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Physical Review A

Changes in potentials due to changes in the point spectrum: Anharmonic oscillators with exact solutions

Phillip B. Abraham, H. E. Moses

Published Oct 1, 1980• 187 citations

Abstract

Assume a solved quantum-mechanical problem for the one-dimensional Schr\"odinger equation, which has a discrete spectrum. An algorithm is presented to calculate exactly the change in the original potential and eigenfunctions brought about by arbitrary changes in the positions of the original eigenvalues and/or the normalizations of the corresponding eigenfunctions. As a first example, we consider the modification of the harmonic oscillator and thereby obtain potentials for anharmonic oscillators with exact eigenvalues and eigenfunctions. Next, we introduce potentials in the one-dimensional box with rigid walls which alter the usual spectrum in a predetermined way. We also sketch the process of obtaining a change in the Coulomb potential which deletes the lowest eigenvalue for the zero-angular-momentum equation.

DOI: 10.1103/physreva.22.1333

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Assume a solved quantum-mechanical problem for the one-dimensional Schr"odinger equation, which has a discrete spectrum. An algorithm is presented to calculate exactly the change in the original potential and eigenfunctions brought about by arbitrary changes in the positions of the original eigenvalues and/or the normalizations of the corresponding eigenfunctions. As a first example, we consider the modification of the harmonic oscillator and thereby obtain potentials for anharmonic oscillators with exact eigenvalues and eigenfunctions. Next, we introduce potentials in the one-dimensional box with rigid walls which alter the usual spectrum in a predetermined way. We also sketch the process of obtaining a change in the Coulomb potential which deletes the lowest eigenvalue for the zero-angular-momentum equation.

This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).

Courses on undergraduate quantum mechanics usually focus on solutions of the Schrödinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced, only academic examples are used, such as the matrix representation of Dirac’s raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.

A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ϕ n ( x ) and eigenvalues α n corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ϕ n ( x ) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.