Relativistic Corrections to a Schrodinger Problem with a Kratzer Potential and a Pseudoscalar Coulomb Term
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint eJournal |
| Terbitan: |
, 2019
|
| Subjects: | |
| Online Access: |
https://zenodo.org/record/3483592 |
Daftar Isi:
- In (1), an exact solution is found for the Dirac equations for a 1D problem with a scalar potential of -2D( a/|x| -.5 qaa/ (xx) and a vector potential of -b/ |x| . The Dirac equations are first decoupled to obtain two second order DEs and these are solved as hypergeometric equations, after a transformation. An expression for energy is found which depends on rest mass. The objective of this note is to try to use the method of (2),(3) to find the corresponding nonrelativistic Schrodinger equation and then a relativistic correction to its energy. We show the ground state Schrodinger solution may be obtained by inspection. We also obtain the first order relativistic energy correction. The form of the DE needed to compute this correction, however, is the same as the Schrodinger equation and the full relativistic case for this example, although constant coefficients are different. Thus, one may use the approach for solving the Schrodinger equation for the full problem and obtain a ground state result. An interesting result which emerges is the dependence of the Schrodinger energy on a series of various orders of m. The leading term is a constant (in terms of m) and the second term of order 1/sqrt(m). The next term is of order m-3/2. Finding the first relativistic correction yields a term of order 1/m which is in between these two values. Thus, the Schrodinger result seems to contain some “relativistic corrections”, but not others, as has been pointed out in the literature before.