# 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.

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)