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Consider a system consisting of a spin onehalf particle with no orbital
angular momentum (e.g., a bound electron)
placed in a uniform directed magnetic field, and then subject
to a small timedependent magnetic field rotating in the  plane at the
angular frequency .
Thus,

(1045) 
where and are constants, with . The rotating
magnetic field usually represents the magnetic component of an electromagnetic wave propagating along the axis. In this system,
the electric component of the wave has no effect. The Hamiltonian
is written

(1046) 
where

(1047) 
and

(1048) 
Here, and are the gyromagnetic ratio [see Eq. (1004)] and mass of the particle in question, respectively.
The eigenstates of the unperturbed Hamiltonian are the ``spin up'' and ``spin
down'' states, denoted and , respectively. Of course,
these states are the eigenstates of corresponding to the
eigenvalues and respectively (see Sect. 10).
Thus, we have

(1049) 
The timedependent Hamiltonian can be written

(1050) 
where and are the conventional raising and lowering
operators for spin angular momentum (see Sect. 10). It follows
that

(1051) 
and

(1052) 
It can be seen that this system is exactly the same as the twostate
system discussed in the previous subsection, provided that the
make the following indentifications:



(1053) 



(1054) 



(1055) 



(1056) 
The resonant frequency, , is simply the spin precession
frequency in a uniform magnetic field of strength (see Sect. 10.6). In the absence of the perturbation, the
expectation values of and oscillate because of the spin
precession, but the expectation value of remains invariant. If we
now apply a magnetic perturbation rotating at the resonant frequency then,
according to the analysis of the previous subsection, the system undergoes
a succession of spin flips,
, in addition
to the spin precession. We also know that if the oscillation frequency
of the applied field is very different from the resonant frequency
then there is virtually zero probability of the field triggering a
spin flip. The width of the resonance (in frequency) is determined by
the strength of the oscillating magnetic perturbation. Experimentalists
are able to measure the gyromagnetic ratios of spin onehalf
particles to a high degree of accuracy by placing the particles
in a uniform magnetic field of known strength, and then subjecting them to an oscillating
magnetic field whose frequency is gradually scanned. By determining the
resonant frequency (i.e., the frequency at which the particles
absorb energy from the oscillating field), it is possible
to determine the gyromagnetic ratio (assuming that the mass is known).
Next: Perturbation Expansion
Up: TimeDependent Perturbation Theory
Previous: TwoState System
Richard Fitzpatrick
20100720