An atom or a molecule can absorb light and undergo a transition from
one quantum state to another. The oscillator strength is a dimensionless
quantity to express the strength of the transition.
The oscillator strength of a transition from a lower state
to an upper state may be defined by
where is the mass of an electron and is
the reduced Planck constant. The quantum states 1,2,..., are assumed to have several
degenerate sub-states, which are labeled by . "Degenerate" means
that they all have the same energy .
The operator is the sum of the x-coordinates
of all electrons in the system, etc.:
The oscillator strength is the same for each sub-state .

Thomas-Reiche-Kuhn sum rule

The sum of the oscillator strength from one sub-state to all
other states is equal to the number of electrons :