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"name": "Quantum Spatial Probability and Time Intervals in the High Energy Limit Part III",
"author": {
"name": "Francesco R. Ruggeri",
"givenName": "Francesco R.",
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"description": " Classical statistical mechanics is often derived using counting methods applied to all possible configurations a system may take subject to a total volume and a total energy constraint. Given that there are numerous configurations, there should also be an average of these configurations described by f(e) for the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) distributions. In previous notes we have argued that one may apply a dynamical constraint to the average configuration. In particular, we argued that there should be a time reversal balance for two body elastic collisions. This immediately yields the various distributions with only consideration given to the average state and no counting techniques used. The counting of configurations, however, emphasizes that statistical mechanical systems contain “fluctuations” and that a specific dynamic constraint (i.e. time reversal two body scattering) applies only to the average. Quantum bound states, we argue, are very similar. They represent a statistical system which has many fluctuations (i.e. a particle may be knocked out with an momentum p with probability a*(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx)). There is, however, also a dynamical constraint which applies to the average, namely KE(x)+V(x)=E which is a classical mechanical constraint. Here KE(x)= [-1/2m d/dx dW/dx /W] ((1)). If fluctuations disappear in the high energy scenario, one may assume there is an almost fixed momentum p at each x. ((1)) still applies and there is still a wavefunction which is often in the form of crests and troughs. If an almost unique p exists at each x, one expects W(x) to vary in height at peak values. This may be verified from actual W(x) solutions. Given that the constraint on the statistical average is KE(x)+V(x)=E, one would expect that as p becomes an almost unique value at each x, that the quantum expectation Integral dx WW KE(x) with KE(x) given by ((1)) should also approach a classical value in other words Integral dx WW —> integral dt. One does not need to impose the correspondence principle to arrive at this result, one only needs to consider the case in which fluctuations in a quantum system disappear allowing the average p at each x to consist of almost a single p value.",
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"version": "1",
"keywords": "quantum density, Jacobian, disappearance of fluctuations",
"datePublished": "2021-12-13",
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