Quantum Entropy Describing Bound Momentum and Wave Features?

Classical statistical mechanics (for example a Maxwell-Boltzmann gas) seems to be governed by interactions which affect the momentum of a particle, but do not involve its wavelength hbar/p actively in a calculation i.e. there are no interference effects. On the other hand, a quantum bound state involves both impulse hits which change the momentum (in a one dimensional problem), but also the associated wavelength which restricts the particle’s presence for certain x regions i.e. there is a resolution.

  In this note, we focus on the latter situation and attempt to formulate an entropy expression which incorporates both the momentum (impulse hits) and wave features in a manner different from: Sp = -Sum over p a*(p)a(p) ln( |a*(p)a(p)|) and Sx = -Integral dx W*(x)W(x) ln(|W*(x)W(x)|) where W(x)=wavefunction = Sum over p a(p)exp(ipx). We use the ground state wavefunction of the quantum oscillator as an example and first present an approximation based on resolution. For a quantum bound state, if the potential delivers an impulse which changes the particle into a p momentum state, then by translational invariance we consider it as having probabilities (resolution) to be at the peaks of cos(px). The overall probability for a given p, using p=hbar/wavelength, is then approximated as:

hbar/ (L2p)   *  |a(p)| / {Sum over |a(p)| }   (where  hbar/(L2p) is 1/ (the number of half wavelengths which fit into L).

This may be used in a Shannon’s entropy scheme to yield:

S =- Sum over p  hbar/(2Lp) |a(p)|B {  -ln(B) + ln(C)-ln(p) + ln(|a(p)| }       

Where B=1/{Sum over |a(p)| }  and C=hbar/2L with L being a cutoff length in the problem (to avoid using infinite).  We apply this approach for the ground state of the harmonic oscillator.

   Next we try to refine the approach by replacing hbar/(2Lp) with an x dependent probability:

|cos(px)|/ A(p)   where A(p) is a normalization over one crest = A/p

We also consider the ground state oscillator result with this modification and then suggest a refinement i.e replace constant1/p with |cos(px)|/p * constant2.