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The partial-wave subspaces

Consider two functions / (r) and / (r) that are square-integrable on the interval 0 r oo. We can use them to define a function [Pg.86]

This is a function in the so-called partial wave subspace (or angular momentum subspace) L ((0,oo),dr) S fCmj,Kj- The norm of this function is given by [Pg.86]

Making the transition back to cartesian coordinates, we obtain a square inte-grable function (x) = (l/r)t/ (r, d, (f) (whose norm is again given by the above expression). An arbitrary function is a linear combination of the form [Pg.86]

We put everything together and obtain the following result Consider the Dirac operator with the potential [Pg.87]

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. [Pg.87]


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) ... [Pg.86]


See other pages where The partial-wave subspaces is mentioned: [Pg.86]    [Pg.87]   


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