Thermodynamics properties of confined inverted harmonic oscillator in a quantum well

Abstract We present a pioneering investigation into the thermodynamic properties of the one-dimensional (1-D) inverted harmonic oscillator confined within a quantum well. Utilizing the linear variational method, we compute the eigen-energy of the confined inverted harmonic oscillator (CIHO) and find good agreement with exact solutions. Our analysis unveils an intriguing phenomenon: the energy of the CIHO can exhibit both positive and negative values due to the interplay between quantum confinement and the inverted harmonic potential. This anomalous behavior directly affects the system’s quantum statistics thermodynamic properties. Our calculation reveals that, for a fixed quantum well width, the average energy, free energy, and entropy all decrease as the inverted harmonic frequency increases. However, the variation of heat capacity ( C V ) with harmonic frequency is remarkably peculiar. Specifically, in the low-temperature region, C V diminishes with increasing harmonic frequency, while the opposite trend occurs at higher temperatures. We also investigate the influence of quantum confinement on the thermodynamic properties of the CIHO. Our results indicate that the thermodynamic properties of the CIHO can be effectively controlled by adjusting both the size of the quantum well and the harmonic frequency. This work can guide the future research on the thermodynamic property of the confined harmonic oscillator and may find applications in various fields, including high-energy physics, condensed state physics, statistical physics, and cold atomic physics.