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Physical Review

Application of the Rayleigh-Schrödinger Perturbation Theory to the Hydrogen Atom

Richard E. Trees

Published Jun 15, 1956• 38 citations
Publisher

Abstract

Wigner has calculated the ground-state energy of the hydrogen atom with second-order perturbation theory, the whole electrostatic potential being considered as the perturbation. Though his result is finite, it does not agree with the known energy for hydrogen. A fact implicit in the literature, but not usually appreciated in this connection, is pointed out---that if part of the electrostatic energy is retained in the zero-order problem, a correct result is obtained, even if the part of the energy retained is given a zero magnitude after the calculation is carried out. It is also shown that if the calculation is carried out in an Einstein hypersphere, the entire "electrostatic potential" can be regarded as the perturbation and the correct result is obtained. Perturbation theory leads to the exact energy eigenvalue in all approximations higher than the first, but the expansion of the eigenfunction converges slowly if the perturbation is made large.

DOI
10.1103/physrev.102.1553

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Wigner has calculated the ground-state energy of the hydrogen atom with second-order perturbation theory, the whole electrostatic potential being considered as the perturbation. Though his result is finite, it does not agree with the known energy for hydrogen. A fact implicit in the literature, but not usually appreciated in this connection, is pointed out---that if part of the electrostatic energy is retained in the zero-order problem, a correct result is obtained, even if the part of the energy retained is given a zero magnitude after the calculation is carried out. It is also shown that if the calculation is carried out in an Einstein hypersphere, the entire "electrostatic potential" can be regarded as the perturbation and the correct result is obtained. Perturbation theory leads to the exact energy eigenvalue in all approximations higher than the first, but the expansion of the eigenfunction converges slowly if the perturbation is made large.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
A formulation of Schr\"odinger perturbation theory is developed that gives a unified treatment of non-degenerate and degenerate cases, is unique, and has a nonzero radius of convergence under very general conditions. Two alternative procedures are given for finding perturbed eigenvectors, one of which is simpler for the nondegenerate case or for small finite degeneracy, the other simpler for infinite or large finite degeneracy. The low-order terms in the perturbation expansions of quantities used in applications are given. The perturbation theory formulated in this paper has the following advantages over the conventional Schr\"odinger and Brillouin-Wigner perturbation theories: (i) When the convergence criterion is satisfied, bounds on the error made in replacing an appropriate infinite perturbation series by its first nnn terms can be obtained. (ii) For the case of degeneracy, the conventional Schr\"odinger perturbation theory can break down under conditions to which the convergence of the perturbation theory developed in this paper are insensitive. (iii) There is no implicit dependence on the eigenvalue, such as appears in the Brillouin-Wigner perturbation theory. (iv) For the case of degeneracy, statistical information about the distribution of certain eigenvalues can be obtained without finding the individual eigenvalues. (v) The theory is applicable to a wider class of problems than the conventional Schr\"odinger and Brillouin-Wigner perturbation theories.
The second-order perturbative Stark effect on the ground state of hydrogen is a typical example presented in many standard texts on quantum mechanics. Some texts miss the fact that the scattering states are significant contributors to the perturbative energy correction. The inclusion of scattering states has wider applicability than to just the Stark effect. An explicit calculation involving a finite-square well with a perturbation is used to illustrate the importance of including scattering states into the calculation. The second-order correction to the ground-state energy is obtained in three distinct ways. The first involves altering the problem by imposing additional boundary conditions at large distances to make the positive-energy spectrum discrete. The second makes use of the continuum scattering states directly. The third bypasses the use of scattering states by solving a differential equation for the first-order correction to the wave function.
This paper discusses some practical problems in quantum mechanics, particularly finding solutions of the Schrödinger equation for some cases, as it is a pivotal part of this area. The article focuses on using perturbation theory to calculate the approximate ground state energy for both helium atom and quartic anharmonic oscillator. For the helium atom, conventional perturbation theory is used to solve for the approximate energy, whilst for the anharmonic oscillator, a slightly different approach, that is perturbation theory incorporating a parameter, is chosen to improve the precision of the approximation. The final relative errors for the ground state energy of helium atom model are, for the first order perturbation, , and for the second order, , and that for the anharmonic oscillator is . The study of perturbation theory gives insights into understandings of how it may be modified further to pursue a better level of precision. This facilitates development of new approximation techniques for solving quantum-mechanical issues.
A straightforward analytical calculation of the s-like energy spectrum of the hydrogen atom is performed entirely within Feynman's path integral formalism. For this purpose the integral transform W = ∫d3rβK(rβ,0), where K(rβ,0) is the density matrix of the hydrogen atom written as a path integral, is calculated by means of the exact summation of a ``modified'' perturbation expansion (W is expanded as a power series in β with β = 1/kT). Performing this summation is equivalent to solving a problem of moments with infinite moments. For a wide class of potentials the perturbation expansion for W converges faster than the power-series expansion for the exponential function (for the Coulomb potential the convergence rate of both expansions is the same). It is shown how the complete energy spectrum can be obtained by this method. It is also illustrated how the wavefunctions might be obtained by transforming W.