Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 110, 12 pages      arXiv:0711.3544      https://doi.org/10.3842/SIGMA.2007.110
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Alternative Method for Determining the Feynman Propagator of a Non-Relativistic Quantum Mechanical Problem

Marcos Moshinsky a, Emerson Sadurní a and Adolfo del Campo b
a) Instituto de Física Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México D.F., México
b) Departamento de Química-Física, Universidad del País Vasco, Apdo. 644, Bilbao, Spain

Received August 21, 2007, in final form November 13, 2007; Published online November 22, 2007; Misprints are corrected December 06, 2007

Abstract
A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman. The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later. In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform. The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be generalized to the derivation of more complicated propagators.

Key words: propagator; Green functions; harmonic oscillator.

pdf (272 kb)   ps (187 kb)   tex (97 kb)

References

  1. Grosche C., Steiner F., Handbook of Feynman path integrals, Springer Tracts in Modern Physics, Vol. 145, Springer, Berlin, 1998.
  2. Jaeger J.C., An introduction to Laplace transform, Methuen, London, 1965.
  3. Gradshteyn I., Ryzhik I., Table of integrals, series and products, Academic Press, 1965.
  4. Sakurai J.J., Modern quantum mechanics, Addison-Wesley, 1994.
  5. Holstein B.R., The linear potential propagator, Amer. J. Phys. 65 (1997), 414-418.
    Holstein B.R., The harmonic oscillator propagator, Amer. J. Phys. 66 (1998), 583-589.
  6. Cohen S.M., Path integral for the quantum harmonic oscillator using elementary methods, Amer. J. Phys. 66 (1998), 537-540.

Previous article   Next article   Contents of Volume 3 (2007)