Quantum versus Classical Bound State Entropy

In (1), the spatial quantum Shannon’s entropy: - Integral dx Wn(x)Wn(x) ln[ Wn(x)Wn(x)] ((A)) for a bound state is considered, where Wn(x) is the nth energy eigenstate. In the limit of n very high, quantum entropy is approximated by: - Integral C/v(x) ln{ C/v(x)} dx ((B)) where C/v(x) is the normalized classical entropy with .5mv(x)v(x) + V(x) = En. In such a case, W(x)W(x) still has peaks and zero points, but if resolution in the classical world is low, the integral is approximated by the classical C/v(x) which is said to pass through the peaks of W(x)W(x).

     In this note, we consider a bound classical particle moving first to the right and then to the left. In such a case, it seems entropy is zero. The argument in (1) is that the time a particle spends in a fixed dx is dx/v(x) and that this should represent an unnormalized probability. We argue this may represent an unnormalized probability for an ensemble of classical systems with the same energy, but different starting points x,t. If one wishes to know the entropy of a single system, we argue it is 0 and not ((B)). 

     The goal of this note is to find a quantum expression for spatial entropy which tends to 0 in the high n limit. Our approach is to introduce a cut off-length dx which is very small with respect to the system length and is also small in the classical world. Next, W(x)W(x) which has humps and zero values is examined. If the base of any hump is within dx, entropy density is set to 0 within this dx, otherwise it is given by ((A)). As humps become narrower, the entropy becomes smaller as more and more dx units are replaced with 0. In this way, we try to study the change in quantum bound entropy as the system becomes more and more classical with higher En values. Thus one has S(dx, n) which may be studied for various dx and n values.