Spin relevant invariants and the general solution of the Dirac equation for the Coulomb fields. (English) Zbl 1487.81081
Summary: The Dirac equation with the Coulomb potential is studied. A new spin relevant invariant is introduced, which does not commute with the Dirac and Johnson-Lippmann integrals of motion. The solutions of the Dirac equation compatible with each of the above three invariants are found and shown to be different. The explicit expressions for the Dirac bispinors are found for each of the three spin related invariants. The generalized invariant is constructed and corresponding to it general solution of the Dirac equation with the Coulomb potential is obtained. The general solution contains free parameters, whose particular values reduce it to any of the three particular solution. It is shown that the distributions of the electron probability density and expectation values of the electron spin direction in the hydrogen-like energy spectrum depend essentially on the spin relevant invariant. This demonstrates physical difference of the states corresponding to different invariants, which can be experimentally manifested.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81R25 | Spinor and twistor methods applied to problems in quantum theory |
| 81R20 | Covariant wave equations in quantum theory, relativistic quantum mechanics |
| 35L65 | Hyperbolic conservation laws |
| 15A66 | Clifford algebras, spinors |
| 81V10 | Electromagnetic interaction; quantum electrodynamics |
| 81V45 | Atomic physics |
| 35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |
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