Time-Dependent and Independent Quantum Entropy

Entropy in quantum mechanics is often based on von Neuman’s prescription (1) which yields zero for a pure eigenstate. Some alternative approaches also yield this result (2). In other words, it seems this type of entropy deals with transitions between different energy eigenstates and excludes the intrinsic entropy of a pure energy eigenstate. There is, however, literature (3) dealing with both spatial and momentum entropy of a single energy eigenstate. In particular, Shannon’s entropy equation yields - W*W ln[W*W] and -a(p)*a(p) ln[a*(p)a(p)] respectively where W, the wavefunction, is Sum over p a(p)exp(ipx). In a previous note (4), we argued that one may use Wn(x) a(p) exp(-ipx) as a probability in Shannon’s entropy equation to obtain an entropy which is the sum of Shannon’s spatial and momentum entropies for a single energy eigenstate. A third term proportional to ipx  integrates to zero. 

    In this note, we wish to extend this approach to a time-dependent wavefunction (for example: W(x,t)= Sum over n bn(En) exp(-iEn t) Wn(x).] We suggest W(x,t) bn(En)an(p) exp(iEn t) exp(-ipx) as a candidate probability for Shannon’s entropy and obtain:   

S= Sum over n  bn(En)bn(En) ln(bn(En)  + Sum over n  bn(En)bn(En) {Sum over p an(p)an(p) ln(an(p))}

•  Sum over x, t W*W ln(W(x,t)) + Sum over x,t,n,p   i (Ent-px) W bn(En)an(p)

For simplicity, we consider bn(En) and an(p) as being real.

For the case of a single n present (i.e. a pure eigenstate) one obtains the entropy of such a state and not zero as in the von Neuman or other approaches. We argue that it might be good to have a single entropy formulation which works for both a single energy eigenstate and multiple states and not the two seemingly independent approaches which seem to be generally used. Furthermore, the approach here includes the intrinsic entropy of pure states together with that of transitions between states in the general W(x,t) (wavefunction) case.