Dirac theory of one-electron systems

At a first glance, the method for finding the relativistic counterpart to the Schrödinger equation seems straight forward. We know, from classical theory of special relativity, that the total energy of an electron in an electromagnetic field defined by the scalar and vector potentials $\phi({\bf r})$ and $\bi{A}({\bf r})$ is given by

$$\{E+e\phi({\bf r})\}^2=c^2\{\bi{p}+e\bi{A}({\bf r})\}^2+m^2c^4.$$ (139)

Interpreting this relation quantum mechanically, we obtain the equation

$$\{ E +e\phi({\bf r})\}^2 \psi({\bf r}) = ( c^2 \{ {\bf p} + e\bi{A}({\bf r})\}^2 +m^2 c^4 ) \psi({\bf r})$$ (140)

where, as usual, ${\bf p} = -i\hbar \bnabla$ and $E=-i\hbar \frac{\partial }{\partial t}$This wave equation, known as the Klein-Gordon equation, is, however, unsatisfactory for a number of reasons. It does not incorporate the electron spin, and, more fundamentally, the time derivative enters this equation in second order. Therefore the solutions will not obey the superposition principle, and the wave function at the time t=0will not completely determine the system for all times. The square root of this equation is not Lorentz invariant, and time and spatial co-ordinates will not enter on equal footing.

Dirac (1928) showed the way out of this dilemma by postulating that the fundamental equation should have the more general form

$${\cal H}_D \psi({\bf r}) = E \psi({\bf r})$$ (141)

where

$${\cal H}_D = c \balpha \bdot \{{\bf p} + e\bi{A}({\bf r})\}- e\phi({\bf r}) +\beta mc^2$$ (142)

is the one-electron Dirac Hamiltonian. As in the non-relativistic case the Hamiltonian is required to be Hermitian and so

$$\balpha = \balpha^{\dagger}, \hspace{5 mm} \beta = \beta^{\dagger}.$$ (143)

In addition the solutions to the Dirac equation should, in accordance with the relation eq:REL1, be solutions also to the Klein-Gordon equation (the converse need not be true) leading to the commutation relations

 

$$\alpha_x^2 = \alpha_y^2 = \alpha_z^2 = \beta^2 = 1$$ (144)

$$[{\alpha}_x,{\alpha}_y]_{+} = [{\alpha}_y,\alpha_z]_{+} = [ \alpha_z,\alpha_x]_{+} = 0$$

$$[\alpha_x,\beta]_{+} = [\alpha_y,\beta]_{+} = [\alpha_z,\beta]_{+} = 0$$

where [A,B]₊ denotes the anti-commutator

[A,B]₊ = AB + BA. (145)

It can be shown that the conditions eq:cond1 and eq:cond2 may be satisfied if $\balpha$ and $\beta$ are taken as $(4 \times 4)$ matrices. Within this representation there is an infinite set of solutions to eq:cond1 and eq:cond2 all leading to the same physical result. A particularly useful set of matices for studying the non-relativistic limit is given by

$$\balpha=\left(\begin{array}{cc} {\bf0} & {\bsigma}\\ {\bsi... ...{\bi I} & {\bf0} \\ {\bf0} & {-\bi I}\\ \end{array}\right)$$ (146)

where the ${\bsigma}$ are the $(2\times 2)$ Pauli spin matrices, ${\bi I}$ and ${\bf0}$ the $(2\times 2)$ unit and zero matrices, respectively. Since $\balpha$ and $\beta$ are represented by $(4 \times 4)$ matrices, the relativistic wave function must be a $(4\times 1)$ colum matrix, or four-component spinor. In many cases it is convenient to split up the four-component spinor into two two-component spinors $\xi(\bf r)$ and $\eta(\bf r)$

$$\psi({\bf r}) =\left(\begin{array}{c} \xi({\bf r}) \\ \eta({\bf r}) \\ \end{array}\right)$$ (147)

where

$$\xi({\bf r}) =\left(\begin{array}{c} \psi_1({\bf r}) \\ \psi_2({... ...eft(\begin{array}{c} \psi_3({\bf r}) \\ \psi_4({\bf r}) \\ \end{array}\right)$$ (148)

in which case the Dirac equation takes the form

$$(E - mc^2) \xi({\bf r}) = c\bsigma \bdot \{{\bf p} +e {\bf A}({\bf r})\} \eta({\bf r}) -e \phi({\bf r})~ \xi({\bf r})$$

$$(E + mc^2) \eta({\bf r}) = c \bsigma \bdot \{ {\bf p} +e {\bf A}({\bf r})\} \xi({\bf r}) -e \phi({\bf r})~ \eta({\bf r})$$ (149)