© 2006 Krishna Myneni

The alkali atoms are those which reside in the first column of the periodic table. These atoms have a single valence electron in the shell, and their quantum mechanical treatment is effectively that of a single electron atom, e.g. the hydrogen atom. The atomic level structure of the alkali atoms, including various internal interactions, and under external perturbations such as static electric and magnetic fields, may then be treated analytically. Thus, the alkali atoms have been important, historically, in the experimental validation of the predictions of quantum mechanics. Alkali atoms are also important from a technological standpoint. Their ground hyperfine states, which we discuss here, are used in setting the standard for time and frequency measurements. In addition, these states have importance in astrophysics, since much of the information obtained from interstellar clouds is due to radio emissions at 21 cm, emitted in transitions between hyperfine levels of the hydrogen atom.

In this paper we discuss how the various internal angular momenta of an atom couple to give its total angular momentum, and the corresponding angular momentum eigenstates of the atom. The hyperfine levels are associated with these eigenstates. A specific example is worked out for the ground state of the alkali atom, Rb.

The total angular momentum of an atom consists of various parts: the
orbital angular momentum of each electron, , the intrinsic spin of each
electron, , and the intrinsic spin of the nucleus, , which is
actually a combined angular momentum of the nucleons, except for hydrogen.
Due to interactions inside the atom, the individual momenta are not
conserved separately -- only the total angular momentum of the atom is
conserved. The total angular momentum is given by[1]

If there were no interaction between the nucleus and the electrons, other than
the Coulomb force, the set of quantum numbers
for the operators
and , and operators for the -components of
these angular momenta, and , would be sufficient and complete for
uniquely labeling the angular momentum states of the atom. However, the
magnetic dipole field arising from the nuclear spin exerts torque on the
electrons, coupling and . Only the magnitude of the total
angular momentum , and its -component, , can be considered as
strictly conserved quantities. This coupling is known as the *hyperfine
interaction*, because it removes the degeneracy of the sub-levels and
leads to very fine structure in the spectra of atoms. Hyperfine structure in
the spectra of atoms can be observed only by very high resolution instruments
(large grating spectrometers, Fabry-Perot interferometers, or laser spectrometers).

The hyperfine coupling causes the -components of and
to not be conserved. Therefore, the quantum numbers and are
not ``good'' quantum numbers, *i.e.* the eigenstates of and
are no longer angular momentum eigenstates of the atom. The new angular
momentum eigenstates, in the presence of the hyperfine interaction, are
given by the simultaneous eigenstates of the operators and .
These eigenstates, which give the hyperfine levels of
the atom, are labeled by the new set of quantum numbers . In
practice, the hyperfine interaction is very weak, and the quantum numbers
and are still useful to label distinct states of the atom, so the
angular momentum eigenstates are denoted by
.

For given and , the allowed values of the quantum number for the
total angular momentum, , are given by the coupling rules for two
angular momenta[2][3]:

The angular momentum eigenstates of the atom,
,
can be written as superpositions of the uncoupled states
:

where the matrix term denotes a Wigner symbol. For a set of coupled and uncoupled quantum numbers, , the Wigner symbol is a number which can either be evaluated from a lengthy formula, or which can be looked up in tables -- the interested reader can see Chapter 5 or Appendix C of reference [1] for both methods.

Since angular momentum conservation requires

the double sum in equation 3 can be replaced by a single sum over , and using eq 4 we can write

Equations 3-5 are valid in general for any atom
in which the angular momenta are coupled as described in eq 1. Thus
far, we have not imposed any
requirements which are particular to alkali atoms. For the particular case of
the ground state of an alkali atom, only the valence -shell electron
contributes to . It follows that, in this case, the quantum number
is given by

and there are then only two terms in the sum for eq 5. For this particular case, we may write explicitly,

For Rb, in the ground state, and .
Therefore,

and

giving 8 possible uncoupled angular momentum basis states . These states are represented compactly below using the notation , where . The values of and have been omitted for clarity below, since they are fixed.

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) | |||

(13) | |||

(14) |

Using eq 6 with , consulting a table of
symbols[1], and, from eq 2, noting that the allowed
values of are

we obtain the coupled hyperfine states ,

(15) | |||

(16) | |||

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) | |||

(22) |

For Rb, the difference in energy between the and hyperfine levels is , with GHz[4]. The three sublevels for : , , and are degenerate, as are the five sublevels for . The degeneracy of the sublevels can be removed by placing the atom in an external magnetic field.

- 1
- R. D. Cowan,
*The Theory of Atomic Structure and Spectra*, (University of California Press, 1981). - 2
- S. Gasiorowicz,
*Quantum Physics*, 3 ed., (Wiley and Sons, 2003). - 3
- E. Merzbacher,
*Quantum Mechanics*, 3 ed., (Wiley and Sons, 1998). - 4
- D. Steck,
*Rubidium 87 D Line Data, rev 1.6*,`http://steck.us/alkalidata`(2003).