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Journal of Physics A Mathematical and General

Harmonic oscillator Green functions

D. B. Khrebtukov, J. H. Macek

Published Mar 27, 1998• 10 citations
Publisher

Abstract

The Green function for the Schrödinger equation with an isotropic, three-dimensional harmonic-oscillator potential is given in closed form. A similar closed form is obtained when the Schrödinger equation also contains a magnetic interaction and the magnetic field is such that the precession and oscillation frequencies are equal. The latter Green function is used to obtain energy and Sturmian eigenvalues that occur in the theory of atom-atom collisions.

DOI
10.1088/0305-4470/31/12/010

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The Green function for the Schrödinger equation with an isotropic, three-dimensional harmonic-oscillator potential is given in closed form. A similar closed form is obtained when the Schrödinger equation also contains a magnetic interaction and the magnetic field is such that the precession and oscillation frequencies are equal. The latter Green function is used to obtain energy and Sturmian eigenvalues that occur in the theory of atom-atom collisions.
In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.
The eigenfunctions and eigenvalues of the three-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square interaction are obtained. A realization of the ladder operators for the wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1,1) group. The closed analytical expressions for the matrix elements of different functions ρ and ρd/dρ with ρ = r 2 are evaluated. Another hidden symmetry explores the relations between the eigenvalues and eigenfunctions for substituting r→ ir. PACS number(s): 03. 65. Fd, 03. 65.Ge and 02. 20.Qs.
Quantum systems interacting with a dissipative environment can effectively be described by a nonlinear modification of the Schrodinger equation. For some standard problems that possess analytical solutions, e.g. the free motion and the harmonic oscillator, also the nonlinear equation can be solved in a closed form and has Gaussian-shaped wave packet solutions. The equation determining the time-evolution of the wave-packet width is - already in the case of the usual conservative Schrodinger equation - nonlinear and provides the quantities that are necessary to construct the Green’s function for the respective system. The Green’s functions for the conservative and dissipative cases will be given explicitly in position and momentum space.
Abstract The energy eigenvalue spectrum for a conservative dynamical system is contained implicitly in its Green's function. It becomes explicit in the Fourier transform of either the Green's function or its trace. The trace exists only when the spectrum is entirely discrete. Applications are made to the free particle, the linear harmonic oscillator, and the hydrogen atom. In the latter two cases determination of the Green's function can be considerably simplified by similarity transformations on the Hamiltonian operator.