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Ferroelectrics

Interacting protons in two double minimum potentials: Quantum mechanical calculations

H. Blaurock, R. Siems

Published Dec 1, 1991• 1 citation
Publisher

Abstract

A system consisting of two protons in two double minimum potentials coupled by a bilinear interaction was studied. Explicit calculations were performed for local potentials described by a combination of harmonic oscillator and Gaussian terms. To determine eigenfunctions and -values variational procedures were employed which made use of numerically determined 1-particle solutions. Eigenvalues and (graphical representations of) 2d eigenfunctions are presented, classified according to their symmetry, and used to determine the time evolution of the system for given initial conditions.

DOI
10.1080/00150199108209409

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A system consisting of two protons in two double minimum potentials coupled by a bilinear interaction was studied. Explicit calculations were performed for local potentials described by a combination of harmonic oscillator and Gaussian terms. To determine eigenfunctions and -values variational procedures were employed which made use of numerically determined 1-particle solutions. Eigenvalues and (graphical representations of) 2d eigenfunctions are presented, classified according to their symmetry, and used to determine the time evolution of the system for given initial conditions.
Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.
Accurate eigenvalues and eigenfunctions of a two-level system interacting with a one-mode quantum field are calculated numerically. A special iteration procedure based on the operator method permits one to consider the solution within a wide range of the Hamiltonian parameters and to find the uniformly approximating analytical formula for the eigenvalues. Characteristic features of the model are considered, such as the level intersections, the population of the field states and the chaotization in the system through the doubling of the frequencies.