Semi-classical limit of wave functions

Abstract

We study in one dimension the semi-classical limit of the exact eigenfunction Ψ E ( N , h ) h \Psi _{E(N,h)}^{h} of the Hamiltonian H = − 1 2 h 2 Δ + V ( x ) H=-\frac {1}{2} h^{2} \Delta +V(x) , for a potential V V being analytic, bounded below and lim | x | → ∞ V ( x ) = + ∞ \lim _{|x|\to \infty }V(x)=+\infty . The main result of this paper is that, for any given E > min x ∈ R 1 V ( x ) E>\min _{x\in R^{1}} V(x) with two turning points, the exact L 2 L^{2} normalized eigenfunction | Ψ E ( N , h ) h ( q ) | 2 |\Psi ^{h}_{E(N,h)}(q)|^{2} converges to the classical probability density, and the momentum distribution | Ψ ^ E ( N , h ) h ( p ) | 2 |\hat \Psi ^{h}_{E(N,h)}(p)|^{2} converges to the classical momentum density in the sense of distribution, as h → 0 h\to 0 and N → ∞ N\to \infty with ( N + 1 2 ) h = 1 π ∫ V ( x ) > E 2 ( E − V ( x ) ) d x (N+\frac {1}{2} )h =\frac {1}{\pi } \int _{V(x)>E} \sqrt {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-probability density and using the generating function of the Laguerre polynomials, we give a complete mathematical proof of the Correspondence Principle.