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11.5: Electron Spin

  • Page ID
    1257
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    According to Equation \ref{1132}, the relativistic Hamiltonian of an electron in an electromagnetic field is

    \( \mbox{\boldmath\)\( \cdot({\bf p}+e\,{\bf A})+ \beta\,m_e\,c^2.\) \ref{1185}

    Hence,

    $ \left(\frac{H}{c}+\frac{e}{c}\,\phi\right)^2 = \left[\mbox{\boldm...
...mbox{\boldmath$\alpha$}\cdot({\bf p}+e\,{\bf A})\right]^{\,2} + m_e^{\,2}\,c^2,$ \ref{1186}

    where use has been made of Equations \ref{1118} and \ref{1119}. Now, we can write

    \( i=1,3\), where \( \Sigma_i = \left(\begin{array}{cc} \sigma_i& 0\\ [0.5ex]0& \sigma_i\end{array}\right).\) \ref{1189}

    Here, 0 and \( 1\) denote \( \sigma_i\) are conventional \( \gamma^5\,\gamma^5=1\) , and

    \( \left(\frac{H}{c}+\frac{e}{c}\,\phi\right)^2 = \left[\mbox{\boldmath\) \ref{1191}

    Now, a straightforward generalization of Equation \ref{508} gives

    \( \mbox{\boldmath\)\( \mbox{\boldmath\)\( \mbox{\boldmath\)\( {\bf a}\) and \( \Sigma\) . It follows that $ \left[\mbox{\boldmath$\Sigma$}\cdot({\bf p}+e\,{\bf A})\right]^{\...
...,\mbox{\boldmath$\Sigma$}\cdot ({\bf p}+e\,{\bf A})\times ({\bf p}+e\,{\bf A}).$ \ref{1193}

    However,

    $ ({\bf p}+e\,{\bf A})\times ({\bf p}+e\,{\bf A}) = e\,{\bf p}\time...
...\bf A} -{ \rm i}\,e\,\hbar\,{\bf A}\times \nabla = -{\rm i}\,e\,\hbar\,{\bf B},$ \ref{1194}

    where \( \left(\frac{H}{c}+\frac{e}{c}\,\phi\right)^2 = ({\bf p}+e\,{\bf A})^2 + m_e^{\,2}\,c^2+e\,\hbar\,\)\( \cdot{\bf B}.\)

    \ref{1195}

    Consider the non-relativistic limit. In this case, we can write

    \( \delta H\) is small compared to \( \delta H^{\,2}\) , and other terms involving \( \delta H\simeq -e\,\phi + \frac{1}{2\,m_e}\,({\bf p} + e\,{\bf A})^2 + \frac{e\,\hbar}{2\,m_e}\,\)\( \cdot{\bf B}.\) \ref{1197}

    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

    \( = - \frac{e\,\hbar}{2\,m_e}\,\)\( .\) \ref{1198}

    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

    \( \mbox{\boldmath\)\( x\) component of the electron's orbital angular momentum, \( {\rm i}\,\hbar\,\dot{L}_x = [L_x,H].\) \ref{1201}

    However, it is easily demonstrated that

    \( = 0,\) \ref{1202} \( = 0,\) \ref{1203} \( = {\rm i}\,\hbar\,p_z,\) \ref{1204} \( = -{\rm i}\,\hbar\,p_y.\) \ref{1205}

    Hence, we obtain

    \( \dot{L}_x = c\,\gamma^5\,(\Sigma_2\,p_z-\Sigma_3\,p_y).\) \ref{1207}

    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,

    \( [\Sigma_1,\gamma^5]\) \( [\Sigma_1,\Sigma_1]\) \( [\Sigma_1,\Sigma_2]\) \( [\Sigma_1,\Sigma_3]\) \( [\Sigma_1,H] = 2\,{\rm i}\,c\,\gamma^5\,(\Sigma_3\,p_y-\Sigma_2\,p_z),\) \ref{1213}

    which implies that

    \( \dot{L}_x +\frac{\hbar}{2}\,\dot{\Sigma}_1 = 0.\) \ref{1215}

    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

    \( = - \frac{e\,g}{2\,m_e}\,{\bf S},\) \ref{1216}

    where the gyromagnetic ratio \( g= 2.\)

    \ref{1217}

    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)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)

    This page titled 11.5: Electron Spin is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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