In this work, we present a detailed study of a one-dimensional Schrodinger equation in the presence of quantum Gaussian well interaction. Further, we investigate the approximate solutions by using the harmonic oscillator approximation, variational principle, four-parameter potential fitting, and also numerical solution using finite difference method. The parabolic approximation yields an excellent energy value compared with the numerical solution of the Gaussian system only for the ground state, while for the excited states it provides a higher approximation. Also the analytical bound state energies of the four parameters potential under the framework of the Nikiforov-Uvarov (NU) method have been used after getting the suitable values of the potential parameters using the numerical fitting. The present results of the system states are found to be in high agreement with the well-known numerical results of the Gaussian potential.