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International Journal of Science and Research (IJSR)

On the Extensions of Virial and Feynman - Hellman Theorem

Pratyusha Biswas Deb, S Brajamani Deceased, HC Mohanto, Soumya Das, P S Majumdar

Published Feb 5, 2022• 0 citations
Publisher

Abstract

In the present work off diagonal and higher order extensions of Feynman-Hellman theorem is studied Rayleigh-Schrodinger perturbation theory for the degenerate case is redefined using a new technique. We also consider off-diagonal extension of quantum mechanical virial theorem by considering the case of linear harmonic oscillator.
DOI
10.21275/sr22202145556

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Epstein’s [Am. J. Phys. 22, 613 (1954)] off-diagonal and higher-order extensions of the Feynman–Hellmann theorem, obtained by using the basic technique of parameter differentiation under the integral sign, are further pursued. Epstein’s rederivation of the Rayleigh–Schrödinger perturbation expansion is also extended to include the degenerate case. The same approach is also used to obtain the Lennard-Jones–Brillouin–Wigner perturbation theory. The quantum virial theorem and its off-diagonal generalization is deduced and its application is illustrated by taking the example of the linear harmonic oscillator. The semiclassical expression for the kinetic energy is obtained directly from the quantization condition.
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This report discusses two new ideas for using perturbation methods to solve the time-independent Schrödinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix diagonalization methods. That allows a simple, compact derivation of the standard (Rayleigh-Schrödinger) equations. But it also leads to a new iterative solution method that is not based on the usual power series expansion. The iterative method can also be used when the unperturbed system is two-fold degenerate. The second concept is quite different from the first but compatible with it. It is based on the fact that one can replace the true Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This approach allows one to cancel part of the perturbation and to reduce the size of the off-diagonal matrix elements, giving better convergence of the series. These methods are illustrated by application to three perturbed harmonic oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which involves degenerate states. The iterative method and the synthetic Hamiltonian method are shown to give significantly better results than the standard method.
There exist several forms of perturbation theory in Quantum Mechanics, namely: non-degenerate perturbation theory, degenerate perturbation theory, time-dependent perturbation theory, time-constant perturbation theory and time-harmonic perturbation theory. This paper presents an extension of the perturbation theory for non-degenerate states. It has been observed that most texts and journal papers treat only the first –order and second-order non-degenerate perturbation theory. Therefore, this paper attempts to treat and present the third order and fourth order non-degenerate perturbation theory in quantum mechanics. It can thus be asserted that the higher the order of perturbations of quantum systems that we know, the more successful will be our efforts in suppressing or eliminating them.