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Solution of Schrödinger equation by Laplace transform

Published online by Cambridge University Press:  09 April 2009

M. J. Englefield
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Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 3 , August 1968 , pp. 557 - 567
Copyright
Copyright © Australian Mathematical Society 1968

References

[1]Cremonesi, A., ‘On the orthonormalization of a particular system of functions’, Atti Accad. Naz. Lincei XX (1956), 168–171.Google Scholar
[2]Kac, A. M., ‘On the evaluation of the quadratic criterion for effectiveness of control’, Akad. Nauk. SSSR Prikl. Mat. Meh. 16 (1952), 362–364; summarized in Math. Reviews 14 (1953), 372.Google Scholar
[3]Murnaghan, F. D., The Laplace transformation (Spartan Book, Washington D.C., 1962).Google Scholar
[4]Puri, N. N. and Weygandt, C. N., ‘Contribution to the transform calculus used in the synthesis of control systems’, Journ. Frank. Inst. 277 (1964), 337–348.CrossRefGoogle Scholar
[5]Richardson, C. H., The calculus of finite differences (Van Nostrand, London, 1954).Google Scholar
[6]Schiff, L. I., Quantum mechanics (McGraw-Hill, New York, 1955), 2nd edition, chapter IV.Google Scholar
[7]Smith, M. G., Laplace transform theory (Van Nostrand, London, 1966). Also [8], chapter VII, and [3], chapter 9.Google Scholar
[8]Van der Pol, B. and Bremmer, H., Operational calculus based on the two-sided Laplace integral (Cambridge, 1959), 2nd Edition, p. 201.Google Scholar
[9]Yamaguchi, Y., ‘Two nucleon problem when the potential is nonlocal but separable’. Phys. Rev. 95 (1954), 1628–1634.CrossRefGoogle Scholar