Abstract
Considering the Feynman path integral representation for the configuration-space density matrix for an ensemble of anharmonic oscillators, we determine the ‘stationary paths’ near which the integrand remains stationary. By taking the path integral to be saturated by contributions from the neighborhood of the path which maximizes the integrand we evaluate the density matrix explicitly in analytic form. This seems to be the first such evaluation of a path integral for a system not describable by a quadratic Hamiltonian. We also comment briefly on the question of analyticity with respect to the perturbation parameter.
Similar content being viewed by others
References
Bender, C. M., and Wu, T. T. (1968).Physical Review Letters 21, 406.
Bender, C. M., and Wu, T. T. (1969).Physical Review 184, 1231.
Bowman, F. (1953).Introduction to Elliptic Functions John Wiley, New York.
Brush, S. G. (1961).Reviews of Modern Physics 33, 79.
Feynman, R. P. (1948).Reviews of Modern Physics 20, 267.
Feynman, R. P., and Hibbs, A. R. (1965).Quantum Mechanics and Path Integrals McGraw Hill, New York.
Gradshteyn, I.S., and Ryzhik, I. M. (1965).Tables of Integrals, Series and Products Academic Press, New York.
Graffi, S., Grecchi, V., and Simon, B. (1970).Physics Letters B 32, 631.
Jaffe, A. (1965).Communications in Mathematical Physics 1, 127.
Lam, C. S. (1967).Nuovo Cimento 47, 451.
Löffel, B., Martin, A., Simon, B., and Wightman, A. S. (1969).Physics Letters B30, 656.
Mathews, P. M., and Eswaran, K. (1972).Nuovo Cimento Letters 5, 15.
Sarkar, S. (1973).Physical Review D8, 1060.
Siegel, A., and Burke, T. (1972).Journal of Mathematical Physics 13, 1681.
Simon, B. (1970).Annals of Physics (N.Y.),58, 76.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mathews, P.M., Seshadri, M.S. Evaluation of the density matrix for an ensemble of anharmonic oscillators by a path integral approach. Int J Theor Phys 13, 279–288 (1975). https://doi.org/10.1007/BF01807431
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01807431