Time in the Time-Independent Schrodinger Equation release_qrznpliuafgjpk2ujdkxk6zasy

Abstract

<strong>In classical mechanics, the spatial density d(x) of a particle moving under a conservation of energy law .5mv(x)v(x) + V(x) = E is proportional to 1/v(x). This is obtained from d(x)= dt/T, i.e. the probability to be at x equals the time spent at x divided by the cycle time T. Given that dx=vdt to first order, density is proportional to 1/v(x). Now, d(x) is supposed to match the time independent quantum mechanical density W(x)W(x), where W(x) is the wavefunction, for high energy levels. Thus, one would expect the time a p wave (plane wave) spends at a point should be part of time independent quantum mechanics, but it does not seem to be overtly present. Furthermore, there is the question of T for the quantum mechanical problem. It is known that even for a time-independent solution W(x) has a factor exp(iEt). Thus, it seems time is present, even if it disappears in the density W*(x)W(x). It is argued that W*(x)W(x) = P(x) where P is probability, but W(x) is related to conditional probabilities such as P(x/p) or P(p/x). Thus, these conditional probabilities should be related to time. In an earlier note, we tried to show the time-independent Schrodinger equation implies all p plane waves are cycling with the same frequency E. In this note, we try to find a quantum mechanical representation for P(x,p)=dt/T where dt(x,p) is the time spent at x by a plane wave p and also try to evaluate T.</strong>
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