High Energy Limit of the Wigner Transform Without The WKB Approximation

 In (1) the high energy limit of the Wigner transform W1(p,x,n) is investigated for the quantum oscillator. The WKB semiclassical approximation is used and the high energy wavefunction approximated by:  1/ p(x) { exp(i p(x) x) + exp(-i p(x) x) } leading to the result of delta ( PP/2m+kXX/2 - E) where p(x)= sqrt(2m(E-.5kxx)), P,X are classical momentum and position and PP/2m +kXX/2 energy. 

    In this note, we first observe that the WKB approximation may be used for any potential V(x) to find a similar result. We then focus on obtaining the result without using the WKB approximation, but instead utilize:   H(p,x) ** W1(p,x,n) = En W1(p,x,n)  (from (2)) where H(p,x) is the classical Hamiltonian, ** the Moyal star product involving exp(ihbar/2 Operator) and W(p,x,n) the Wigner function (as used in (2)). The main idea is that in the classical limit, hbar—>0 so exp(ihbar/2 Operator) —>1. Thus H(p,x)W1(p,x,n→large) = En W1n(p,x,n -> large). In (2) it is shown that Integral dp dx W1(p,x,n) = 1, so Integral H(p,x) W1(p,x,n->large) = E. As a result,  W1(p,x,n->large) acts as a delta function enforcing pp/2m + V(x)=E.