Different Initial Assumptions for Statistical and Quantum Mechanics?
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint eJournal |
| Terbitan: |
, 2019
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/3555811 |
Daftar Isi:
- Statistical mechanics seems to be based on the idea of distributing a fixed amount of energy among a number of particles. (Usually this number is very large.) Alternatively, one may think of a single particle changing its state (e.g. momentum) during many time intervals. In both cases, one takes an average of the energy, but this means one needs weights for each single particle state. These weights seem to be based on the idea that each ‘vector’ of states has the same probability of occurrence. Thus, if one has N particles and an energy eN, one vector involves one particle having eN and the rest 0. Many vectors, however, may be created with N particles having energy near e. Thus, the weight for a single particle to have energy e is much higher than for it to have energy E. The weight is ultimately taken to be the Boltzmann factor. In quantum mechanics, it seems a very different assumption is made. Unlike classical physics where one follows a particle in space and time, knowing its acceleration, position and velocity at each x, in quantum mechanics, collisions with the potential seem to be granular, and the main assumption seems to be that a quantum particle may be in any momentum plane wave state at a given time. Like statistical mechanics, there is a set of weights, but one no longer uses the scenario of distributing energy among different states to determine these weights. Instead, the weights are obtained by noting that a quantum average at a point x is taken using fp exp(ipx)/ (Sum over fp exp(ipx)). The weights must ensure that: -1/2m (Sum over p p2/2m fp exp(ipx)) / (Sum over fp exp(ipx)) = E-V(x) at each x point. Thus, the plane wave behavior of quantum particles becomes a dominant factor. For the case of V(x)=0, the plane wave nature of the possible single particle momentum states leads to an overall (Sum over fp exp(ipx)) which has the appearance to two plane waves. This is very different from the classical statistical mechanical case. In this note, we attempt to examine the difference of these two statistical theories, and understand why in one case, the ground state of an oscillator, the two become the same.