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 \( x^2\) , \( x\) , \( z\) , respectively, and \( c\,t\) . The time dependent wavefunction then takes the form \( x\) 's as \( \mu= 0,1,2,3\) . A space-time vector with four components that transforms under Lorentz transformation in an analogous manner to the four space-time coordinates \( a^{\,\mu}\) (i.e., with an upper Greek suffix). We can lower the suffix according to the rules

\( = a^0,\) \ref{1098} \( =-a^1,\) \ref{1099} \( = -a^2,\) \ref{1100} \( = -a^3.\) \ref{1101}

Here, the \( a\) , whereas the \( a^{\,\mu}\) and \( a^0\,b^0-a^1\,b^1-a^2\,a^2-a^3\,b^3 = a^{\,\mu}\,b_\mu= a_\mu\,b^{\,\mu},\) \ref{1102}

a summation being implied over a repeated letter suffix. The metric tenor \( g_{00}\) \( g_{11}\) \( g_{22}\) \( g_{33}\) \( a_{\mu} = g_{\mu\,\nu}\,a^\nu.\) \ref{1107}

Likewise,

\( g^{00}=1\) , \( g_\nu^{~\mu}=g^{\,\mu}_{~\nu} =1\) if \( g_\nu^{~\mu}=g^{\,\mu}_{~\nu}=0\) otherwise.

In the Schrödinger representation, the momentum of a particle, whose components are written \( p_y\) , \( p^1\) , \( p^3\) , is represented by the operators

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

for \( \partial /\partial x_{\mu}\) . So, to make expression \ref{1109} consistent with relativistic theory, we must first write it with its suffixes balanced,

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

and then extend it to the complete 4-vector equation

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

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)

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