where

(727) |

where , and . We can understand the origin of the delta-function term in the above expression by thinking of the proton as a tiny current loop centred on the origin. All magnetic field-lines 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 delta-function term. Now, the Hamiltonian of the electron in the magnetic field generated by the proton is simply

(728) |

where

(729) |

Here,

(730) |

Note that, because 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 first-order perturbation theory, the energy-shift induced by spin-spin coupling between the proton and the electron is the expectation value of the perturbing Hamiltonian. Hence,

(731) |

For the ground-state 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

(732) |

Let

(733) |

be the total spin. We can show that

(734) |

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 one-half particles. According to Chapter 6, when two spin one-half 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,

(735) |

for the singlet state, and

(736) |

for the triplet states.

It follows, from the above analysis, that spin-spin coupling breaks
the degeneracy of the two
states of the hydrogen atom, 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

(737) |

where is the (magnitude of the) ground-state energy. Note that the hyperfine energy-shift is much smaller, by a factor , than a typical fine structure energy-shift (see Exercise 4). If we convert the above energy into a wavelength then we obtain

(738) |

This is the wavelength of the radiation emitted by a hydrogen atom that 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.