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Other One-dimensional Systems

Owen L. de Lange, R. E. Raab

Published Nov 28, 1991• 0 citations
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Abstract

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.

DOI
10.1093/oso/9780198539612.003.0003

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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.
Abstract In both classical and quantum physics the harmonic oscillator is one of the most important physical systems [1–4]. Furthermore, because of the simplicity and symmetry of the Hamiltonian operator, the quantum mechanical oscillator is particularly suited to analysis by abstract operator methods. For this reason the study of the one dimensional oscillator is a useful introduction to the application of operator methods in quantum mechanics. In Sections 2.1–2.4 we use these methods to determine energy eigenvalues and eigenkets, and matrix elements; the analysis is extended to isotropic multidimensional oscillators in Section 2.10. Quasiclassical states of the oscillator are discussed in Section 2.5. In Section 2.6 shift operators are used to calculate wave functions. In Sections 2.7–2.11 we discuss some applications of the theory of the oscillator.
Abstract Quantum mechanical problems capable of exact solution are traditionally solved in a few instances only (such as the harmonic oscillator and angular momentum) by operator methods, but mainly by means of Schrodinger's wave mechanics. The present volume shows that a large range of one- and three-dimensional problems, including certain relativistic ones, are solvable by algebraic, representation-independent methods using commutation relations, shift operators, the virial, hypervirial, and Hellman-Feyman theorems. Applications of these operator methods to the calculation of eigenvalues, matrix elements, and wavefunctions are discussed in detail. This book is an introduction to the use of operator methods in quantum mechanics and also a reference work with numerous problmes solved. It is suitable for use by students of intermediate quantum mechanics and also more advanced postgraduate students who wish to study the algebraic method of solving quantum mechanical problems.
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 In Chapter 5 we considered the factorization of a radial Hamiltonian for a particle in a spherically symmetric potential which is independent of angular momentum. This factorization provides shift operators for energy and angular momentum in the eigenkets of the radial Hamiltonian for the oscillator and Coulomb potentials. These operators are linear functions of the momentum operator *p and they are also nonlinear functions of the position operator r: in the coordinate representation of wave mechanics they yield first-order differential operators. In this chapter we show that for the three-dimensional isotropic harmonic oscillator and the Coulomb potential, one can also construct shift operators for energy and angular momentum that are linear functions* of r and nonlinear functions of p. In the momentum representation of wave mechanics the shift operators yield first-order differential operators. Applications to the calculation of energy eigenvalues, matrix elements, and radial momentum-space wavefunctions are considered.
Two applications of the quantum mechanical hypervirial theorem are discussed. This states that for any system having eigenstates A and B of Hamiltonian H, an operator W has the matrix element relation (A,(HW andminus; WH) B) = (EA andminus; EB) (A, W B) providing W is such that H is Hermitean between A and W B. The first application uses the operator W = eitx to derive a further relationship between certain matrix elements. For one dimensional systems in which the potential V(x) involves only positive powers of x, this relationship becomes a differential equation in f(t) = (A, eitx B) This differential equation has been solved analytically for the harmonic oscillator and in series for the quartic oscillator which were taken as example systems. The second application is the off-diagonal hypervirial method, which seeks to require a wave function A to satisfy a number of hypervirial relations. In particular the wave function A is forced to obey as many conditions as it has variable parameters. To accomplish this a computational algorithm was developed which allowed one of these parameters to be non-linear. The results for oscillator test systems are not encouraging. The systems tested were the quartic and sixth power oscillators with a trial basis of harmonic oscillator eigenfunctions. In general the results were not as good as those obtainable by the variational method.