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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study in one dimension the semi-classical limit of the exact eigenfunction of the Hamiltonian , for a potential being analytic, bounded below and . The main result of this paper is that, for any given with two turning points, the exact normalized eigenfunction converges to the classical probability density, and the momentum distribution converges to the classical momentum density in the sense of distribution, as and with 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.

**[1]**M.V. Berry,*Waves near Stokes lines. I*, Proc. R. Soc. Lond. A**427**(1990), 265-280. MR**91g:34027****[2]**P.R. Holland,*The quantum theory of motion. I*, Cambridge University Press, Cambridge, 1993. MR**97e:81001****[3]**R.E. Langer,*On the connection formulas and the solutions of the wave equation. I*, Physical Review**51**(1937), 669-676.**[4]**R.L. Liboff,*Introductory quantum mechanics. I*, Addison-Wesley Publishing Company, 1980.**[5]**V.P. Maslov and M.V. Fedoriuk,*Semi-classical approximations in quantum mechanics. I*, Reidel, Dordrecht, 1981. MR**84k:58226****[6]**M. Reed and B. Simon,*Methods of modern mathematical physics II: Fourier analysis, self-adjointness. I*, Academic Press, New York, 1975. MR**58:12429b****[7]**G. Szegoe,*Orthogonal Polynomials. 3-rd edition. l*, Providence, 1967.**[8]**M. Sirugue, M. Sirugue-Collin and A. Truman,*Semi-classical approximation and microcanonical ensemble. I*, Annales De L Institut Henri Poincare-Physique Theorique,**41**(1984), 429-444. MR**86g:81039****[9]**A. Truman and H.Z. Zhao,*WKB-Langer asymptotic expansions and convergence of eigenfunctions and their derivatives. I*, Preprint (1998).**[10]**E.T. Whittaker and G.N. Watson,*Modern analysis, 2-nd edition. I*, Cambridge University Press, Cambridge, 1915.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35Q40,
81Q20

Retrieve articles in all journals with MSC (2000): 35Q40, 81Q20

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