Perturbation theory via the Ricatti equation

✍ Scribed by C. K. Au

• Book ID: 104580570
• Publisher: John Wiley and Sons
• Year: 1982
• Tongue: English
• Weight: 531 KB
• Volume: 21
• Category: Article
• ISSN: 0020-7608
• DOI: 10.1002/qua.560210114

✦ Synopsis

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.