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International Journal of Quantum Chemistry

Perturbation theory via the Ricatti equation

C. K. Au

Published Jan 1, 1982• 7 citations
Publisher

Abstract

Abstract We present summary results of a bound‐state perturbation theory for a one‐space and one‐time dimension nonrelativistic spinless (Schrödinger) particle, a relativistic spinless (Klein‐Gordon) particle, and a relativistic spin‐half (Dirac) particle in central fields due to scalar or fourth‐component vector‐type interactions for an arbitrary bound state. This is accomplished by the reduction of the wave equations to Ricatti form. This enables a decoupling between the pair of coupled first order differential equations on the large and small component Dirac wave functions or a decoupling of the second order differential equation in the Schrödinger or Klein‐Gordon equations. All corrections to the energies and wave functions, including corrections to the positions of the nodes in excited states, are expressed in quadratures in a hierarchial scheme, without the use of either the Green's function or the sum over intermediate states. For the ground states of a Schrödinger particle, it is possible to extend this technique to multidimension in the case where the perturbation is due to noncentral fields, for example, in the problem of a nonrelativistic hydrogen atom in a linear combination of multiple fields.

DOI
10.1002/qua.560210114

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Abstract We present summary results of a bound‐state perturbation theory for a one‐space and one‐time dimension nonrelativistic spinless (Schrödinger) particle, a relativistic spinless (Klein‐Gordon) particle, and a relativistic spin‐half (Dirac) particle in central fields due to scalar or fourth‐component vector‐type interactions for an arbitrary bound state. This is accomplished by the reduction of the wave equations to Ricatti form. This enables a decoupling between the pair of coupled first order differential equations on the large and small component Dirac wave functions or a decoupling of the second order differential equation in the Schrödinger or Klein‐Gordon equations. All corrections to the energies and wave functions, including corrections to the positions of the nodes in excited states, are expressed in quadratures in a hierarchial scheme, without the use of either the Green's function or the sum over intermediate states. For the ground states of a Schrödinger particle, it is possible to extend this technique to multidimension in the case where the perturbation is due to noncentral fields, for example, in the problem of a nonrelativistic hydrogen atom in a linear combination of multiple fields.
The Schrödinger, Klein–Gordon, and Dirac equations for the hydrogen atom are solved in D dimensions, where D may be noninteger. For D=1 the nondegenerate ground state with infinite binding energy is obtained as a limit as D tends to 1 for the Schrödinger equation. The corresponding solution of the Klein–Gordon equation is shown to be unacceptable, contradicting previous work. Unexpectedly, no bound state solutions of the Dirac equation with D=1 are found at all. A few comments are made on the D=3 and 2 cases.
Abstract Using the concept of distributions, the square root of the operator - Δ + m2 is taken in a mathematically well defined way for one component wave functions. A new representation of proper Lorentz transformations for one component wave functions makes it possible to construct a relativistic quantum mechanics for spin 0, comprising a Lorentz invariant wave equation, a scalar product, and a positive definite density satisfying, together with a current, a continuity equation, and coupling of scalar and vector potentials. Some interesting consequences of the theory concerning the concept of particle trajectory and velocity of propagation of the probability amplitude are discussed in detail. As an example of practical application a perturbation theory for discrete states is set up.
We discuss an alternative method of performing coupling constant perturbation expansions in nonrelativistic quantum mechanics. This method, called logarithmic perturbation theory, yields new expressions for any-order corrections En to an unperturbed bound-state energy which do not involve cumbersome sums over intermediate unperturbed states. In one dimension, these corrections En can be evaluated using a simple explicit form containing a small number of integrals. In more than one dimension the approach is systematic but computations require the solution of well-defined partial differential equations. For order n=2 this equation is identical to that appearing in the method of Dalgarno and Lewis. Numerous illustrative examples are presented.