Spatial Quantum Entropy

In the literature, Shannon’s entropy with spatial density as probability i.e. density(x) ln(density(x)) appears frequently. It seems, however, that this calculation of entropy assumes no knowledge of momentum, yet density=W(x)W(x) (for a bound state) where W(x) = Sum over p a(p) exp(ipx). In other words, the wavefunction W(x) contains information about the momentum distribution associated with P(p/x)= a(p) exp(ipx)/W(x).

  In classical statistical mechanics, entropy is calculated using exp[ - (p*p/2m + V(x))/T) which includes both momentum and spatial considerations. In a previous note (1), we calculated a possible expression for entropy using P(p/x)P(x). This includes Shannon’s spatial entropy and other terms even if one sums over p after forming Shannon’s entropy with P(p/x)P(x). In this note, we try to consider the interpretation of these two forms of entropy.