Thermodynamics of quantum oscillators

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Computational physics

Background:

• Calculating quantum partition functions for systems of multiple oscillators is computationally challenging.
• Existing methods often struggle with anharmonic potentials or large numbers of coupled oscillators.

Purpose of the Study:

• To develop a general and accurate analytical approximation for the quantum partition function.
• To provide a computationally efficient method for determining thermodynamic properties of quantum oscillator systems.

Main Methods:

• Utilized path-integral formulation to derive an approximate quantum partition function.
• Employed a temperature-dependent Gaussian approximation for the high-temperature propagator.
• Applied the principle of minimal sensitivity to optimize Gaussian parameters via coupled nonlinear equations.

Main Results:

• The analytical approximation accurately reproduces thermodynamic quantities (free energy, average energy, specific heat) with 1%-5% error.
• The method remains accurate even at zero temperature and with increased anharmonicity and coupling.
• Validated against Hamiltonian diagonalization and path-integral Monte Carlo simulations for systems up to ten oscillators.

Conclusions:

• The proposed compact analytical approximation offers a reliable and efficient approach for quantum oscillator systems.
• This method significantly simplifies the calculation of thermodynamic properties for complex quantum systems.