Semi-classical limit of wave functions

We study in one dimension the semi-classical limit of the exact eigenfunctionΨE(N,h)h\Psi _{E(N,h)}^{h}of the HamiltonianH=−12h2Δ+V(x)H=-\frac {1}{2} h^{2} \Delta +V(x), for a potentialVVbeing analytic, bounded below andlim|x|→∞V(x)=+∞\lim _{|x|\to \infty }V(x)=+\infty. The main result of this paper is that, for any givenE>minx∈R1V(x)E>\min _{x\in R^{1}} V(x)with two turning points, the exactL2L^{2}normalized eigenfunction|ΨE(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, ash→0h\to 0andN→∞N\to \inftywith(N+12)h=1π∫V(x)>E2(E−V(x))dx(N+\frac {1}{2} )h =\frac {1}{\pi } \int _{V(x)>E} \sqrt {2(E-V(x))}dxfixed. 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.