Perturbative and Numerical Approach on The Quantum Harmonic Oscillator with a Negative Gaussian Potential

Modeling a system in a harmonic oscillator approach still beneficial for the researcher to elucidate the physical mechanism in a small system such as a quantum dot. In this paper, we investigated the quantum harmonic oscillator system with the addition of a negative Gaussian potential (QHONGP). The effect of this Gaussian term was calculated analytically using a perturbative approach, and numerically using the Numerov and shooting method. The results showed that both approaches were able to show the unique feature of a pairing formation, i.e the decreasing energy of the system and the localization of particles in all (odd and even) states, at the lower pairing strength. For the higher pairing strength, the perturbative results highly depend on the number of energy levels and the expansion terms included in the calculation. The limited number of energy levels and the expansion terms resulted in the inaccuracy of the perturbative approach in giving the solution of QHONGP system as compared to the shooting method.