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International Journal of Mathematical Education in Science and Technology

Demonstration of systematic improvements in the application of the variational method to anharmonic oscillator potentials

W. N. Mei

Published Jul 1, 1997• 3 citations
Publisher

Abstract

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.

DOI
10.1080/0020739970280404

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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.
When applying the variational method, it is demonstrated that the systematic improvements in calculating the eigen‐energies of quantum mechanical systems can be achieved by careful study of the general features of the trial wave functions. Two sets of trial wave functions were used for the harmonic oscillator potential. In the first example, it is shown, even with a crude initial guess, that accurate results will be obtained by adjusting the wave function without introducing any additional variational parameters. Even though the second example requires more algebraic manipulation, it is shown that it converges to the exact result much faster. It is suggested that these calculations be incorporated in computer algebra software packages for the teaching of quantum mechanics.
In applying the variational method, we demonstrate that, with careful study of the general features of the trial wave functions, we can achieve systematic improvements in calculating the eigenvalues of quantum mechanical systems. We apply five different sets of trial wave functions to calculate the ground state and first excited state energies of weakly bound potentials, such as V(x) = ∣x∣, √∣x∣ and ln (∣x∣). We demonstrate that accurate results can be obtained through a thorough understanding of the asymptotic behaviours. With proper adjustments, we show how to improve trial wave functions. Finally, by calculating to the fifth excited states, we compare the exact and variational results with those obtained from the semi-classical method, and discuss the merits of the different approaches.
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.
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.