A new simple class of superpotentials in SUSY quantum mechanics

Abstract

In this work we introduce the class of quantum mechanics superpotentials W (x) = gε(x)x 2n and study in details the cases n = 0 and 1. The n = 0 superpotential is shown to lead to the known problem of two supersymmetrically related Dirac delta potentials (well and barrier). The n = 1 case result in the potentialsFor V − we present the exact ground state solution and study the excited states by a variational technic. Starting from the ground state of V − and using logarithmic perturbation theory we study the ground states of V + and also of V (x) = g 2 x 4 and compare the result got by this new way with other results for this last potential in the literature. I. INTRODUCTION Supersymmetric quantum mechanics (SUSY QM) was first introduced by E. Witten [1][2], as a simplified model (a 0 + 1 dimensional field theory) to study the possibility of SUSY breaking. Soon it became a research branch in itself, a way of getting new solutions to problems in quantum mechanics [3] [4] [5] [6]. Of particular interest, to our work below, we must cite the many papers in the literature [7] [8] [9] [10] [11] [12] [13] [14] devoted to the development of technics for treating the anharmonic oscillator V (x) = ω 2 x 2 + g 2 x 4 , and other related potentials, which in general do not have exact solutions.In this work we present a new simple class of superpotentials in SUSY QM, in the form W (x) = gε(x)x 2n with n = 0, 1, 2, . . .. The first example of this class, i.e., the case n = 0, *

Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanics

“…However, we are not aware of any proof that either of the basis sets (3) or (4) is complete. We see that present results are smaller, and therefore more accurate, than those of Marques et al [1]. In addition, the results in their tables 1 and 2 exhibit unchanged entries when the size of the basis set increases from to + 1 which reflects the fact that the added basis function improves the even (odd) state but has no effect on the odd (even) one.…”

“…In a recent paper Marques et al [1] derived two new simple supersymmetric partner potentials and showed that the ground-state eigenfunction of one of the Hamiltonian operators can be obtained exactly. In order to calculate the remaining states they resorted to the Rayleigh-Ritz variational method.…”

“…First we improve the variational method proposed by Marques et al [1] and, second, we show that the Riccati-Padé method (RPM) [2][3][4] provides highly accurate results for the eigenvalues of those partner Hamiltonian operators. In addition, we show that exactly the same Rayleigh-Ritz variational method applied to the SUSY partner Hamiltonians is also suitable for the calculation of the eigenvalues of the quartic anharmonic oscillator.…”

“…In Section 2 we apply the Rayleigh-Ritz variational method and the RPM to the pair of SUSY Hamiltonians proposed by Marques et al [1] and also to the quartic oscillator. In Section 3 we comment on those approaches and draw conclusions.…”

“…In Section 3 we comment on those approaches and draw conclusions. The partner Hamiltonian operators are given by [1]:…”