Quantum Entropy, XP Covariance and Dynamics
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint Journal |
| Terbitan: |
, 2020
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/4056944 |
Daftar Isi:
- In a previous note, we suggested quantum Shannon’s entropy should utilize the probability a(p1)exp(ip1x)a(p2)exp(ip2x) ((1)) which is an addend of spatial density WW where W is the wavefunction. This yields an entropy expression of: - Integral dx W x d/dx W -Sum over p a(p)a(-p). The first term is of the form of a covariance of x and average p, but it is also the average information or entropy of an exp(ipx) type term. In quantum bound states, density is based entirely on joint relative probabilities of conditional momentum probabilities i.e. ((1)). Thus, spatial density is dynamic, while in classical statistical mechanics it is represented by the number of particles in a box at x / volume of the box. The average momentum at a point x = Sum over p pP(p/x)P(x) = -iWW d ln(W)/dx is also equivalent to: -i .5 d/dx (WW) i.e. half the change in spatial density. The operator d/dx used to calculate average momentum is the inverse of the position operator x from the point of view of integration by parts. An entropy term of: -Integral dx x d[WW]/dx is equivalent to an integral of the density which is 1. Covariance of x and average p at x is a constant regardless of the wavefunction. This covariance, however, is also the portion of Shannon’s entropy of a wave exp(ipx). Thus, the average information of a quantum wave portion (or its entropy) is a constant, regardless of the wavefunction bound system.