier series of V(x) with an ensemble of quantum particles Sum over p fp exp(ipx) yields: V(x) Sum over p fp exp(ipx) ((1)) A point which arises, however, is that Sum over p fp exp(ipx) cannot represent physical density because it may be negative at certain x . Thus, ((1)) is a departure from classical physics in which the potential energy is V(x) for one particle or V(x) density(x). ((1)) represents a kind of potential energy interaction, but introduces a new ensemble object Sum over p fp exp(ipx) which does not seem to exist in classical physics. This ensemble also seems to be a vector. The objective of this note is to try to determine properties and conditions which arise from this merge. In particular, we wish to find an equation satisfied by W(x) and also see how the classical picture is linked to this alternative scenario.