Path-integral approach to the Wigner-Kirkwood expansion.

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Computational physics

Background:

• Quantum-mechanical path integrals are crucial for understanding systems at finite temperatures.
• The Feynman-Kac formula provides a link between quantum mechanics and classical statistical mechanics.
• Calculating thermodynamic properties of quantum systems often involves complex path integral formulations.

Purpose of the Study:

• To derive a functional representation of the Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities.
• To demonstrate the efficiency of this expansion in obtaining analytic higher-order expansion coefficients.
• To apply the expansion to calculate thermodynamic functions for a one-dimensional anharmonic oscillator.

Main Methods:

• Utilizing the Feynman-Kac formula as a starting point.
• Developing a functional representation for the Wigner-Kirkwood perturbation expansion.
• Applying the derived expansion to the one-dimensional anharmonic oscillator model.
• Discussing generalizations to the Bloch density matrix and comparing with world-line formulations.

Main Results:

• An efficient Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities was successfully derived.
• The expansion provides analytic forms for higher-order expansion coefficients, simplifying calculations.
• Basic thermodynamic functions for the one-dimensional anharmonic oscillator were obtained using this method.

Conclusions:

• The derived Wigner-Kirkwood expansion offers an efficient approach for studying high-temperature quantum systems.
• This method facilitates the analytic calculation of thermodynamic properties for various potentials.
• Further research can explore generalizations and comparisons with alternative quantum mechanical formulations.