Let us examine a phenomenon known as *fine structure*, which is due to
interaction between the spin and orbital angular momenta of the outermost
electron. This electron experiences an electric field

(683) |

(684) |

(685) |

where is the orbital angular momentum. When the above expression is compared to the observed spin-orbit interaction, it is found to be too large by a factor of two. There is a classical explanation for this, due to spin precession, which we need not go into. The correct quantum mechanical explanation requires a relativistically covariant treatment of electron dynamics (this is achieved using the so-called

Let us now apply perturbation theory to a hydrogen-like atom, using
as the perturbation (with taking one half of the value given above), and

(687) |

It is fairly obvious that the first group of operators ( and )

We now need to find the simultaneous eigenstates of
and .
This is equivalent to finding the eigenstates of the total angular momentum
resulting from the addition of two angular momenta: , and .
According to Eq. (572), the allowed values of the total angular
momentum are and . We can write

(689) | |||

(690) |

Here, the kets on the left-hand side are kets, whereas those on the right-hand side are kets (the labels have been dropped, for the sake of clarity). We have made use of the fact that the Clebsch-Gordon coefficients are automatically zero unless . We have also made use of the fact that both the and kets are orthonormal, and have unit lengths. We now need to determine

(691) |

Let us now employ the recursion relation for Clebsch-Gordon coefficients, Eq. (578),
with
(lower sign).
We obtain

(692) |

which reduces to

(693) |

(694) |

This procedure can be continued until attains its maximum possible value, . Thus,

Consider the situation in which and both take their maximum values,
and , respectively. The corresponding value of is
. This value is possible when , but not when .
Thus, the
ket must be equal to
the ket
, up to an arbitrary phase-factor.
By convention, this factor is taken to be unity, giving

(696) |

(697) |

(698) |

We now need to determine the sign of . A careful examination
of the recursion relation, Eq. (578), shows that the plus sign is
appropriate. Thus,

It is convenient to define so called

(701) |

These functions are eigenfunctions of the total angular momentum for spin one-half particles, just as the spherical harmonics are eigenfunctions of the orbital angular momentum. A general wave-function for an energy eigenstate in a hydrogen-like atom is written

The radial part of the wave-function, , depends on the radial quantum number and the angular quantum number . The wave-function is also labeled by , which is the quantum number associated with . For a given choice of , the quantum number (

The
kets are eigenstates of
,
according to Eq. (688).
Thus,

(703) |

(704) | |||

(705) |

It follows that

where the integrals are over all solid angle.

Let us now apply degenerate perturbation theory to evaluate the
shift in energy of a state whose wave-function is
due to the spin-orbit Hamiltonian . To first-order, the energy-shift is given by

(708) |

where

Equations (709)-(710) are known as

Let us now apply the above result to the case of a sodium atom.
In chemist's notation, the ground state is written

(712) |

(713) |