Question

With the standard Dirac Hamiltonian plus Coulomb potential below:

$H=\vec{\alpha}\cdot \vec{p}+m\gamma ^{o}+V(r),\; \; \; \vec{j}=\vec{L}+\frac{1}{2}\vec{\Sigma }$

a) Show that $\left [ \vec{j},\; H \right ]= 0$ .

b) Show that $\left [ K,\; H \right ]= 0$ , where $K=\gamma ^{o}\left ( \vec{\Sigma }\cdot \vec{j} -\frac{1}{2}\right )$ .

c) Show that $\vec{j^{2}}=K^{2}-\frac{1}{4}$ .

d) Since $H,\, \vec{J^{2}},\, J_{z},\, K,$ all mutually commute, they should have common eigenfunctions, and thus using (c), find the eigenvalues of K² and K, in terms of j.