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Journal of Science Natural Science

STRONG-COUPLING PERTURBATION THEORY IN A SIMPLE AND EXACTLY SOLVABLE MODEL

Le Duc Anh, Doan Thi Hai Quynh, Ngo Thi Ha Trang

Published Mar 31, 2025• 0 citations
Publisher

Abstract

We study strong-coupling perturbation theory in a simple and exactly solvable model. The divergence of the perturbative series is effectively addressed using the technique of continued fractions. The results obtained from perturbation theory show good agreement with the exact solutions. The calculations are straightforward, making them suitable for teaching quantum mechanics or other theoretical physics courses, as well as for application in more complex research studies.

DOI
10.18173/2354-1059.2025-0005

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We study strong-coupling perturbation theory in a simple and exactly solvable model. The divergence of the perturbative series is effectively addressed using the technique of continued fractions. The results obtained from perturbation theory show good agreement with the exact solutions. The calculations are straightforward, making them suitable for teaching quantum mechanics or other theoretical physics courses, as well as for application in more complex research studies.
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Exactly solvable models in quantum mechanics are very rare.However, perturbation theory is a powerful tool to research the complex quantum systems and is widely used in theoretical physics and chemistry.For the particle weakly perturbed by the surrounding environment, its energy eigenstate and eigenvalue are separately calculated up to the first-order and second-order perturbation approximations in the undergraduate textbook of quantum mechanics.Whereas in reality, we have to calculate the high-order perturbation correction terms occasionally.Along the same lines of the low-order perturbation correction terms of energy eigenvalues and eigenstate for stationary state perturbation theory in the non-degenerate case, the third-order and fourth-order perturbation correction terms are strictly attained in this paper.Moreover, it points out the essence of perturbation theory and provides its derivation ideas.Students are no longer afraid of tedious derivation and can derive any-order perturbation correction.These results can serve as an extension and supplement for the quantum mechanics textbooks.Furthermore, it is very helpful for students to deeply understand quantum perturbation theory.