Talk:Supersymmetric quantum mechanics

Relativity

I removed a comment about SUSY QM only applying to non-relativistic quantum mechanics, as this is not entirely accurate. As described in Thaller's textbook The Dirac Equation, problems involving relativistic charged particles in various potentials can be connected, through SUSY, to non-relativistic cases that are more easily soluble. Both Thaller and the 1995 review article cited on supersymmetry describe how to apply SUSY to an electron in a Coulomb potential, accounting for all relativistic effects. This gives an expression for the energy levels accurate to terms in α⁴, I believe, including lots of fine and hyperfine structure. (I have to check to be sure, but I believe this expression is too precise for its own good. Whether derived using SUSY trickery or not, the smallest term—of highest order in α—may be smaller than the Lamb shift, which is due to EM field quantization and isn't included in the Dirac formulation. I don't have all the equations in front of me, so don't take my word for it. Anyway, this "problem" isn't with the machinery used to get the answer, but with the answer itself, so it's really tangential to the point under discussion.)

The SUSY inherent in the Dirac Equation is most apparent in the case of a massless electron, which I admit is slightly unphysical. Still, the properties are there, and they do provide a practical aid to calculation.

Hey! The first snow of winter is falling outside.

Anville 17:02, 12 Nov 2004 (UTC)