Solutions of the Dirac equation

The commutation relations established in the previous section implies that ${\cal H}_D, {\bf j}^2, j_z$ and $\beta \Pi$ define a set of mutually commuting operators. The simultaneous eigenfunctions to these operators can be written

$$\psi({\bf r}) =\left(\begin{array}{c} \xi({\bf r}) \\ \eta({\bf ... ...heta,\varphi) \\ iQ(r) \Omega_{l'sjm_j}(\theta,\varphi) \\ \end{array}\right)$$ (158)

where $\Omega_{lsjm_j}(\theta,\varphi)$ are spherical two-spinors

$$\Omega_{lsjm_j}(\theta,\varphi) = \sum_{m_l m_s} \langle lm_... ...ert jm \rangle Y_{lm_l}(\theta,\varphi) \chi_{\frac{1}{2}m_s}.$$ (159)

The spherical two-spinors are obtained by coupling the spherical harmonics $Y_{lm_l}(\theta,\varphi)$ and spin functions

$$\chi_{\frac{1}{2} \frac{1}{2}} = \left(\begin{array}{c} 1\\ 0\\ ... ...frac{1}{2} -\frac{1}{2}} = \left(\begin{array}{c} 0\\ 1\\ \end{array}\right),$$ (160)

and thus $\vert l-1/2\vert \le j \le l+1/2$ and m_j = -j,-j+1,..,j-i,j. In order for $\psi({\bf r})$ to be an eigenfunction of the parity operator l' must be related to l according to

$$l' = \left\{ \begin{array}{c} l+1 \hspace{3 mm}\mbox{for} \hspace... ... l-1 \hspace{3 mm}\mbox{for} \hspace{3 mm} j = l - 1/2 \\ \end{array}\right. .$$ (161)

In relativistic theory it is convenient to define a quantum number $\kappa$

$$\kappa = \pm \left( j + \frac{1}{2} \right)\hspace{3 mm} \mbox{for} \hspace{3 mm} l \pm \left( l + \frac{1}{2} \right)$$ (162)

and to express the eigenfunctions in terms of this quantum number in which case we have

$$\psi({\bf r}) = \frac{1}{r} \left(\begin{array}{c} P(r) \Omega_{\... ...a,\varphi) \\ iQ(r) \Omega_{-\kappa m}(\theta,\varphi) \\ \end{array}\right).$$ (163)

Inserting this expression into the Dirac equation and using the properties of the spherical two-spinors, we obtain a system of equations for the radial functions

$$\hbar \left( \frac{d}{dr} + \frac{\kappa}{r} \right) P(r) - \frac{1}{c} \left( E + mc^2 -V(r) \right) Q(r) = 0$$

$$\hbar \left( \frac{d}{dr} - \frac{\kappa}{r} \right) Q(r) + \frac{1}{c} \left( E - mc^2 -V(r) \right) P(r) = 0.$$ (164)

Since $\psi({\bf r})$ must be everywhere finite, P(r) and Q(r) should satisfy the boundary conditions P(0)=Q(0)=0. For $\psi({\bf r})$ to be square integrable, it is necessary that both P(r) and Q(r) go to zero as r goes to infinity. Solutions to the system of radial equations satisfying the boundary conditions at the origin and at the infinity exist only for certain discrete energy values. Since the electron rest mass energy mc² is included in the Dirac Hamiltonian, the discrete energy values are shifted with roughly E=mc² compared to the non-relativistic case. For E > mc² the electron is no longer bound to the nucleus and it behaves, for large r, essentially like a free particle. In this case there is a solution to the radial equations for every E, and a continuous spectrum of eigenvalues adjoins the discrete levels. For wave functions belonging to the discrete and the positive continous spectrum it can be shown that P(r) is of the order c larger than Q(r). These components of the radial functions are therefore referred to as the large and small components.

In addition to the positive continuous eigenvalue spectrum there is a corresponding negative one in the range $-\infty < E < -mc^2$. The wave functions corresponding to the negative energies describe positron states. That is, states of a particle with the same mass as the electron but with the opposite charge. We will not be concerned with the negative energy solutions but refer the reader to Greiner (1990) Chap. 12 for a comprehensive treatement.