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# 11.1: 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 , , to represent the Cartesian coordinates , , , respectively, and to represent . The time dependent wavefunction then takes the form . Adopting standard relativistic notation, we write the four 's as , for . A space-time vector with four components that transforms under Lorentz transformation in an analogous manner to the four space-time coordinates is termed a 4-vector, and its components are written like (i.e., with an upper Greek suffix). We can lower the suffix according to the rules

 (1098) (1099) (1100) (1101)

Here, the are called the contravariant components of the vector , whereas the are termed the covariant components. Two 4-vectors and have the Lorentz invariant scalar product

 (1102)

a summation being implied over a repeated letter suffix. The metric tenor is defined

 (1103) (1104) (1105) (1106)

with all other components zero. Thus,

 (1107)

Likewise,

 (1108)

where , , with all other components zero. Finally, if , and otherwise.

In the Schrödinger representation, the momentum of a particle, whose components are written , , , or , , , is represented by the operators

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

for . Now, the four operators $$\partial/\partial x^{\,\mu}$$ form the covariant components of a 4-vector whose contravariant components are written . 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.