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# 5.6: Spin Precession

The Hamiltonian for an electron at rest in a -directed magnetic field, , is

 (463)

where

 (464)

According to Equation (247), the time evolution operator for this system is

 (465)

It can be seen, by comparison with Equation (440), that the time evolution operator is precisely the same as the rotation operator for spin, with set equal to . It is immediately clear that the Hamiltonian (463) causes the electron spin to precess about the -axis with angular frequency . In fact, Equations (451)-(453) imply that

 (466) (467) (468)

The time evolution of the state ket is given by analogy with Equation (456):

 (469)

Note that it takes time for the state ket to return to its original state. By contrast, it only takes times for the spin vector to point in its original direction.

We now describe an experiment to detect the minus sign in Equation (457). An almost monoenergetic beam of neutrons is split in two, sent along two different paths, and , and then recombined. Path goes through a magnetic field free region. However, path enters a small region where a static magnetic field is present. As a result, a neutron state ket going along path acquires a phase-shift (the signs correspond to states). Here, is the time spent in the magnetic field, and is the spin precession frequency

 (470)

This frequency is defined in an analogous manner to Equation (464). The gyromagnetic ratio for a neutron is found experimentally to be . (The magnetic moment of a neutron is entirely a quantum field effect). When neutrons from path and path meet they undergo interference. We expect the observed neutron intensity in the interference region to exhibit a variation, where is the phase difference between paths and in the absence of a magnetic field. In experiments, the time of flight through the magnetic field region is kept constant, while the field-strength is varied. It follows that the change in magnetic field required to produce successive maxima is

 (471)

where is the path-length through the magnetic field region, and is the de Broglie wavelength over 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 rotation to return a state ket to its original state. If it only took a rotation then would be half of the value given above, which does not agree with the experimental data.