Asymptotic Functional Form Preservation Method with Supersymmetric Quantum Mechanics for Anharmonic Oscillators

Abstract

The asymptotic functional form preservation method is developed in the framework of supersymmetric quantum mechanics to obtain the energies and wave functions of anharmonic oscillators. For each of the ground states in the hierarchy of supersymmetric partner Hamiltonians, we derive a series expansion of the superpotential for . Employing a transformation containing an unphysical parameter, we convert the series expansion of the superpotential into a new series expansion applicable to all the range of x. The unphysical parameter is determined by the principle of minimal sensitivity. Requiring the preservation of the correct asymptotic functional form of the full-range series expansion as x tends to infinity, we obtain the ground and excited state energies. The truncated full-range series expansion for the superpotential provides an approximate analytical expression for the wave function. In addition, several ansatz functional forms are also proposed for the superpotential to obtain the ground and excited state energies with high accuracy. Excellent computational results for the anharmonic oscillator demonstrate that the method proposed here is suitable for solving similar quantum mechanical problems.