Bohm Aharonov Effect Part II

 In this note, we consider the Hamiltonian 1/2m (p-e/c A)(p-e/c A) W = E W ((1)) in the situation where there is no magnetic field even though A(x) still exists. Thus, if there is no force, one might expect to have a free particle which satisfies -1/2m d/dx d/dx W = E W and yields a free particle spatial density. Thus, one might guess the solution to ((1)) should be a free particle wavefunction exp(ikx) with perhaps a phase added. Such a phase does not affect spatial density W*W. One may solve ((1)) using the assumption that d/dx A(x) = 0 (used in the literature (1)) to find: W= Wfree(x) exp(ie/hbar c Integral x1=0 to x A(x1)dx1). Thus, one treats this phase as if one had exp(i K x) where K=A(x). A second derivative d/dx brings down i A(x) again. In previous notes, we suggested that a potential is stochastic and may be written as:  V(x)= Sum over k V(k) exp(ikx). If one expands the terms in ((1)) and uses d/dx A =0, one obtains a momentum dependent potential. This potential should be physical, we argue, because it applies to cases where there is a magnetic field as well as to the case here where there is not. Thus, the potential should give hits of the form V(k) exp(ikx) as well as couple to d/dx W. In other words, we argue these hits are real whether or not average force is zero. The effect of such hits is to create a momentum distribution at each x: P(p/x) = a(p) exp(ipx) / W(x). There is forward and backward motion, but the average kinetic energy(x) + V(x)= E at each point where:   kinetic energy(x) = [Sum over p p*p/2m a(p) exp(ipx)]/W(x). Given that average kinetic energy should be p*p/2m for a free particle (if the potential yields no hits), one may consider a phase exp(i K(x) x) multiplied by Wfree(x) i.e. exp(ipx) . This seems to imply a shifted velocity i.e. a local collective velocity of K(x) where K(x) happens to equal A(x) in this case. Matters, however, seem to be more complicated as we consider a microscopic coupling of exp(ikx) terms from both Wfree(x) and exp(i Bohm Aharonov) phase just as there may be a coupling between W=WoW1 where Wo(x) is the ground state and W1(x) represents the effects of an interacting substance (e.g. a photon or phonon).