Effect of a Stochastic Potential in Bound Quantum Mechanics Part III
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint |
| Terbitan: |
, 2020
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/3986717 |
Daftar Isi:
- In a previous note (1), we examined the effects of postulating that: V(x)=Sum over k Vk exp(ikx). This is a Fourier expansion, but we argued it exists for physical reasons i.e. at different times, forces corresponding to different Vk exp(ikx) pieces i.e. -d/dx Vk exp(ikx) act on a quantum particle. If one considers k/-k, the force is proportional to sin(kx) if V(k)=V(-k) which may be positive or negative in different x regions even though -d/dx V(x) may be strictly positive. Thus, for an average flow in one direction (caused by -d/dx V(x)), there may be pockets of force which act in the forward or negative direction. (One has to keep in mind that some averaging over time is being done.) We also noted that P(p/x) = a(p) exp(ipx) / W(x) and so P(p/x)+P(-p/x) is proportional to cos(px) if a(-p)=a(p). This too may be positive or negative leading to p*p/2m [P(p/x)+P(-p/x) ] being positive or negative in different regions. Thus, if one calculates the average kinetic energy at x: Sum over p p*p/2m a(p) exp(ipx)/ W(x), some p/-p pairs contribute “negative” kinetic energy. It seems there should be a physical explanation. In (2), we argued that the bound state Schrodinger equation was equivalent to a hydrodynamical equation if one replaces spatial density with W(x), the wavefunction. Thus, it seems negative contributions to average kinetic energy at x and also momentum “pockets” with the spatial density W*(x)W(x) which are negative for certain x ranges, might be thought of as being due to non-average motion or acceleration in the opposite direction of the average motion dictated by -d/dxV(x). I.e. the positive/negative forces of the Vk exp(ikx)’s stir up reverse motion which may be seen as “particles’ and “kinetic energy” leaving the forward flow. This, we argue, may account for negative contributions to local spatial density and negative kinetic energies. Later, the particle reverses direction or acceleration and contributes to the forward flow what was lost in a different x region. Thus, there may be a buildup of spatial density and kinetic energy density in different regions. In this note, we try to examine this in more detail and also consider an interpretation of the density average W*(x) Operator W(x) in this model.