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Exact response of the non-relativistic harmonic oscillator

Pace, E, Salmè, G, Rinat, A S

Published Jan 1, 1994• 8 citations
OpenAlex

Abstract

Using Green$'$s function and operator techniques we give a closed expression for the response of a non-relativistic system interacting through confining, harmonic forces. The expression for the incoherent part permits rapid evaluation of coefficients in a 1/q expansion. A comparison is made with standard approximation methods.

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Using Green′'′s function and operator techniques we give a closed expression for the response of a non-relativistic system interacting through confining, harmonic forces. The expression for the incoherent part permits rapid evaluation of coefficients in a 1/q expansion. A comparison is made with standard approximation methods.
Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
Abstract We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel–Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant k and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to ∞, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.
A new technique is presented for evaluating the electron Green function Gk(t) in an insulating electron–phonon system. The technique is particularly suitable for those electron–phonon interactions in which the long wavelength phonons are of prime importance such as the polaron and piezoelectric interactions.An exact formal expression is obtained for the time and temperature dependent Green function in the form[Formula: see text]where S is a functional of the differential operator d/dk. The Green function is then expressed as [Formula: see text], and from the exact result a series expansion is derived for the action function Ak(t).This series expansion has a very special property: to whatever order Ak(t) is evaluated the resulting expression for Gk(t) includes, in some way, every perturbation theory diagram. By this we mean that as in conventional techniques (the Hartree–Fock approximation, for example) an infinite subset of diagrams are summed exactly and in contrast to these techniques the remaining infinity of diagrams are not discarded but are evaluated approximately.The technique is applied to the polaron interaction.