Some Properties of Quantum Shannon's Entropy Density

In this note, we examine the behaviour of Shannon’s entropy density for some examples in quantum mechanics.

 We start with a quasiparticle probability  P(x and p) = W(x) fp exp(ipx) (where W(x) is the wavefunction given by Sum over p fp exp(ipx)) and obtain a Shannon’s entropy (after summing on p) :

 Total entropy density =  -P(x) ln(P(x)) - P(p) ln(P(p)) - W(x) x d/dx W(x)   ((1))

It may be noted that if f(p)=f(-p) ((1)) may also be written as:

Total entropy density = -P(x) ln(P(x)) - P(p) ln(P(p)) -  fp p d/dp fp

We consider the cases of a particle in a box with an infinite potential at the walls and the harmonic oscillator ground state. Finally we consider the influence of u(x)= [d/dx W(x)]/W(x) on the spatial change of Shannon’s spatial entropy as W(x) and u(x) are often periodic and this leads to a periodic entropy density. In general, entropy is obtained by integrating the entropy density and one may question whether one should examine the density at all. In (1), however, entropy density is considered and it seems that there are some interesting properties of Shannon’s entropy density in quantum systems. Furthermore, there has been some recent work in the literature which makes use of entropy density to create an extra potential term for the Schrodinger equation in cases of temperature.