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International Journal of Quantum Chemistry

A discrete look at localization

GEORGE G. HALL, David H. E. Rees

Published Jan 15, 1995• 10 citations
Publisher

Abstract

Abstract This article introduces a set of localized orthonormal functions to serve as basis functions for quantum calculations. They are defined to be eigenfunctions of the position operator in a function space. Their properties, including their variances, for a one‐dimensional system are developed. The application to simple harmonic motion is considered as an example and, in particular, the time evolution of an initially localized function is calculated and shown to be periodic. The theory can be interpreted as producing a discrete quantization of space with Hamiltonian interactions that are predominantly between nearest neighbors. These functions can also be used in approximate calculations. To illustrate their accuracy, the example of a Morse oscillator treated as a perturbation of a harmonic oscillator is reconsidered. It is shown that the localized functions in a variational calculation lead to a result that is a good approximation for the lowest states. Furthermore, the use of a wave function that is defined only at discrete points can be justified as the first approximation to this, so that its accuracy can also be discussed. © 1995 John Wiley & Sons, Inc.

DOI
10.1002/qua.560530206

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Abstract This article introduces a set of localized orthonormal functions to serve as basis functions for quantum calculations. They are defined to be eigenfunctions of the position operator in a function space. Their properties, including their variances, for a one‐dimensional system are developed. The application to simple harmonic motion is considered as an example and, in particular, the time evolution of an initially localized function is calculated and shown to be periodic. The theory can be interpreted as producing a discrete quantization of space with Hamiltonian interactions that are predominantly between nearest neighbors. These functions can also be used in approximate calculations. To illustrate their accuracy, the example of a Morse oscillator treated as a perturbation of a harmonic oscillator is reconsidered. It is shown that the localized functions in a variational calculation lead to a result that is a good approximation for the lowest states. Furthermore, the use of a wave function that is defined only at discrete points can be justified as the first approximation to this, so that its accuracy can also be discussed. © 1995 John Wiley & Sons, Inc.
Abstract The derivation of hybrids as localized equivalent functions in the plane is discussed using the simultaneous eigenfunctions of the x and y position operators, as represented in a finite basis. It proves helpful, initially, to use complex exponentials as basis functions, but the transformation to a real basis is made later. The introduction of alias functions to produce commuting matrices is described. Full results are obtained for any number of functions in the plane. © 1995 John Wiley & Sons, Inc.
The authors show how the perturbative-variational approximation of eigenvalues and eigenvectors, that has provided very good results for one-dimensional problems, can be generalized to Hamiltonians with degenerate states. This generalization will be illustrated by applying it to a pair of coupled harmonic oscillators. The obtained results for the ground and first excited states have a very good precision when they are compared with the results of direct diagonalization of the Hamiltonian with 190 states in the same auxiliary basis used in the perturbative-variational approximation.
We consider the nature of the wave function using the example of a harmonic oscillator. We show that the eigenfunctions ψn=znψ_n{=}z^nψn​=zn of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation with z∈Cz\in\mathbb Cz∈C are the coordinates of a classical oscillator with energy En=ℏωnE_n=\hbarωnEn​=ℏωn, n=0,1,2,... n=0,1,2,...\,n=0,1,2,.... They are defined on conical spaces C/Zn{\mathbb C}/{\mathbb Z}_nC/Zn​ with cone angles 2π/n2π/n2π/n, which are embedded as subspaces in the phase space C\mathbb CC of the classical oscillator. Here Zn{\mathbb Z}_nZn​ is the finite cyclic group of rotations of the space C\mathbb CC by an angle 2π/n2π/n2π/n. The superposition ψ=∑ncnψnψ=\sum_n c_nψ_nψ=∑n​cn​ψn​ of the eigenfunctions ψnψ_nψn​ arises only with incomplete knowledge of the initial data for solving the Schrödinger equation, when the conditions of invariance with respect to the discrete groups Zn{\mathbb Z}_nZn​ are not imposed and the general solution takes into account all possible initial data parametrized by the numbers n∈Nn\in\mathbb Nn∈N.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.