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Proceedings of the Physical Society

Two theorems on the variation perturbation method for the excited states of stationary quantum systems

C. S. Sharma

Published May 1, 1967• 14 citations
Publisher

Abstract

Two theorems are proved which are of considerable use in understanding the situation which arises while applying the variation perturbation method for calculating the state vectors and the energies of stationary quantum systems. These theorems also enable us to ensure that the second-order energy calculated by the use of the method is a rigorous upper bound to the true value The earlier results of Sinanoulu and Miller follow as immediate corollaries of these theorems.

DOI
10.1088/0370-1328/91/1/310

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Two theorems are proved which are of considerable use in understanding the situation which arises while applying the variation perturbation method for calculating the state vectors and the energies of stationary quantum systems. These theorems also enable us to ensure that the second-order energy calculated by the use of the method is a rigorous upper bound to the true value The earlier results of Sinanoulu and Miller follow as immediate corollaries of these theorems.
It is pointed out that the notation of the theory of functionals and the theorem in it analogous to Taylor's theorem facilitates a compact discussion of perturbation and variation methods in quantum mechanics, both for bound state and scattering problems. With this aid it is shown (i) how stationary expressions required for variation methods may be sought and found, (ii) that the perturbation and variation methods are virtually identical, (iii) that second order perturbations and expressions stationary to the second order may be found similarly, though these are complicated. The form found for the second order perturbation of bound states differs from that commonly used in that it does not explicitly involve the eigenfunctions of states other than that perturbed.
In this paper, pursuing a new advised method called Delta method which is basically similar to variational method, we find the ground and excited states, according to a typical quantum Hamiltonian. Moreover, applying this method, the upper bound values for the eigenenergies of the so-called ground and excited states are estimated. We will show that this new method, is as beneficial as the traditional variational method which is common in deriving eigenenergies of some of the quantum Hamiltonians. This method helps physics students to broaden their knowledge about the possible mathematical ways; they can use to obtain eigenenergies of some quantum Hamiltonians. The advantage of Delta method to variational method is in its simplicity and reduction of the calculation procedures.
In this chapter, we present some fundamental issues about approximation methods that are often used when a quantum-mechanical system is perturbed and about the relationship between classical and quantum mechanics. In Sec. 10.1 we introduce the stationary perturbation theory, while Sec. 10.2 is devoted to time-dependent perturbations. In Sec. 10.3 we briefly examine the adiabatic theorem. In Sec. 10.4 we introduce the variation method, an approximation method that is not based on perturbation theory. In Sec. 10.5 we discuss the classical limit of the quantum-mechanical equations, whereas in Sec. 10.6 we deal with the semiclassical approximation, in particular the WKB method. On the basis of the previous approximation methods in Sec. 10.7 we present scattering theory. Finally, in Sec. 10.8 we treat a method that has a wide range of applications: the path-integral method. Stationary perturbation theory Perturbation theory is a rather general approximation method that may be applied when a small additional force (the perturbation) acts on a system (the unperturbed system), whose quantum dynamics is fully known. If the disturbance is small, it modifies both the energy levels and the stationary states. This allows us to make an expansion in power series of a perturbation parameter, which is assumed to be small. Perturbation theory may be applied both to the case where the additional force is time-independent (in which case a stationary treatment suffices – the subject of the present section) as well as to the case where it explicitly depends on time.
Chapter 13 is devoted to two important approximation methods used to solve time-independent problems, namely stationary perturbation theory and the variational method. Perturbation theory is developed for both non-degenerate states and degenerate states, with illustrative examples. The Stark effect is considered separately. The dielectric constant of a transparent medium as a function of frequency is shown to result from the modification of the wave function due to the perturbation of the radiation field. Two complementary topics are presented: the canonical transformation method and the Feynman-Hellman method. Finally, the possibilities of the variational method are illustrated by using it first to determine an upper bound for the energy of the stationary states of a quantum system, and then to study more complicated cases, including the partially shielded Coulomb potential produced by a neutral atom.
Abstract As a preliminary to a wide range of applications, conventional perturbation theory is re-examined and a number of useful properties emphasized. It is shown in particular that the total, kinetic and potential energies can be obtained to the (2s + 1)th order from a knowledge of only the sth-order wave function and that similar though less powerful theorems hold for other diagonal matrix elements. A combination of variational and perturbation techniques is suggested as the best method of calculating small disturbances of a stationary system.