Semi-classical Limit of Wave Functions

A bstract . We study in one dimension the semi-classical limit of the exact eigenfunction Ψh E(N,h) of the Hamiltonian H = − 1 2 h2∆ + V (x), for a potential V being analytic, bounded below and lim|x|→∞ V (x) = +∞. The main result of this paper is that for any given E > minx∈R1 V (x) with two turning points, the exact L2 normalized eigenfunction |Ψh E(N,h) (q)|2 converges to the classical probability density, and the momentum distribution |Ψ̂h E(N,h) (p)|2 converges to the classical momentum density in the sense of distribution, as h → 0 and N → ∞ with (N + 1 2 )h = 1 π ∫