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Journal of Tsinghua University(Science and Technology)

New application of supersymmetric operator method to one dimensional bound states

Zhan Wei Xu

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

Starting from the operator factorization in Quantum Mechanics and combined with the shape invariance of superpotential, the supersymmetric operator method can be used to exactly solve the energies and wavefunctions of bound states in a one dimensional potential well, Scarf(Ⅱ) potential. And the existence of its bound states was discussed. The similar one dimensional square well barrier was also investigated and conditions for existence of bound states were also found. Comparing other various square potential wells with continuous ones, the general discussion for existence condition of one dimensional bound states was given and it is found that this existence condition is closely related with the shapes of one dimensional potentials.

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Starting from the operator factorization in Quantum Mechanics and combined with the shape invariance of superpotential, the supersymmetric operator method can be used to exactly solve the energies and wavefunctions of bound states in a one dimensional potential well, Scarf(Ⅱ) potential. And the existence of its bound states was discussed. The similar one dimensional square well barrier was also investigated and conditions for existence of bound states were also found. Comparing other various square potential wells with continuous ones, the general discussion for existence condition of one dimensional bound states was given and it is found that this existence condition is closely related with the shapes of one dimensional potentials.
We review SUSY in non-relativistic quantum mechanics as a powerful tool for studying one dimensional potentials and discuss some applications. The Dirac operatorial factorization method for the harmonic oscillator in QM is generalized to a broader scope. A Schr¨odinger second order operator can be factorized into two first-order ones by solving a Riccati equation and determining the corresponding superpotential. We apply the technique to square well, P¨oschl-Teller and finite barrier potentials, analyzing the resulting energy spectra and the transmission and reflection coefficients. Both discrete and continuous energy spectrum cases are discussed.
Abstract We discuss the application of operator methods to one-dimensional systems other than the harmonic oscillator. In Section 3.1 the method of ‘oneway factorization’ is used to obtain generalized quantum harmonic potentials. In Section 3.2 factorization and shift operators for an arbitrary one-dimensional potential are considered, and a classification is presented of the problems that can be treated by this method [1]. The use of the hypervirial theorem in the calculation of expectation values for one-dimensional systems is discussed in Section 3.3. In Sections 3.4–3.7 four examples are used to illustrate the applications of the preceding material. In these examples we consider in some detail how shift operators for energy can be derived and we also use these shift operators to calculate wavefunctions. Supersymmetric quantum mechanics and shape invariance are discussed in Sections 3.8 and 3.9.
In this work we present an introduction to Supersymmetry in the context of 1-dimensional Quantum Mechanics. For that purpose we develop the concept of hamiltonians factorization using the simple harmonic oscillator as an example, we introduce the supersymmetric oscilator and, next, we generalize these concepts to introduce the fundamentals of Supersymmetric Quantum Mechanics. We also discuss useful tools to solve problems in Quantum Mechanics which are intrinsecally related to Supersymmetry as hierarchy of hamiltonians and shape invariance. We present two approximation methods which will be specially useful: the well known Variational Method and the Logarithmic Perturbation Theory, the latter being closely related to the concepts of superpotentials and hierarchy of hamiltonians. Finally, we present problems related to superpotentials which are monomials in even powers of the x coordinate multiplied by the sign function epsilon(x), which seems to be a new class of problems in Supersymmetric Quantum Mechanics.
The angular part of 4-dimensional Schrodinger equation for Scarf potential was solved by using the Nikiforov-Uvarov method and Supersymmetric Quantum Mechanic method. The determination of the ground state wave function has been used Nikiforov-Uvarov method and by applying the parametric generalization of the hypergeometric type equation. By using manipulation of the properties and operators of the Supersymmetric Quantum Mechanic method the partner potential was constructed. The ground state wave functions of original Scarf potential is different than the ground state wave functions of the construction result potential.
In this paper, we show that the exact energy eigenvalues and eigenfunctions of the Schrödinger equation for charged particles moving in a certain class of noncentral potentials can be easily calculated analytically in a simple and elegant manner by using Supersymmetric method (SUSYQM). We discuss the Poschl-Teller plus Scarf non-central potential systems. Then, by operating the lowering operator we get the ground state wave function, and the excited state wave functions are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the closed form obtained using the shape invariant properties. The results are in exact agreement with other methods.Keyword: supersymmetry, non-central potentials, poschl teller plus scarf.