A quantum gas in an external field at finite temperatures. An exact expression for density and exited states

Authors

Keywords:

квантовая статистика, интегралы по траекториям, матрица плотности, сеточные методы, квантовый газ, гармоническое поле

Abstract

A general expression for the density of a quantum system at finite temperature is obtained in the form of the variational derivative of the canonical partition function. Cyclic expansions for density are obtained in the absence and in the presence of interparticle interaction. In the first case the density is expressed in terms of single-particle characteristics for a set of increasing temperatures. The successive squaring method for the density matrix is considered; with the use of the cyclic density expansions, the densities of 1, 2,..., and 10 noninteracting fermi-particles with spin 1/2 and spin 0 and of 1, 2, 3,4, and 5 spinless noninteracting particles in the Morse potential are evaluated. From the low-temperature data, 10 states in a harmonic field and all 5 bound states in the Morse potential are reproduced with high accuracy. The system of two spinless fermions with and without the Coulomb repulsion is considered. One- and two-dimensional densities of the system and their energies are evaluated. The recurrent state subtraction method is discussed for the evaluation of excited states of quantum systems. The successive squaring method for the density matrix is used to obtain more than 20 states for the one-dimensional harmonic oscillator and all bound states for the Morse oscillator.

Author Biographies

E.A. Polyakov

P.N. Vorontsov-Velyaminov

References

  1. Lyubartsev A.P. Simulation of excited states and the sign problem in path integral Monte Carlo method // J. Phys. A.: Math. Gen. 2005. 38. 6659-6674.
  2. Feynman R.P. Statistical Mechanics. New York: McGraw-Hill, 1972.
  3. Vorontsov-Velyaminov P.N., Voznesenski M.A., Malakhov D.V., Lyubartsev A.P., Broukhno A.V. Path integral method in quantum statistics problems: generalized ensemble Monte Carlo and density functional approach // J. Phys. A: Math. Gen. 2006. 39. 4711-4716.
  4. Vorontsov-Velyaminov P.N., Ivanov S.D., Gorbunov R.I. Quantum gas in an external field: exact grand canonical expressions and numerical treatment // Phys. Rev. E. 1999. 59. 168-176.
  5. Эллиот Дж., Добер П. Симметрия в физике. М.: Мир, 1983.
  6. Хаммермеш М. Теория групп и ее применение в физике. М.: Мир, 1966.
  7. Цвелик А.М. Квантовая теория поля в физике конденсированного состояния. М.: Физматлит, 2002.
  8. Heerman M.F., Bruskin E.J., Berne B.J. On path integral Monte Carlo simulations // J. Chem. Phys. 1982. 76, N 10. 5150-5155.
  9. Broukhno A., Vorontsov-Velyaminov P.N., Bohr H. Polymer density functional approach to efficient evaluation of path integrals // Phys. Rev. E. 2005. 72, N 4. Art. № 046703.
  10. Lyubartsev A.P., Vorontsov-Velyaminov P.N. Path-integral Monte Carlo method in quantum statistics for a system of N identical fermions // Phys. Rev. A. 1993. 48. 4075-4083.
  11. Mitchell W.F. Adaptive refinement for arbitrary finite-element spaces with hierarchical bases // J. Comp. Appl. 1991. 36. 65-78.
  12. Mitchell W.F. The full domain partition approach to parallel adaptive refinement // Grid Generation and Adaptive Algorithms. IMA Volumes in Mathematics and its Applications. Vol. 113. Berlin: Springer-Verlag, 1998. 151-162.
  13. Sim E., Makri N. Time-dependent discrete variable representations for quantum wave packet propagation // J. Chem. Phys. 1995. 102. 5616-5625.
  14. Kosloff R., Tal-Ezer H. A direct relaxation method for calculating eigenfunctions and eigenvalues of the Schrödinger equation on a grid // Chem. Phys. Lett. 1986. 127. 223-230.
  15. Ландау Л.Д., Лившиц Е.М. Теоретическая физика. Том III. М.: Физматлит, 2002.

Published

16-11-2007

How to Cite

Поляков Е., Воронцов-Вельяминов П. A Quantum Gas in an External Field at Finite Temperatures. An Exact Expression for Density and Exited States // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2007. 8. 334-351

Issue

Section

Section 1. Numerical methods and applications