Wavefunction and the Identity of a Momentum State in a Quantum Bound State Part II

 In Part I of this note, we argued that in classical statistical mechanics, the probability P(p), where p is momentum, is well defined because there is a balance of two-body elastic collisions and their time-reversed reactions. In particular, this may be applied to a point x where one imagines a tiny box with many particles. In quantum bound states, collisions are with a potential and one does not have a balance of reactions and their time reversed cases. Thus, the notion of p at x does not seem to be clearly defined. A particle may be a x with momentum p, but at the same time may be reacting with V(x) and transforming into another pj. Thus, there is an ambiguity which does not exist in classical statistical mechanics. There, a particle may be colliding and so a p appears to disappear, but a new p is created by a time reversed reaction.

P(p), we argue, must hold over time in an equilibrium. Due to this issue, it appears that the best one may do is to define P(p) for the whole x region in quantum mechanics. This P(p) is measurable by taking an ensemble of identical single particles, knocking them out and measuring their momentum. As a result, knowing “p”  exactly means one does not know “x”.

   As a result, a different view must be taken to dealing with probabilities in a quantum bound state. We use the idea of invariance in x and argue that a wave-like behaviour in probability may be used to describe results. This is unusual because it appears as if particles are appearing or disappearing at different x points. If one has periodic motion, a question which arises is: What is the spatial frequency? We argue that the answer follows from classical physics. There:  delta p = delta t  (-dV/dx). If p initial =0 and delta t is constant, one sees that a large p final corresponds to a large change in V(x) in a tiny dx or that there is very good resolution in x. Thus a large p (linear) implies high resolution in x. Thus, we use px as the argument of the periodic probability function.