Bohm's Formalism, Bound States and Component Plane Waves
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint |
| Terbitan: |
, 2019
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/2599560 |
Daftar Isi:
- Recently, there have been numerous articles in the literature dealing with Bohm’s formulation of quantum mechanics obtained by writing the wavefunction W(x) in polar complex notation, namely R(x,t)exp(iS(x,t)). In general, however, this approach has been applied to time dependent quantum mechanics. Reducing the result to the time independent case, yields the usual time-independent Schrodinger equation, but the current in Bohm’s approach disappears. In this note, we try to see if the Bohm formalism may be used to obtain information about the quantum bound state and its relationship to plane waves. Traditionally, this relationship is based on the wavefunction being written as a Fourier transform W(x)=Sum over p fp exp(ip) with the exp(ip) being identified as plane waves. By examining the Bohm formalism, it seems one can see the transformation of exp(ikx-Et), exp(-ikx-iEt) plane waves for which an S(x,t) exists, to the case where the plane waves combine to yield only R(x) (with respect to x), leaving only exp(iEt) for exp(iS). More importantly, one may try to see how quantum mechanics is based on a constant energy density EW(x)W(x) at each point x, while classical statistical mechanics is based on constant temperature at each point. For the ground state of an oscillator, the time-independent Schrodinger equation seems to become an equation of free energy equaling EW(x)W(x), with kinetic energy representing energy in the free energy expression. This helps to link quantum mechanics and classical statistical mechanics as two different statistical models which become similar in the ground state oscillator case.