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Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials
As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Inc...

Lie theory and separation of variables. 6. The equation iUt + Delta2U = 0

J. Math. Phys. 16, 499 (1975); doi:10.1063/1.522573

Issue Date: March 1975

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C. P. Boyer
Centro de Investigación en Matemáticas Applicadas y en Sistemas, Universidad Nacional de México, México,20 D.F., Mexico

E. G. Kalnins and W. Miller, Jr.
Centre de recherches mathématiques, Université de Montréal, Montréal 101, P.Q., Canada
This paper constitutes a detailed study of the nine−parameter symmetry group of the time−dependent free particle Schrödinger equation in two space dimensions. It is shown that this equation separates in exactly 26 coordinate systems and that each system corresponds to an orbit consisting of a commuting pair of first− and second−order symmetry operators. The study yields a unified treatment of the (attractive and repulsive) harmonic oscillator, linear potential and free particle Hamiltonians in a time−dependent formalism. Use of representation theory for the symmetry group permits simple derivations of addition and expansion theorems relating various solutions of the Schrödinger equation, many of which are new. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
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PACS

  • 03.65.Kh
    Classical and quantum physics; mechanics and fields Quantum mechanics Group-theoretical methods
  • YEAR: 1975

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (30)
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