Quantum Waves, Entropy Density and Conservation?

 Quantum free particles are represented by exp(ipx) which we argue is a dynamical probability which maps into a static probability exp(-ipx)exp(ipx) i.e. the usual spatial density. Shannon’s entropy = - Sum over i P(i) ln(P(i)) is often associated with the static probability P(x)=W*(x)W(x) (1), but in (2) we argue that one may introduce a different probability C1 Real(W(x)). In this note we consider both in different examples. In particular we argue that the two different probability definitions are ultimately linked with momentum conservation. The form W*(x)W(x) introduces local nonconstant spatial densities if there are mixed exp(ip1x) and exp(ip2x)’s within W(x). If there are not, this form W*(x)W(x) does not distinguish space i.e. exp(ipx)exp(ipx+ib) is still a constant. Nevertheless interference may still occur which may be linked with entropy.

  Thus we first examine the case of exp(ipx) + exp(ipx + ib) where b is a constant phase shift. W*(x)W(x)=constant which does not lead to a structure different from exp(-ipx)exp(ipx), yet W(x)=C2{exp(ipx) + exp(ipx+ib)} interfere. On the other hand |cos(px)| and | cos(px)(1+cos(b)) - sin(px)sin(b) | are not the same distributions (one is a parity eigenstate, the other is not) and so we argue there is a different local entropy density. Thus introducing extra information i.e. b changes local density and entropy density if one adjusts the definition of probability used.

   We next consider W(x)= exp(ip1x)+exp(ip2x). One may also use the probability C3 Real|W(x)|, but here we are interested in an association with conservation of momentum. For a given exp(ipx), exp(-ipx)exp(ipx)=1. We argue that this represents a steady state picture i.e. a kind of equilibrium in which all x points are treated the same. W*(x)W(x) breaks this equilibrium scenario by introducing a term 2 cos( (p1-p2)x ). Thus both local spatial density and local entropy (based on W*W) are different and do not represent the “equilibrium” scenario. We further argue that momentum conservation may be linked with this equilibrium type of density and entropy density at least in the case of the transition amplitude <p1 V(x) p2>. V(x)exp(ipx) yields a sum of exp(ipx)’s, but only exp(ip1x) survives the integral over dx. The others create spatial densities which are not constant and so represent local entropy densities which do not treat all x points the same i.e .is not an equilibrium one.

   Thus we try to use Shannon’s entropy with two different probability definitions to analyse different quantum interference scenarios and argue that momentum conservation (and possibly others) may be linked with local entropy density in some situations.