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American Journal of Physics

Calculation of a bound state wavefunction using free state wavefunctions only

Kenneth R. Brownstein

Published Feb 1, 1975• 23 citations
Publisher

Abstract

A pedagogical example is presented in which the completeness of the free state wavefunctions of a Hamiltonian can be explicitly checked. For certain values of the potential strength the free states prove to be complete. For other values of the potential strength the free states are shown to be incomplete and the extent of this incompleteness is shown to consist of precisely one function. Using the fact that the totality of all energy eigenfunctions must be complete, this single bound state wavefunction is calculated using the free state wavefunctions only.

DOI
10.1119/1.9892

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A pedagogical example is presented in which the completeness of the free state wavefunctions of a Hamiltonian can be explicitly checked. For certain values of the potential strength the free states prove to be complete. For other values of the potential strength the free states are shown to be incomplete and the extent of this incompleteness is shown to consist of precisely one function. Using the fact that the totality of all energy eigenfunctions must be complete, this single bound state wavefunction is calculated using the free state wavefunctions only.
There are a few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians.
We propose a simple and straightforward method based on Wronskians for the calculation of bound-state energies and wavefunctions of one-dimensional quantum-mechanical problems. We explicitly discuss the asymptotic behaviour of the wavefunction and show that the allowed energies make the divergent part vanish. As illustrative examples we consider an exactly solvable model,the Gaussian potential well, and a two-well potential proposed earlier for the interpretation of the infrared spectrum of ammonia.
We propose a simple and straightforward method based on Wronskians for the calculation of bound-state energies and wavefunctions of one-dimensional quantum-mechanical problems. We explicitly discuss the asymptotic behaviour of the wavefunction and show that the allowed energies make the divergent part vanish. As illustrative examples we consider an exactly solvable model, the Gaussian potential well, and a two-well potential proposed earlier for the interpretation of the infrared spectrum of ammonia.
In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.