Zeeman Effect

Consider a hydrogen-like atom placed in a uniform -directed magnetic field, . The change in energy of the outermost electron is [49]

(7.124) |

where

(7.125) |

is its magnetic moment, including both the spin and orbital contributions. (See Section 5.5.) Thus,

(7.126) |

Suppose that the energy-shifts induced by the magnetic field are much smaller than those induced by spin-orbit interaction. In this situation, we can treat as a small perturbation acting on the eigenstates of . Of course, these states are the simultaneous eigenstates of and . Let us consider one of these states, labeled by the quantum numbers and , where . From standard perturbation theory, the first-order energy-shift in the presence of a magnetic field is

(7.127) |

Because

(7.128) |

we find that

(7.129) |

Now, from Equations (7.113)-(7.114),

(7.130) |

It follows that

(7.131) |

Thus, we obtain the so-called

where

(7.133) |

is called the

Let us apply this theory to the sodium atom. We have already seen that the non-Coulomb potential splits the degeneracy of the and states, the latter states acquiring a higher energy. The spin-orbit interaction splits the six (i.e., ) states into two groups, with four states lying at a slightly higher energy than two states. According to Equation (7.135), a magnetic field splits the quadruplet of states, each state acquiring a different energy. In fact, the energy of each state becomes dependent on the quantum number , which measures the projection of the total angular momentum along the -axis. States with higher values have higher energies. A magnetic field also splits the doublet of states. However, it is evident from Equation (7.135) that these states are split by a lesser amount than the states. In fact, the splitting is for the quadruplet, and for the doublet.

Suppose that we increase the strength of the magnetic
field until the energy-shift due to the magnetic field becomes
comparable to the energy-shift induced by spin-orbit interaction.
Clearly, in this situation, it does not make much sense to think
of
as a small interaction term operating on the eigenstates
of
. In fact, this intermediate case is very difficult
to analyze. Let us, instead, consider the extreme limit in which the energy-shift
due to the magnetic field greatly exceeds that induced by spin-orbit effects.
This is called the *Paschen-Back limit* [78].

In the Paschen-Back limit, we can think of the spin-orbit Hamiltonian, , as a small interaction term operating on the eigenstates of . Note that the magnetic Hamiltonian, , commutes with , but does not commute with . Thus, in an intense magnetic field, the energy eigenstates of a hydrogen-like atom are approximate eigenstates of the spin and orbital angular momenta, but are not eigenstates of the total angular momentum. We can label each state by the quantum numbers (the principal quantum number), , , and . (Here, and are the quantum numbers corresponding to and , respectively.) Thus, our energy eigenkets are written . The unperturbed Hamiltonian, , causes states with different values of the quantum numbers and to have different energies. However, states with the same value of and , but different values of and , are degenerate. The shift in energy due to the magnetic field is simply

(7.134) |

Thus, states with different values of acquire different energies.

Let us apply this result to a sodium atom. In the absence of a magnetic field, the six states form two groups of four and two states, depending on the values of their total angular momentum. In the presence of an intense magnetic field, the states are split into five groups. There is a state with , a state with , two states with , a state with , and a state with . These groups are equally spaced in energy, the energy difference between adjacent groups being .

The energy-shift induced by the spin-orbit Hamiltonian is given by

(7.135) |

where

(7.136) |

Now,

(7.137) |

because

(7.138) |

for expectation values taken between the simultaneous eigenkets of and . Thus,

(7.139) |

Note, finally, that the eigenstates of are not eigenstates of . However, only couples non-degenerate eigenstates of . Hence, there is no danger of singular terms arising in the perturbation expansion to second order.

Let us apply the previous result to a sodium atom. In the presence of an intense magnetic field, the states are split into five groups with ( ) quantum numbers , , or , , and , respectively, in order of decreasing energy. The spin-orbit term increases the energy of the highest energy state (because ), does not affect the next highest energy state, decreases, but does not split, the energy of the doublet, does not affect the next lowest energy state, and increases the energy of the lowest energy state. (See Exercise 18.) The net result is that the five groups of states are no longer equally spaced in energy.

The typical magnetic field-strength needed to access the Paschen-Back limit is

where is the fine structure constant, and the Bohr radius. (See Exercise 19.)