High Energy Spatial Entropy Utilizing the Correspondence Principle

In (1), Shannon’s spatial entropy is computed using - [W*W] ln[W*W], where W(x) is the wavefunction and a classical envelope approximation: W*(x)W(x)=C/v(x), where v(x) is classical velocity, is used (according to the idea of the correspondence principle). A momentum entropy: - Integral dp a(p)a(p) ln[a(p)a(p)]  (one dimension) is also calculated. Recently in (2), we argued that one should form Shannon’s entropy using the probability = a(p1)exp(ip1 x) a(p2)exp(ip2 x). This leads to the following if one sums the first term over p1,p2 and integrates the second over x:

Entropy= Integral dx  -2W x d/dx W  -   Integral dp  a(p)a(-p) ln[a(p)a(-p)]    ((1)) one dimension

  In (1), the spatial entropy is found to behave as ln(n) for high n whereas for ((2)), spatial entropy is always 1. Thus, there is a very large difference between the two. In (1), a momentum entropy -a(p)a(p)ln[a(p)a(p)] is added to spatial entropy and in ((1)) a very similar momentum piece appears. For the case of the oscillator, W(x) and a(p) are both Gaussians and so the overall result for entropy from (1) and from this note both behave as ln(n) when n is large, but the factor of ln(n) differs. Ours is half as large for the case of a quantum oscillator. For our case, all of the entropy comes from the momentum term in the large n limit.

 In this note, we consider these ideas in more detail.