11.5: Electron Spin
- Page ID
- 1257
According to Equation \ref{1132}, the relativistic Hamiltonian of an electron in an electromagnetic field is
Hence,
where use has been made of Equations \ref{1118} and \ref{1119}. Now, we can write
Here, 0 and \( 1\) denote \( \sigma_i\) are conventional \( \gamma^5\,\gamma^5=1\) , and
Now, a straightforward generalization of Equation \ref{508} gives
However,
where \( \left(\frac{H}{c}+\frac{e}{c}\,\phi\right)^2 = ({\bf p}+e\,{\bf A})^2 + m_e^{\,2}\,c^2+e\,\hbar\,\)\( \cdot{\bf B}.\)
Consider the non-relativistic limit. In this case, we can write
This Hamiltonian is the same as the classical Hamiltonian of a non-relativistic electron, except for the final term. This term may be interpreted as arising from the electron having an intrinsic magnetic moment
In order to demonstrate that the electron's intrinsic magnetic moment is associated with an intrinsic angular momentum, consider the motion of an electron in a central electrostatic potential: i.e., \( {\bf A}={\bf0}\) . In this case, the Hamiltonian \ref{1185} becomes
However, it is easily demonstrated that
Hence, we obtain
It can be seen that \( x\) -component of the total angular momentum of the system must be a constant of the motion (because a central electrostatic potential exerts zero torque on the system). Hence, we deduce that the electron possesses additional angular momentum that is not connected with its motion through space. Now,
which implies that
Since there is nothing special about the \( {\bf L} + (\hbar/2)\,\)\( \Sigma\) is a constant of the motion. We can interpret this result by saying that the electron has a spin angular momentum \( \Sigma\) , which must be added to its orbital angular momentum in order to obtain a constant of the motion. According to \ref{1198}, the relationship between the electron's spin angular momentum and its intrinsic (i.e., non-orbital) magnetic moment is
where the gyromagnetic ratio \( g= 2.\)
As explained in Section 5.5, this is twice the value one would naively predict by analogy with classical physics.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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