Schr\"odinger Equation with a Non-Central Potential: Some Statistical Quantities

TL;DR

This paper investigates how statistical quantities like free energy, entropy, and specific heat depend on temperature in the Schr"odinger equation with a non-central potential, using analytical methods and exploring effects of angular momentum and phase transitions.

Contribution

It introduces a method based on the Euler-Maclaurin formula to compute the partition function for non-central potentials in quantum systems.

Findings

Analytical expressions for thermal functions in 1D and 3D cases.

Effect of angular momentum on statistical quantities.

Evidence of phase transition behavior.

Abstract

In this paper, we search the dependence of some statistical quantities such as the free energy, the mean energy, the entropy, and the specific heat for the Schr\"odinger equation on the temperature, particularly the case of a non-central potential. The basic point is to find the partition function which is obtained by a method based on the Euler-Maclaurin formula. At first, we present the analytical results by supporting with some plots for the thermal functions for one- and three-dimensional cases to find out the effect of the angular momentum. We also search then the effect of the angle-dependent part of the non-central potential. We discuss the results briefly for a phase transition for the system. We also present our results for three-dimesional harmonic oscillator.