Semi-classical limit of wave functions

Abstract:

We study in one dimension the semi-classical limit of the exact eigenfunction $\Psi _{E(N,h)}^{h}$ of the Hamiltonian $H=-\frac {1}{2} h^{2} \Delta +V(x)$, for a potential $V$ being analytic, bounded below and $\lim _{|x|\to \infty }V(x)=+\infty$. The main result of this paper is that, for any given $E>\min _{x\in R^{1}} V(x)$ with two turning points, the exact $L^{2}$ normalized eigenfunction $|\Psi ^{h}_{E(N,h)}(q)|^{2}$ converges to the classical probability density, and the momentum distribution $|\hat \Psi ^{h}_{E(N,h)}(p)|^{2}$ converges to the classical momentum density in the sense of distribution, as $h\to 0$ and $N\to \infty$ with $(N+\frac {1}{2} )h =\frac {1}{\pi } \int _{V(x)<E} \sqrt {2(E-V(x))}dx$ 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.