Radial Density and Expectation Values

Owing to the radial symmetry of atoms, all angular- and spin-dependent terms can be treated analytically so that these degrees of freedom can be integrated out. Only the radial-dependent terms remain in expectation values. This still holds in the case of many-electron atoms as we shall see in chapter 9. 

If we can transform a one-electron operator d to spherical coordinates in such a way that 

After integration of the angular- and spin-dependent terms, 

Since the radial functions Pj(r) and Qi r) do not, in general, feature nodes at the same radial position, the resulting radial density will not show any nodes — apart from the one at the origin r = 0. This is in sharp contrast to the nonrelativistic radial density which is the square of the nonrelativistic 

It is occasionally argued that the missing nodes in the Dirac radial density explain how electrons get across the nodes [129] as there are no longer any nodes. However, one must keep in mind that this is a somewhat artificial question as time has been eliminated from the stationary Dirac and Schrodinger equations. The question remains why the density dramatically 

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