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American Journal of Physics

Derivation of the perturbation formulas for the energy in quantum mechanics

Luis Gottdiener

Published Sep 1, 1978• 1 citation
Publisher

Abstract

A method to derive the formulas for the energy appearing in the Rayleigh-Schrödinger perturbation theory in quantum mechanics is presented. The procedure, which we consider to have some advantages over the usual approach, aims at solving the secular equation D (E) =0. The first-, second-, and third-order formulas are derived by this method.

DOI
10.1119/1.11511

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A method to derive the formulas for the energy appearing in the Rayleigh-Schrödinger perturbation theory in quantum mechanics is presented. The procedure, which we consider to have some advantages over the usual approach, aims at solving the secular equation D (E) =0. The first-, second-, and third-order formulas are derived by this method.
It is shown that repeated differentiation, with respect to the perturbation parameter λ, of the equation 〈ψm (λ)‖H0+λV‖ψn(λ)〉 =En(λ)δmn, and methodical application of the closure relation ∑‖ψn〉〈ψn‖=1, provide a painless technique for deriving the energy corrections in the Rayleigh–Schrödinger perturbation theory. The procedure is compared with one based on solving the secular equation D(E)=0, where D is the determinant whose elements are 〈ψi‖H0+V‖ψj〉−Eδij.
The authors develop a simple alternative approach to perturbation theory in one-dimensional nonrelativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve cumbersome sums over intermediate states which appear in the usual Rayleigh-Schrodinger expansion. Unlike standard logarithmic perturbation theory, the approach does not utilize taking the logarithm of the wavefunction, and is therefore applicable in the same form to both the ground state and excited bound states.
Rayleigh-Schrödinger time-independent perturbation theory is a key method in quantum mechanics for solving problems where the Hamiltonian of a quantum system cannot be exactly solved. By expanding the solutions in series, this method allows one to obtain approximations for quantum states and eigen-energies, especially for systems where the interactions or perturbing forces are small compared to the principal part of the Hamiltonian. This is crucial for understanding complex phenomena, predicting the behaviours of atomic and molecular systems, and calculating corrections to basic theoretical models, thereby improving the accuracy of quantum predictions. In this work, we will show that using the unitary operators, we arrive at the equations giving the different order of corrections of both energy as well as wave functions of the time-independent perturbation theory. Our approach closely mirrors the famous formalism of Rayleigh-Schrodinger perturbation theory (RSPT). The use of unitary operators simplifies the derivation of higher-order corrections, making the process more accurate and accessible to researchers. This technique harnesses the potential of unitary operators within a perturbation series framework. By combining these two powerful techniques, we offer a comprehensive framework for addressing normalization challenges in quantum mechanical calculations. This methodology not only facilitates a deeper understanding of perturbation theory but also serves as a pedagogical stepping stone, encouraging postgraduate researchers to apply their foundational knowledge and acquired skills in practical problem-solving situations with confidence. This is therefore a second proof or a new method to derive the fundamental equations of Rayleigh Schrodinger perturbation Theory RSPT using the unitary transformations.
We propose a new derivation of Time-Independent Perturbation Theory (PT) that has a fundamental advantage over the usual derivations presented in textbooks on Quantum Mechanics (QM): it is simpler and much shorter. As such, it can provide an easier and quicker way for students to learn PT, than afforded by current methods. In spite of that, our approach does not require the potentials to be energy independent or the inverse free Green function G0−1(E)G_0^{-1}(E)G0−1​(E) to be a linear function of energy EEE, as is the case in QM, and can be applied directly to various extensions of QM including Relativistic QM, the Bethe-Salpeter equation, and all kinds of quasipotential approaches in Quantum Field Theory.