Statistical Mechanical Properties of the Quantum Oscillator?

In a previous note on the quantum harmonic oscillator, it was seen that for the ground state (-1/2m) d/dx d/dx Wo(x) + .5kx*x Wo(x) = .5 w Wo(x) with Wo(x)= C exp(-.5 sqrt(km) x*x). Thus, a Gaussian wavefunction cancels the potential term .5k x*x Wo(x). For an excited state of the form Hn(x)Wo(x) = wavefunction, the second derivative with respect to x yields a term proportional to Hn(x) d/dx d/dx Wo(x) which will again cancel .5k x*x Wo(x)Hn(x). Thus, the cancellation of V(x) is due to Wo(x). Hn(x) (a hermite polynomial) on the other hand couples with Wo in the kinetic energy calculation to create more energy. In order for the Schrodinger equation to hold, this coupling must be proportional to: constant Wo(x)Hn(x), with the constant being the extra energy i.e. that beyond the ground state. In this note, we wish to examine the quantum oscillator in terms of statistical mechanical ideas. For the ground state, Wo(x)W(x) V(x), in the Schrodinger equation, is equivalent to - b Wo(x)Wo(x) ln(Wo(x)Wo(x)). Thus, one has an equation containing Shannon’s entropy and exp(-.5sqrt(km) x*x) has the appearance of an extremum solution. The kinetic energy and potential energy densities may also be written in terms of statistical density functions such as entropy and free energy. When one considers higher energy levels, it seems the statistical mechanical forms of the ground state remain, although they are modified by an overall factor of Hn(x)Hn(x). Thus, even though the full Shannon’s entropy for the excited state is -Hn(x)Hn(x)Wo(x)Wo(x) ln[ Hn(x)Hn(x)Wo(x)Wo(x) ] only a piece of this is included in quantum mechanical expressions. The term proportional to ln[ Hn(x)Hn(x) ] is dropped. Again, Wo(x)=C exp(-.5sqrt(km) x*x ), even though the energy is higher, and this still has the appearance of an extremum solution, but only of an entropy involving Wo(x). We suggest that in quantum mechanics (at least in the oscillator case), there may be extremization of a portion of entropy rather than the full Shannon’s entropy. This seems to make it complicated to discern the statistical mechanical nature of quantum mechanics