Non-relativistic energy analysis of class of shape invariant potentials using Dong proper quantization and variable transformation in SUSY WKB

The analysis of non-relativistic energy has been an important research area because they contain all important information about quantum systems. This research was aimed to obtain and analyze the energy of a non-relativistic system under the influence of a class of shape invariant potentials. This energy equation is important because it can be used to determine the thermodynamical properties and the superstatistic mechanics of the system. The lack of research in this area makes an opportunity to explore more about the application of the energy equation. Non-relativistic energy of a three-dimensional Harmonic Oscillator was obtained by using Supersymmetry WKB quantization rule through Dong Proper Quantization. By using appropriate variable transformation in the SUSY WKB quantization condition formula, a class of shape invariant potentials was reduced to the Coulombic potential ones. The non-relativistic energy of these potentials was obtained by comparing the constant parameters in the SUSY WKB quantization rule scheme between the transformed ones and the Coulombic potential. The result shows that the energy spectra were linear to the change of orbital quantum number, but not linear with the change of the radial quantum number. The variation of the effective mass of the system only gives a small effect to the energy spectra. The energy equation is also used to determine the partition function of the system. The thermodynamics and superstatistics properties of the system were determined by using the partition function and approximated in semiclassical conditions. Other thermodynamics properties such as internal energy, Helmholtz free energy, entropy, and specific heat also could be obtained. These properties are the key to analyze the characteristic features of a system or particle.