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11.1: Introduction

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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

 

$\displaystyle a_0$ $\displaystyle = a^0,$ (1098)
$\displaystyle a_1$ $\displaystyle =-a^1,$ (1099)
$\displaystyle a_2$ $\displaystyle = -a^2,$ (1100)
$\displaystyle a_3$ $\displaystyle = -a^3.$ (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

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

 

 

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

 

$\displaystyle g_{00}$ $\displaystyle = 1,$ (1103)
$\displaystyle g_{11}$ $\displaystyle =-1,$ (1104)
$\displaystyle g_{22}$ $\displaystyle =-1,$ (1105)
$\displaystyle g_{33}$ $\displaystyle =-1,$ (1106)

 

 

with all other components zero. Thus,

 

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

 

 

Likewise,

 

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

\[p^{\,i} = -{\rm i}\,\hbar\,\frac{\partial}{\partial x^{\,i}}, \tag{1109}\]

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,

\[ p^{\,i} = {\rm i}\,\hbar\,\frac{\partial}{\partial x_i}, \tag{1110}\]

and then extend it to the complete 4-vector equation

\[p^{\,\mu} = {\rm i}\,\hbar\,\frac{\partial}{\partial x_{\mu}}. \tag{1111}\]

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.

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