Semiclassical Approach to Quantum Oscillator Ground State Energy with Moyal Product Part II

In (1), it is argued that if the classical energy of an oscillator E = pp/2 + wwxx/2  (m=1) is written as E = (p+iwx)(p-iwx) and this in turn written using the Moyal star product { d/dp left d/dx right - d/dx left d/dp right } in terms of a parameter ( i hbar /2), one obtains:  E = pp/2 + wwxx/2  + hbar w/2 i.e. a semiclassical expression for the quantum result with the ground state energy hbar w/2.

   In Part I, we showed that this result follows from combining a time-independent Schrodinger equation in x-space with one in p (momentum) - space. The two second order equations yield second order derivatives composed of two parts. One part returns the form pp/2m and k/2 xx and the other returns a constant which is the ground state energy if one considers the ground state p and x wavefunctions which are of the form exp(- (pp/2 + wwxx/2) /b) where b is the ground state energy.. The first part cancels with kinetic energy from the p-space equation and .5kxx from the x-space equation. Thus, this combination of two Schrodinger equations really involves three pieces: ((A)) an energy term in p,x, ((B)) a cancellation term of this energy and ((C)) a constant ground state energy term.

    In (2) it is stated that the Moyal star product ** has the property:

Integral dx dp f(p,x) g(p,x) = Integral dx dp f(p,x) ** g(p,x)  

This is then used to argue that:   H(p,x) ** f(p,x) = E f(p,x) ((1)).  For the ground state oscillator case, only the first and second order operator terms contribute. The first order operator term is zero, while the second order term behaves in exactly the same manner as the combined x-space and p-space time-independent Schrodinger equations used in Part I.  Thus the combined p and x space time-independent Schrodinger equations for the oscillator Hamiltonian pp/2 +wwxx/2 is identical to the Moyal approach.