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International Journal of Mathematical Education in Science and Technology

Combined variational‐perturbative approach to anharmonic oscillator problems

W. N. Mei

Published Nov 1, 1998• 6 citations
Publisher

Abstract

In this work, an unusual approach is taken to calculate the ground state and first excited state energies of the anharmonic oscillator potential. By using the variational method, analytical wave functions which have extremely accurate expectation values for the quartic or sextic oscillators are obtained. The corrections due to the other terms are then calculated perturbatively. It is shown that each approach, which starts with a different choice of zeroth‐order term, provides results that agree with those computed from numerical integration and series solution. Joining the above schemes, yields a combined variational and perturbative treatment of anharmonic oscillator problems.

DOI
10.1080/0020739980290609

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In this work, an unusual approach is taken to calculate the ground state and first excited state energies of the anharmonic oscillator potential. By using the variational method, analytical wave functions which have extremely accurate expectation values for the quartic or sextic oscillators are obtained. The corrections due to the other terms are then calculated perturbatively. It is shown that each approach, which starts with a different choice of zeroth‐order term, provides results that agree with those computed from numerical integration and series solution. Joining the above schemes, yields a combined variational and perturbative treatment of anharmonic oscillator problems.
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.
Using the general anharmonic oscillator as a case study, we examine the coupled-cluster method (CCM) in some detail. Emphasis is specially placed on the accuracy of the standard ground-state energy calculation and the excitation spectrum as derived via the Emrich ansatz. We are particularly interested in problems that can arise from large differences between the exact wave function and its model counterpart that forms the starting point of the CCM. To this end we begin with a variational Hartree approximation applied to three particular anharmonic-oscillator Hamiltonians that contain, respectively, pure quartic, equally weighted cubic and quartic, and double-well perturbations. These are chosen to provide increasingly stringent tests for the CCM in the above sense. Considerable attention is paid to the variational description of the double-well case, where the various possible solutions to the Hartree equations are considered and a further calculation on the energy-level splitting is performed. The CCM is then used to improve systematically upon our chosen starting point and is shown, particularly for the ground state, both to produce extremely accurate results and to be rather resilient to even gross deficiencies in the starting wave function. The main problem is the double-well system, where the CCM shows its lack of intrinsic inbuilt tunneling mechanisms via the absence of level splitting. Even here the CCM still produces very accurate average energies for very deep wells. While clearly needing some modification in such extreme cases, the CCM is shown to be quite adaptable and robust with regard to inaccurate starting wave functions.
In applying the variational method, six different sets of trial wave functions are used to calculate the ground state and first excited state energies of the strongly bound potentials, i.e. V(x) = x 2 m , where m = 4, 5 and 6. It is shown that accurate results can be obtained from thorough analysis of the asymptotic behaviour of the solutions. Consequently, it was found that there is a progressive qualitative change in the wave functions from the harmonic to anharmonic and then strongly bounded region, which explains why using a harmonic oscillator wave function together with the conventional perturbative approach to calculate the energy correction, was unsuccessful.
Three sets of trial wave functions were applied to calculate the ground state and first excited state energies of the x 4 and x 6 anharmonic oscillator potentials. We demonstrate that, even with a rather crude initial guess, results can be improved by proper adjustment of the trial wave functions after careful study of their general features. Meanwhile, we found some interesting and unexpected results when applying the second set of trial wave functions. Once again, we suggest using together with computer algebra software packages, these examples when teaching the variational method in a quantum mechanics course.
This paper discusses some practical problems in quantum mechanics, particularly finding solutions of the Schrödinger equation for some cases, as it is a pivotal part of this area. The article focuses on using perturbation theory to calculate the approximate ground state energy for both helium atom and quartic anharmonic oscillator. For the helium atom, conventional perturbation theory is used to solve for the approximate energy, whilst for the anharmonic oscillator, a slightly different approach, that is perturbation theory incorporating a parameter, is chosen to improve the precision of the approximation. The final relative errors for the ground state energy of helium atom model are, for the first order perturbation, , and for the second order, , and that for the anharmonic oscillator is . The study of perturbation theory gives insights into understandings of how it may be modified further to pursue a better level of precision. This facilitates development of new approximation techniques for solving quantum-mechanical issues.