Commutation relations for the Dirac Hamiltonian

We now consider the commutation relations for the central field Hamiltonian

$${\cal H}_D = c \balpha \bdot {\bf p} + V(r) +\beta mc^2,$$ (150)

which is the Hamiltonian relevant for describing an electron moving in the Coulomb field of an infinitely heavy nucleus. Contrary to the non-relativistic theory, the central field Hamiltonian does not commute with the orbital angular momentum operator. Instead we have

$$[{\cal H}_D,{\bf l}]= - i\hbar c \balpha \times {\bf p}$$ (151)

Defining the four-component operator, ${\bf s}$, in terms of the Pauli spin matrices

$${\bf s} = \frac{\hbar}{2} \left(\begin{array}{cc} {\bsigma} & {\bf0}\\ {\bf0} & {\bsigma}\\ \end{array}\right)$$ (152)

it follows from the commutation relations of the latter that

$$[s_x,s_y]= i\hbar s_z$$ (153)

and

$${\bf s}^2\psi({\bf r}) = \frac{3}{4} \hbar^2 \left(\begin{array}{cc} {\bf I} & {\bf0}\\ {\bf0} & {\bf I}\\ \end{array}\right) \psi({\bf r})$$ (154)

Thus ${\bf s}$ can be interpreted as an angular momentum operator with quantum numbers s=1/2 and $m_s = \pm 1/2$. We will refer to this operator as the spin angular momentum operator. Using the commutation relations of the Pauli matrices, it is easy to show that

$$[{\cal H}_D,{\bf s}]= + i\hbar c \balpha \times {\bf p}.$$ (155)

If we now take the total angular momentum of the electron as the sum of the orbital and spin angular momenta ${\bf j} = {\bf l} + {\bf s}$, it is see that ${\bf j}$ commutes with the Dirac Hamiltonian, that is

$$[{\cal H}_D,{\bf j}]= 0.$$ (156)

The interpretation of these commutation relations is that there now is an interaction, in accordance with experiment, between the orbital and spin angular momenta of the electron.

In addition to the angular symmetry, the Dirac Hamiltonian has an inversion symmetry and we need to investigate the commutation relations for the parity operator. Since this Hamiltonian is linear in the momentum operator ${\bf p}$ it does not commute with the ordinary non-relativistic parity operator $\Pi$. From the relations $[\balpha, \beta]_{+}=0$ it is, however, seen that the desired commutation relation is restored for a parity operator of form $\beta \Pi$and we have

$$[{\cal H}_D,\beta \Pi]= 0.$$ (157)

In addition to the commutation relation with the Hamiltonian, the parity operator commutes with the total angular momentum operator.