Semi-classical limit of wave functions

We study in one dimension the semi-classical limit of the exact eigenfuncti on Psi(E(N,h))(h) of the Hamiltonian H = -1/2 h(2) Delta + V (x), for a pot ential V being analytic, bounded below and lim(\x\ --> infinity) V (x) = +i nfinity. The main result of this paper is that, for any given E > min(x is an element of R1) V (x) with two turning points, the exact L-2 normalized e igenfunction \Psi(E(N,h))(h) (q)\(2) converges to the classical probability density, and the momentum distribution \<(Psi)over cap>(E(N,h)h) (p)\(2) c onverges to the classical momentum density in the sense of distribution, as h --> 0 and N --> infinity with (N + 1/2)h = 1/pi integral(V(x)<E) root 2( E - V (x))dx fixed. In this paper we only consider the harmonic oscillator Hamiltonian. By studying the semi-classical limit of the Wigner's quasi-pro bability density and using the generating function of the Laguerre polynomi als, we give a complete mathematical proof of the Correspondence Principle.