Published January 10, 2023 | Version 1
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Two Kinds of Equilibria: Particle and Wave/Dynamic Based? Part III Combining the Two

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In Parts I and II we examined equilibrium situations based on an additive scalar (distributive variable) such as ei= pi dot pi/2m in a Maxwell-Boltzmann gas and a vector dynamic quantity such as the momentum vector in a reflection/refraction of light situation or a single particle quantum bound state. We argued that a loss of information in the form P(a1)P(a2)=P(a3) holds in both cases, but in the additive scalar situation this leads to P(a1) = C exp(-a1/T) while in the vector case to exp( -i a1 dot conjugate variable). For a one dimensional quantum bound state a1=p and the conjugate variable is x. In such a case, exp(ipx) is a periodic eigenstate of the translational generator multiplied by -i. Within the bound state, stochastic hits from V(x) the potential lead to an ensemble i..e W(x) = Sum over p a(p)exp(ipx) where a(p)’s are fixed by considering average energy at each x to be the same i.e.  { Sum over p   a(p) pp/2m exp(ipx) } / { Sum over p a(p)exp(ipx)} + V(x) = En. 

    In this note, we consider the a(p)’s as also being probabilities like exp(ipx) and ask whether they could exhibit the same loss of information as in the vector p if one associate p in a(p) with the additive scalare pp/2m.  We argue that such considerations lead to the ground state solution of a quantum oscillator.

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