5.6: Spin Precession
- Page ID
- 1209
The Hamiltonian for an electron at rest in a \( z\) -directed magnetic field, , is
According to Equation \ref{247}, the time evolution operator for this system is
The time evolution of the state ket is given by analogy with Equation \ref{456}:
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 \( \omega = \frac{g_n\, e\,B}{m_p}.\) 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\ref{470} 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.\( l\)
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)