Features of Dirac Radial Functions

While the mathematical expressions for the radial functions are the more complicated the larger the principal quantum number is, the graphical representation of basic features can easily be sketched as in the nonrelativistic case. By convention, the large components Pj(r) are chosen to start with positive values from the origin of the r coordinate. 

An important feature of the radial functions of the 4-spinor is the number of (radial) nodes, i.e., the number of positions where these functions are zero. This number is determined by the polynomial prefactor to the exponential function of the radial functions. Hence, the number of nodes solely depends on the quantum numbers that are used to classify the spinor. PnK f) always has n — I — 1) radial nodes like its nonrelativistic analog, the Schrodinger radial function Pni f)- However, QnK ) has as many nodes as PnK ) for negative values of k and one additional node for positive values of jc, [126, p. 754]. Interestingly enough, all nodes of PnK(f) lie before (from the origin s point of view) the absolute extremum of the function. However, the Qhk( ) function has some nodes behind its absolute extremum. 

Finally, we should note that spin-orbit coupling is the reason for the splitting of the p shells into pi/2 and p /2, the d shells into d /2 and 5/2 and so on. 

The effect of spin-orbit splitting is reduced for increasing principal quantum number n (compare the energy eigenvalues given in Table 6.4). 

Spectrum of Dirac Hydrogen-like Atoms with Coulombic Potential 

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