The discussion focuses on the application of the Dirac equation to the hydrogen atom, specifically examining the potential used in this context. Participants explore whether the potential should remain the same as in the non-relativistic Schrödinger equation or if modifications are necessary.
Participants exhibit some agreement on the use of the 1/r potential in non-relativistic quantum mechanics, but there is uncertainty regarding the necessity of modifications for the Dirac equation. Multiple viewpoints on the potential's form and implications remain unresolved.
Some assumptions about the nature of the vector potential and its implications for the Dirac equation are not fully explored, and the discussion does not resolve the question of whether modifications to the potential are necessary.
tom.stoer said:In non-rel. QM one starts with the same 1/r potential. What else do you have in mind?
Choose a gauge in which the vector potential A = 0, leaving just the Coulomb 1/r potential. Separate variables in spherical coordinates as usual, and you'll find that the four components of the Dirac spinor can be written in terms of two radial functions, leding to recursion relations, etc, etc, and solved by confluent hypergeometric functions.I was just wondering that since the time-independent part of the solution is in vector form you might have to consider a potential in the form of a vector field. But maybe i´m wrong?
Bill_K said:Choose a gauge in which the vector potential A = 0, leaving just the Coulomb 1/r potential. Separate variables in spherical coordinates as usual, and you'll find that the four components of the Dirac spinor can be written in terms of two radial functions, leding to recursion relations, etc, etc, and solved by confluent hypergeometric functions.