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5.6: Spin Precession

  • Page ID
    1209
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    The Hamiltonian for an electron at rest in a \( z\) -directed magnetic field, $ {\bf B}=
B\,{\bf e}_z$ , is

    \( \mbox{\boldmath\)\( \omega = \frac{e\,B}{m_e}.\) \ref{464}

    According to Equation \ref{247}, the time evolution operator for this system is

    \( {\mit\Delta}\varphi\) set equal to \( z\) -axis with angular frequency \( \langle S_x\rangle_t\) \( \langle S_y\rangle_t\) \( \langle S_z\rangle_t\) \( = \langle S_z\rangle_{t=0}.\) \ref{468}

    The time evolution of the state ket is given by analogy with Equation \ref{456}:

    $ \vert A, t\rangle = {\rm e}^{-{\rm i}\,\omega \,t/2}\, \langle +\...
...+ {\rm e}^{\,{\rm i}\,\omega \,t/2}\, \langle -\vert A, 0\rangle \vert-\rangle.$ \ref{469}

    Note that it takes time \( t=2\pi/\omega\) for the spin vector to point in its original direction.

    We now describe an experiment to detect the minus sign in Equation \ref{457}. An almost monoenergetic beam of neutrons is split in two, sent along two different paths, \( A\) and \( B\) , and then recombined. Path \( A\) goes through a magnetic field free region. However, path \( B\) enters a small region where a static magnetic field is present. As a result, a neutron state ket going along path \( B\) acquires a phase-shift \( \mp\) signs correspond to \( T\) is the time spent in the magnetic field, and \( \omega = \frac{g_n\, e\,B}{m_p}.\) \ref{470}

    This frequency is defined in an analogous manner to Equation \ref{464}. The gyromagnetic ratio for a neutron is found experimentally to be \( A\) and path \( B\) meet they undergo interference. We expect the observed neutron intensity in the interference region to exhibit a \( \delta\) is the phase difference between paths \( A\) and \( B\) in the absence of a magnetic field. In experiments, the time of flight \( T\) through the magnetic field region is kept constant, while the field-strength \( B\) is varied. It follows that the change in magnetic field required to produce successive maxima is

    \( l\) is the path-length through the magnetic field region, and \( 2\pi\) of the neutrons. The above prediction has been verified experimentally to within a fraction of a percent. This prediction depends crucially on the fact that it takes a \( 2\pi\) rotation then \( {\mit\Delta} B\) would be half of the value given above, which does not agree with the experimental data.

    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 5.6: Spin Precession is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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