The solution of the quantum-mechanical eigenvalue problem is discussed for cases when a series of approximate eigenfunctions is known. If these ``unperturbed'' states are divided into two classes, a perturbation formula is derived giving the influence of one class of states on the other in the final solution. The formula contains as special cases: (i) the Schrödinger-Brillouin formula for the eigenvalue of a nondegenerate state, (ii) a new simple formula for treating a class of degenerate states, and (iii) the splitting of the secular equation in cases where the system naturally consists of two independent parts in mutual interaction.
The solution of the quantum-mechanical eigenvalue problem is discussed for cases when a series of approximate eigenfunctions is known. If these ``unperturbed'' states are divided into two classes, a perturbation formula is derived giving the influence of one class of states on the other in the final solution. The formula contains as special cases: (i) the Schrödinger-Brillouin formula for the eigenvalue of a nondegenerate state, (ii) a new simple formula for treating a class of degenerate states, and (iii) the splitting of the secular equation in cases where the system naturally consists of two independent parts in mutual interaction.
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 n 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.
A new approach to the eigenvalue problem in quantum mechanics is proposed. This approach is based on three propositions: 1) a perturbation theory which does not require knowledge of the entire eigenvalue spectrum of the unperturbed problem and which uses a nonlinearization procedure (leaving some latitude in the choice of a zeroth order approximation); 2) a relationship between the perturbation theory and a variational principle, namely that any variational calculation is none other than the first two terms of some nontrivial perturbation theory which, when developed further, can reveal the accuracy of the variational calculations and can refine them by an iterative procedure; 3) Dyson's argument, which serves as a criterion for the reasonableness of the choice of a zeroth order approximation (the unperturbed problem). The realization of this perturbation theory in a k-dimensional space is equivalent to the solution of a k-dimensional electrostatic problem with a variable dielectric permittivity. In the one-dimensional case and in cases which reduce to the one-dimensional case, all the corrections are written in quadratures. It is shown that the construction of an ordinary perturbation theory (in which the zeroth order approximation is an exactly solvable problem) within the framework of this perturbation theory is a purely algebraic procedure, which reduces to the solution of some simple recurrence relations. An approximation analogous to the leading logarithmic approximation of quantum field theory is constructed. Some standard problems of quantum mechanics—the anharmonic oscillator, the Zeeman effect, and the Stark effect—are treated as examples. It is shown that this new approach makes it possible to develop systematically a theory for strong coupling and large perturbations.
There are few potentials V (r) for which the Schrödinger equation is exactly solvable. We must therefore develop approximation techniques to obtain the eigenvalues and eigenfunctions for potentials that do not lead to exactly soluble equations. One of these methods is the perturbation theory, which was developed both for time independent and time dependent phenomena. We will discuss the time independent perturbation theory. The formulation of the time independent perturbation theory is different for nondegenerate and degenerate states.
A new method for treating the most important eigenvalue problems in quantum mechanics is developed. The solutions can be found immediately once the equations are factorized by means of linear operators. These operators acting on a normalized eigenfunction change it into a new normalized eigenfunction and all solutions can be found once the basic eigenfunction is known. This basic eigenfunction is a solution of a simple differential equation of the first order. The underlying theory is explained more fully on a special case (Section 1) and then the rules of procedure are formulated explicitly (Section 2). The rest of the paper contains applications including the Kepler problem treated according to Dirac's theory.
The vast majority of problems in quantum mechanics cannot be solved exactly. As in all theories of physics approximation methods are therefore of great importance. Here we shall deal with time-independent problems, which means that we seek the eigenvalues and eigenstates of H . In perturbation theory we split H into two pieces, H = H 0 + H 1 , where we know the spectrum and eigenstates of H 0 , and H 1 is, in some sense, small. Needless to say, a split of this type is often not possible, or at least unknown. In that case one frequently replaces the system by a simpler model for which perturbation theory works, and which, it is hoped, captures some of the essential features of the actual problem at hand. This art of finding models is something that can only be taken up within the context of a particular subject (e.g., solid state or nuclear physics). Here we shall treat systems where good zero-order solutions are rather obvious.
Abstract In a previous paper the theory of perturbations was discussed for system which possess only discrete spectra. The theory is now extended to systems with both discrete and continuous spectra, and in addition a slightly more general method of perturbation is considered. The Hamiltonian H of the perturbed systems is split up into three parts H0, H1, H2 (t), two of which can be chosen arbitrarily. The solutions of the perturbed problem can then be worked out in terms of the characteristic functions corresponding to the Hamiltonian H0 + H2 (T), where T is any arbitrary fixed time. In general, however, the series of perturbations does not converge, and different modes of splitting up the Hamiltonian will lead to different formal results. The perturbation theory does usually give the correct physical results, and it is therefore necessary to give some explanation of how this happens. In (I) the asymptotic nature of certain solutions was emphasised, but this does not extend to the more general systems treated here. It is suggested that the usual boundary conditions do not give an adequate description of any but the simplest atomic problems, and that more detailed restrictions, determined by the experimental conditions, ought to be substituted. As this is impossible in practice, an alternative method is to regard the motion as a perturbation on that Hamiltonian H0 + H2 (T) which most nearly corresponds to the experimental conditions. (For example, we may regard an electron as free or bound according as the experiment determines the number of free or bound electrons respectively, whereas in the usual theory the initial Hamiltonian is taken as fundamental.) This criterion, together with the initial conditions, suffices to fix the division of H into the three parts H0, H1, H2 (t) which at first were chosen arbitrarily. The formal solution can be worked out on this basis, and it is then assumed that this solution is asymptotic to the one which would be obtained if the full boundary conditions were used. This suggestion is in accordance with the fact that a hydrogen atom in an electric field gives rise not only to the usual Stark effect, but also to an ionisation effect. In § 2 the perturbation equation are derived in a slightly more general form than has previously been given. The theories of Born,* Dirac and Oppenheimer are particular cases of that given here, and are obtained by particular choices of H0, H1, H2 (t). To obtain Born’s theory we take H1 = 0, and for Oppenheimer’s we take H2 (t) = H2 (T) and T = 0. In Dirac’s theory we have H1 = 0, H2 (0) = 0 and T = 0. Each of these theories is appropriate for the solution of special problems, but the more general method of this paper is required for the full discussion of the perturbation theory.