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5.5: Magnetic Moments

  • Page ID
    1208
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    Consider a particle of electric charge \( v\) performing a circular orbit of radius \( r\) in the \( y\) plane. The charge is equivalent to a current loop of radius \( r\) in the \( y\) plane carrying current \( \mu\) of the loop is of magnitude \( z\) -axis. Thus, we can write

    \( = \frac{q}{2}\, {\bf x} \times {\bf v},\) \ref{458}

    where \( {\bf x}\) and \( {\bf p} = {\bf v} /m\) , where \( m\) is its mass. We also know that \( {\bf L}\) is the orbital angular momentum. It follows that

    \( = \frac{q}{2\,m} \,{\bf L}.\) \ref{459}

    Using the usual analogy between classical and quantum mechanics, we expect the above relation to also hold between the quantum mechanical operators, \( {\bf L}\) , which represent magnetic moment and orbital angular momentum, respectively. This is indeed found to the the case.

    Spin angular momentum also gives rise to a contribution to the magnetic moment of a charged particle. In fact, relativistic quantum mechanics predicts that a charged particle possessing spin must also possess a corresponding magnetic moment (this was first demonstrated by Dirac--see Chapter 11). We can write

    \( = \frac{q}{2\,m} \left({\bf L} + g \,{\bf S}\right),\) \ref{460}

    where \(g\) is called the gyromagnetic ratio. For an electron this ratio is found to be

    \( 1/(2\pi\, 137)\), derived originally by Schwinger, is due to quantum field effects. We shall ignore this correction in the following, so \( \simeq - \frac{e}{2\,m_e} \left({\bf L} + 2 \,{\bf S}\right)\) \ref{462}

    for an electron (here, \(e> 0\)).

    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 5.5: Magnetic Moments is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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