5.6: Spin Precession

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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)

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