Semiclassical Approach to Quantum Oscillator Ground State Energy with Moyal Product

The quantum oscillator ground state energy may be obtained by solving the time-independent Schrodinger equation and selecting the lowest energy eigenvalue. In (1) it is suggested that ground state oscillator energy appears in an semiclassical calculation of oscillator energy using the Moyal product (**) i.e.   (p+iwx) ** (p-iwx) .5 = pp/2 + wwxx/2 + hbar w/2  for m=1. In this note, we try to show that the Moyal product:   (p+iwx)/sqrt(2){  d/dx (left)d/dp (right) - d/dp (left) d/dx (right) } (p-iw)/sqrt(2) follows from the fact that Wo(x) and ao(p) where Wo is the ground state wavefunction and ao the ground state momentum wavefunction have the same functional form and that this form is in fact a Gaussian so that Wo(x)ao(p) = exp(-constant( pp/2 + wwxx/2). By using both, one may create a combined time-independent Schrodinger equation with both p and x present at the same time giving the appearance of  a semiclassical approach. With respect to the ground state energy, d/dx d/dx acting on a Gaussian exp(-axx-bpp) first brings down 2ax followed by 2a from the second derivative (and similar results for d/dp d/dp). The other part of the second derivative is x dependent and cancels the x dependent potential. Thus one may think in terms of a single derivative d/dx acting on ax or a single derivative acting on bp. Combining in a symmetric form leads to the Moyal product  acting on p+iwx and p-iwx to yield the ground state oscillator energy. We argue that this is essentially the same information as the time-independent Schrodinger equation with the special Gaussian form of the ground state wavefunction. Thus derivatives essentially act on pp/2 + wwxx/2 in a product of the x-wavefunction and p-wavefunction and this may be converted into a scheme of single derivative acting on the variable p+iwx and p-iwx because the relative piece of the d/dx d/dx of the Schrodinger equation is:   d/dx d/dx exp(-axx) →d/dx 2ax exp(-axx) → 2a (linear in a) with the other term of the second derivative canceling with .5kxx exp(-axx).