Introduction

The aim of this chapter is to develop a quantum mechanical theory of electron dynamics that is consistent with special relativity. Such a theory is needed to explain the origin of electron spin (which is essentially a relativistic effect), and to account for the fact that the spin contribution to the electron's magnetic moment is twice what we would naively expect by analogy with (non-relativistic) classical physics (see Section 5.5). Relativistic electron theory is also required to fully understand the fine structure of the hydrogen atom energy levels (recall, from Section 7.7, and Exercises 3 and 4, that the modification to the energy levels due to spin-orbit coupling is of the same order of magnitude as the first-order correction due to the electron's relativistic mass increase.)

In the following, we shall use $x^1$ , $x^2$ , $x^3$ to represent the Cartesian coordinates $x$ , $y$ , $z$ , respectively, and $x^0$ to represent $c\,t$ . The time dependent wavefunction then takes the form $\psi(x^0,x^1,x^2,x^3)$ . Adopting standard relativistic notation, we write the four $x$ 's as $x^{\,\mu}$ , for $\mu= 0,1,2,3$ . A space-time vector with four components that transforms under Lorentz transformation in an analogous manner to the four space-time coordinates $x^{\,\mu}$ is termed a 4-vector, and its components are written like $a^{\,\mu}$ (i.e., with an upper Greek suffix). We can lower the suffix according to the rules

$$a_0 = a^0,$$ (1098)

$$a_1 =-a^1,$$ (1099)

$$a_2 = -a^2,$$ (1100)

$$a_3 = -a^3.$$ (1101)

Here, the $a^{\,\mu}$ are called the contravariant components of the vector $a$ , whereas the $a_{\mu}$ are termed the covariant components. Two 4-vectors $a^{\,\mu}$ and $b^{\,\mu}$ have the Lorentz invariant scalar product

$$a^0\,b^0-a^1\,b^1-a^2\,a^2-a^3\,b^3 = a^{\,\mu}\,b_\mu= a_\mu\,b^{\,\mu},$$ (1102)

a summation being implied over a repeated letter suffix. The metric tenor $g_{\mu\,\nu}$ is defined

$$g_{00} = 1,$$ (1103)

$$g_{11} =-1,$$ (1104)

$$g_{22} =-1,$$ (1105)

$$g_{33} =-1,$$ (1106)

with all other components zero. Thus,

$$a_{\mu} = g_{\mu\,\nu}\,a^\nu.$$ (1107)

Likewise,

$$a^{\,\mu} = g^{\,\mu\,\nu}\,a_\nu,$$ (1108)

where $g^{00}=1$ , $g^{11}=g^{22}=g^{33}=-1$ , with all other components zero. Finally, $g_\nu^{~\mu}=g^{\,\mu}_{~\nu} =1$ if $\mu=\nu$ , and $g_\nu^{~\mu}=g^{\,\mu}_{~\nu}=0$ otherwise.

In the Schrödinger representation, the momentum of a particle, whose components are written $p_x$ , $p_y$ , $p_z$ , or $p^1$ , $p^2$ , $p^3$ , is represented by the operators

$$p^{\,i} = -{\rm i}\,\hbar\,\frac{\partial}{\partial x^{\,i}},$$ (1109)

for $i=1,2,3$ . Now, the four operators $\partial/\partial x^{\,\mu}$ form the covariant components of a 4-vector whose contravariant components are written $\partial /\partial x_{\mu}$ . So, to make expression (1109) consistent with relativistic theory, we must first write it with its suffixes balanced,

$$p^{\,i} = {\rm i}\,\hbar\,\frac{\partial}{\partial x_i},$$ (1110)

and then extend it to the complete 4-vector equation

$$p^{\,\mu} = {\rm i}\,\hbar\,\frac{\partial}{\partial x_{\mu}}.$$ (1111)

According to standard relativistic theory, the new operator $p^0={\rm i}\,\hbar\,\partial/\partial x_0$ , which forms a 4-vector when combined with the momenta $p^{\,i}$ , is interpreted as the energy of the particle divided by $c$ , where $c$ is the velocity of light in vacuum.