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Homework Statement
Exact spin symmetry in the Dirac equation occurs when there is both a scalar and a vector potential, and they are equal to each other. What physical effect is absent in this case, that does exist in the Dirac solution for the hydrogen atom (vector potential = Coulomb and scalar potential = 0)?
Homework Equations
Hydrogen atom Dirac equation:
##(\vec{\alpha} \cdot \vec{p}c+\beta mc^2)\psi = (E-V^v_0(r))\psi ##
##V^v_0(r) = -e^2/r##
Spin symmetry Dirac equation:
##(\vec{\alpha} \cdot \vec{p}c+\beta (mc^2+V_s(r)))\psi = (E-V^v_0(r))\psi ##
##V^v_0(r) = V_s(r)##
The Attempt at a Solution
Taking the non relativistic limit of both equations I found that it was hard to solve for the non relativistic energy in the hydrogen atom case, but the spin-symmetry case immediately led to a Schrodinger-type equation with ##E=E_{NR}##. However I don't know what this says about the hydrogen atom case, more specifically what physical effect is missing in the spin symmetry case that makes it easy to solve for the energy.