(992) |

(993) |

(994) |

Suppose that the applied magnetic field is much weaker than the atom's internal
magnetic field (977). Since the magnitude of the internal
field is about 25 tesla, this is a fairly reasonable assumption. In this
situation, we can treat as a small perturbation acting
on the simultaneous eigenstates of the unperturbed Hamiltonian and
the fine structure Hamiltonian. Of course, these states
are the simultaneous eigenstates of , , , and (see
previous section). Hence, from standard perturbation theory, the
first-order energy-shift induced by a weak external magnetic field
is

(995) |

since . Now, according to Eqs. (825) and (826),

when , and

(997) |

It follows from Eqs. (996)-(998), and the orthormality of the , that

(999) |

where the signs correspond to . Here,

(1001) |

Likewise, the Zeeman effect splits degenerate states characterized by into equally spaced states of interstate spacing

In conclusion, in the presence of a weak external magnetic field, the two degenerate states of the hydrogen atom are split by . Likewise, the four degenerate and states are split by , whereas the four degenerate states are split by . This is illustrated in Fig. 24. Note, finally, that since the are not simultaneous eigenstates of the unperturbed and perturbing Hamiltonians, Eqs. (1002) and (1003) can only be regarded as the expectation values of the magnetic-field induced energy-shifts. However, as long as the external magnetic field is much weaker than the internal magnetic field, these expectation values are almost identical to the actual measured values of the energy-shifts.