Quantum Bound States and Constrained Random Walk?
Francesco R. Ruggeri
In quantum mechanics, one often speaks of wave-like behaviour. In a previous note, however, we argued that wavelike behaviour is based on “averaging” of complex, periodic probabilities and so represents an ensemble average taken at different times. If one does not want an average over different times, but rather follows a quantum particle in time, it seems it undergoes a constrained random walk. This random walk differs from usual statistical ones in that the stochastic hits from a potential must on average yield V(x) at each x. We argue this is the source of the complex, periodic terms exp(ipx) which are part of a Fourier series of V(x). The potential influences the momentum distribution of a particle causing it to also exhibit complex, periodic probabilities of the form exp(ipx). Thus, if one considers “or” probabilistic situations for p and -p, one may obtain real periodic functions such as sin(px) and cos(px), but we argue these are time averages. A particle is either in a state p or -p (or some other momentum state) at x at a particular time. Thus, the wave-like behaviour follows from averaging over time exp(ipx) terms, but at each instant it seems a particle follows a constrained random walk type motion possibly changing its direction of motion with each stochastic hit from a potential. Overall, however, there must be a periodicity in the bound system, but we have argued this (based on exp(iEt)) is related to changes of a time averaged ensemble. We argue that this periodicity is also found in the constrained random walk picture if one considers a series of xi points (xi) with associated momenta (pi) and uses P(xi)P(pi/xi) as the probability at each point. Then the average kinetic energy for a constrained random walk added to V(x)W(x)W(x) yields EW(x)W(x). We also argue that W(x)W(x) is related to the idea of constrained random walk. The idea of a constrained random walk in quantum mechanics has no doubt been suggested in the literature, but the main focus of quantum mechanics does not seem to be with such a random walk, but rather with averages (taken at different times) which related to a wave picture that exists only as an ensemble average.
•Aug 3, 2020•0 citations