Partial Wave Dirac Operator

The Dirac operator H is thus unitarily equivalent to a direct sum of all these partial wave Dirac operators. It is therefore sufficient to solve the eigenvalue... 

The partial wave Dirac operator is independent of the quantum number ruj. Hence, if solve the above eigenvalue equation, then there are 2j + 1 orthogonal eigenfunctions of H that belong to the same eigenvalue E. These eigenfunctions are labelled by the quantum number mj = —j, j + 1,..., +j in... 

For example, the action of K is just multiplication by the eigenvalue —Kj. The action of the Dirac matrices / and a in the partial wave subspace is described by (110). Likewise, we can compute the action of a spherically symmetric potential in one of the angular momentum subspaces. It remains to observe that due to the factor r in (102) the operator djdr - 1/r in (which is part of expression for the Dirac operator in polar coordinates) simply becomes d/dr in L (0,oo) ... 

In particular, the Dirac operator leaves each of the partial wave subspaces invariant, that is, the result of its action is a wave function in the same angular momentum wave subspace. 

It was found [19], that the partial wave increments E of the leading relativistic correction E2 converge even more slowly, namely - for the Dirac-Coulomb operator - as... 

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