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American Journal of Physics

Derivation of energy expressions in the Rayleigh–Schrödinger perturbation theory

Kalbe Razi Naqvi

Published Mar 1, 1987• 1 citation
Publisher

Abstract

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.

DOI
10.1119/1.15176

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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 energy function E( lambda ) of a quantum system with Hamiltonian H( lambda )=H 0 + lambda H 1 is determined by an analytical iterative procedure which yields a rich variety of functional forms. The wavefunction iterates include the solutions of Rayleigh-Schrodinger perturbation theory as well as some additional functions, all of which may be obtained sequentially by standard analytical techniques. At each step, an arbitrary ( lambda -dependent) constant may be chosen freely so as to improve the results of low-order iterates, thereby enhancing the convergence of the procedure.
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.
The constrained energy functional, defined by E0(a) =minSa{a〈Ψ‖Ĥ0‖Ψ〉a}, ai=a〈Ψ‖Âi‖Ψ〉a, i=1, ..., M, ‖Ψ〉a∈Sa, with a={ai} a preassigned set of expectation values generated by any state vector ‖Ψ〉a in the class Sa, is shown to satisfy a first order, but nonlinear, partial differential equation. The equation results by replacing the Lagrange multipliers introduced in an earlier paper by the corresponding partial derivatives of the to-be-found energy functional with respect to the constraints. Two alternative expressions for E0(a), one focusing upon the ground state eigenvalue of a modified Hamiltonian operator and the other upon the corresponding eigenvector, lead via standard Rayleigh–Schrödinger perturbation theory to two approximate partial differential equations for the energy functional. Both equations together with the imposed subsidiary conditions lead in turn to the same approximate E0(a). An example based on the linear harmonic oscillator illustrates the concepts presented in the formalism.
Rayleigh–Schrödinger perturbation theory is used to find energy and wavefunction corrections for a particle confined in a box with infinite walls. The resulting perturbation equations have simple, analytic solutions. The procedure for normalization of the wavefunction corrections is demonstrated, and the perturbation energies and normalized wavefunctions are shown to approach the exact values. The method is applied to both the ground state and first excited state and can become a valuable component of a student’s introduction to perturbation analysis.
The treatment of Schrödinger's perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. It is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. The solution of the Schrödinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an alternative insight in the mathematical structure of perturbation theory, particularly with respect to the ``bracketing theorem'' and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately.