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Semi-classical limit of wave functions


Authors: A. Truman and H. Z. Zhao
Journal: Proc. Amer. Math. Soc. 128 (2000), 1003-1009
MSC (2000): Primary 35Q40; Secondary 81Q20
DOI: https://doi.org/10.1090/S0002-9939-99-05469-6
Published electronically: November 23, 1999
MathSciNet review: 1691007
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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.


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Additional Information

A. Truman
Affiliation: Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom
Email: A.Truman@swan.ac.uk

H. Z. Zhao
Affiliation: Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom
Address at time of publication: Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom
Email: h.zhao@lboro.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-99-05469-6
Keywords: Schr\"{o}dinger operator, semi-classical limit, weak convergence, probability density
Received by editor(s): April 9, 1998
Published electronically: November 23, 1999
Additional Notes: The research is supported by the EPSRC grants GR/L37823 and GR/K70397.
Communicated by: James Glimm
Article copyright: © Copyright 2000 American Mathematical Society

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