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Physics of the Solid State

Energy features of a loaded quantum anharmonic oscillator

V. L. Gilyarov, A. I. Slutsker

Published Mar 1, 2010• 7 citations
Publisher

Abstract

A quantum anharmonic oscillator in the ground state has been considered under the conditions of loading with an external force. The wave functions have been calculated for different forces, and the eigenvalues of the energy of the system have been determined as a function of the load. It has been established that the zero-point energy of the oscillator varies linearly with a variation in the force (decreases under tension and increases under compression) and that the average kinetic and potential components of the energy are also characterized by linear variations.

DOI
10.1134/s1063783410030200

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A quantum anharmonic oscillator in the ground state has been considered under the conditions of loading with an external force. The wave functions have been calculated for different forces, and the eigenvalues of the energy of the system have been determined as a function of the load. It has been established that the zero-point energy of the oscillator varies linearly with a variation in the force (decreases under tension and increases under compression) and that the average kinetic and potential components of the energy are also characterized by linear variations.
Abstract An anharmonic oscillator with a perturbed quadratic potential which is coupled with an exponential term is being investigated in this paper by isolating an anharmonic oscillator interaction potential from the actual interaction experienced by the quantum system studied, and using standard perturbative techniques. The anharmonic potential considered is of interest because of its usefulness in the study of non-centrosymmetric materials which have applications in piezoelectricity. The ground state energy eigenvalue and its associated eigenstates were calculated for the quantum system using an analytical approach. Results obtained are compared to those of quantised harmonic oscillator to show the effect of the perturbation.
In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.
This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).
This study aims to apply the matrix method to determine the energy of anharmonic oscillators containing linear, quadratic, cubic, quartic, quintic, sextic, septic, octic, nonic, and decic anharmonicities.Energy calculations were performed analytically using the matrix method with a matrix of size 4×4.The calculations were carried out by finding the Hamiltonian matrix representation of the oscillators, which was then solved using the time-independent Schrödinger equation.As a comparison, computational calculations were also conducted with Wolfram Mathematica for large matrix sizes.Results for small coupling constants and small perturbation terms such as linear anharmonic oscillators showed that energies obtained agree with the exact solution (up to 3 digits).Meanwhile, the accuracy of the analytical matrix method was found to decrease for larger coupling constants.The application of analytical matrix methods with small matrix sizes can produce relatively accurate energies at small coupling constants, while large coupling constants require larger matrix sizes to obtain reasonably accurate results.The calculation of energy by matrix method with a large matrix size will be more effectively done computationally, such as by using Wolfram Mathematica.