High Energy Quantum Bound States and Classical Physics

The classical limit of quantum mechanics is often considered in terms of the time dependent Schrodinger equation. In this note we consider the bound case solution i.e. a time dependence of exp(-iEt) and as in previous notes argue that time has been removed from this statistical equilibrium picture as a the quantum particle tries to establish a kind of impulse potential field based on superpositions of exp(ipx) terms. Such a potential field is linked with a bound state and does not involve following the particle about in time as it undergoes stochastic impulse hits with exp(ikx) terms from V(x)=Sum over k Vk exp(ikx). On average a classical energy balance equation KE(x)+V(x) = En holds for every energy En, but the average is created by a spread of momenta i.e. W(x)=wavefunction= Sum over p a(p)exp(ipx). In this note we consider the case of a particle in an infinite potential well and note how the spatial resolution becomes very high at the same time that the momentum spread becomes very narrow in the case of high n (energy level). Thus as n becomes approaches infinite, it is not just the vrms(x) which is linked with x, but the statistical scenario of quantum mechanics approaches a situation in which x is being resolved to a very high level and p is almost constant. Furthermore quantum entropy becomes a constant (independent of n) as n→ infinite suggesting a theory which is nonstatistical in this limit.This would represent a classical picture except for the fact that time is still removed from the quantum bound state i.e. forward and backward exp(ipx), exp(-ipx) still interfere. Given that x and p are well resolved locally for high n it seems one may follow the high n quantum particle in space and time and manually remove the forward and backward portions to create a theory of a particle moving in one direction under V(x) i.e. classical physics. This bypasses the idea of a wavepacket whose x-spread increases in time and allows one to base classical physics on a bound state solution of a quantum particle in an infinite well