Next: Timedependent perturbation theory
Up: Timeindependent perturbation theory
Previous: The Zeeman effect
Contents
The proton in a hydrogen atom is a spin onehalf charged particle, and therefore
possesses a magnetic moment. By analogy with Eq. (738),
we can write

(983) 
where
is the proton magnetic
moment, is the proton spin, and the proton gyromagnetic ratio is found experimentally to take that value . Note that the
magnetic moment of a proton is much smaller (by a factor of order )
than that of an electron.
According
to classical electromagnetism, the proton's magnetic moment generates a
magnetic field of the form

(984) 
where
. We can understand the origin of the deltafunction term
in the above expression by thinking of the proton as a tiny current loop centred on the origin.
All magnetic fieldlines generated by the loop must pass through the loop.
Hence, if the size of the loop goes to zero, then the field will be infinite at the origin, and this contribution is what is reflected by the deltafunction term. Now, the Hamiltonian of the electron in the magnetic
field generated by the proton is simply

(985) 
where

(986) 
Here,
is the electron magnetic moment [see Eqs. (738)
and (739)], and the electron spin. Thus, the
perturbing Hamiltonian is written

(987) 
Note that, since we have neglected coupling between the proton
spin and the magnetic field generated by the electron's orbital motion,
the above expression is only valid for states.
According to standard firstorder perturbation theory, the energyshift induced
by spinspin coupling between the proton and the electron is the expectation
value of the perturbing Hamiltonian. Hence,

(988) 
For the groundstate of hydrogen, which is spherically symmetric,
the first term in the above expression vanishes by symmetry.
Moreover, it is easily demonstrated that
. Thus, we obtain

(989) 
Let

(990) 
be the total spin. We can show that

(991) 
Thus, the simultaneous eigenstates of the perturbing Hamiltonian
and the main Hamiltonian are the simultaneous eigenstates of ,
, and . However, both the proton and
the electron are spin onehalf particles. According to Sect. 11.4,
when two spin onehalf particles are combined (in the absence of orbital
angular momentum) the net state has either spin 1 or spin 0.
In fact, there are three spin 1 states, known as triplet states, and a single
spin 0 state, known as the singlet state. For all states,
the eigenvalues of and are
.
The eigenvalue of is 0 for the singlet state, and
for the triplet states. Hence,

(992) 
for the singlet state, and

(993) 
for the triplet states.
It follows, from the above analysis, that spinspin coupling breaks
the degeneracy of the two states in hydrogen, lifting the
energy of the triplet configuration, and lowering that of the singlet.
This splitting is known as hyperfine structure.
The net energy difference between the singlet and the triplet states
is

(994) 
where
is the (magnitude of the) groundstate energy.
Note that the hyperfine energyshift is much smaller, by a factor , than
a typical fine structure energyshift.
If we convert the above energy into a wavelength then we obtain

(995) 
This is the wavelength of the radiation emitted by a hydrogen atom
which is collisionally excited from the singlet to the triplet
state, and then decays back to the lower energy singlet state.
The 21cm line is famous in radio astronomy because it was used to
map out the spiral structure of our galaxy in the 1950's.
Next: Timedependent perturbation theory
Up: Timeindependent perturbation theory
Previous: The Zeeman effect
Contents
Richard Fitzpatrick
20061212