Semi-classical limit of wave functions
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- by A. Truman and H. Z. Zhao PDF
- Proc. Amer. Math. Soc. 128 (2000), 1003-1009 Request permission
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.References
- M. V. Berry, Waves near Stokes lines, Proc. Roy. Soc. London Ser. A 427 (1990), no. 1873, 265–280. MR 1039788
- Peter R. Holland, The quantum theory of motion, Cambridge University Press, Cambridge, 1995. An account of the de Broglie-Bohm causal interpretation of quantum mechanics. MR 1341368
- R.E. Langer, On the connection formulas and the solutions of the wave equation. I, Physical Review 51 (1937), 669–676.
- R.L. Liboff, Introductory quantum mechanics. I, Addison-Wesley Publishing Company, 1980.
- V. P. Maslov and M. V. Fedoriuk, Semiclassical approximation in quantum mechanics, Contemporary Mathematics, vol. 5, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. Translated from the Russian by J. Niederle and J. Tolar. MR 634377, DOI 10.1007/978-94-009-8410-3
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
- G. Szegoe, Orthogonal Polynomials. 3-rd edition. l, Providence, 1967.
- M. Sirugue, M. Sirugue-Collin, and A. Truman, Semiclassical approximation and microcanonical ensemble, Ann. Inst. H. Poincaré Phys. Théor. 41 (1984), no. 4, 429–444 (English, with French summary). MR 777915
- A. Truman and H.Z. Zhao, WKB-Langer asymptotic expansions and $L^{2}$ convergence of eigenfunctions and their derivatives. I, Preprint (1998).
- E.T. Whittaker and G.N. Watson, Modern analysis, 2-nd edition. I, Cambridge University Press, Cambridge, 1915.
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
- 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
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1691007