Quantum Mechanics, Classical Phase Space and Free Particles
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint Journal |
| Terbitan: |
, 2021
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/5076028 |
Daftar Isi:
- There have been a number of derivations of the Schrodinger equation from a classical statistical scenario (1) using a joint classical probability function F(p,x,t). A main feature of these approaches is to use 0=dF/dt= dF/dt (partial) + dF/dx p/m + dF/dp (-dV/dx). Thus, these scenarios focus on force and also make use of classical density. Furthermore, derivatives of density are linked to dynamics i.e. to -dV/dx.Towards of the end of these derivations, classical density is set equal to W*(x,t)W(x,t) without apparent motivation. Nevertheless, the approaches do lead to the Schrodinger equation. In some previous notes, we argued that the physics of the problem is contained in W(x). In this note, we particularly point out that a notion of a free particle is directly linked to W(x) if one wishes to have spatial derivatives linked to dynamics which seems to be a main focus of these classical statistical approaches. Even if density is constant i.e. has zero spatial derivatives, one may have another function, linked to density, which contains the dynamics of a free particle. This function may be of a statistical nature and may allow for the construction of an ensemble of free particles (just as in classical physics) In this note, we consider these ideas and link them to a particular derivation of the Schrodinger equation given in (1) based on a Wigner-Weyl (or Wigner-Moyal) transformation.