The magnetic moment
associated with the total electronic angular momentum
of an atom and its nuclear magnetic moment
associated with its nuclear spin
(
is the Bohr magneton and
,
are the electronic and nuclear
-factors, respectively) are coupled by the hyperfine Hamiltonian
. If
the system has
groups of degenerate energy eigenstates, labeled by the total atomic angular momentum
, which takes values
. Each group consists of
degenerate sublevels, yielding a total of
substates, labeled
and
.
When placed in an external magnetic field
the combined hyperfine-Zeeman Hamiltonian reads
,
and its energy eigenvalues are found by diagonalization. If one of the spins, say
, is 1/2, the diagonalization yields an algebraic expression
for the energies of the states
, known as the Breit-Rabi formula.
is the hyperfine splitting for
, that is, the energy difference between the states belonging to the manifolds
.
By introducing the dimensionless parameter
the energies can be written in the dimensionless form
which have an explicit dependence (given by
and
) on the specific atom considered.
Considering that
is in general on the order of a few times
of
, one can neglect the second term to obtain a universal expression
with
that is valid for all Zeeman-hyperfine problems in which one of the spins is 1/2. It is the latter equation that is displayed in the present Demonstration.
