For a single, nonrelativistic particle of mass m, in zero magnetic field, the transition dipole moment between two energy eigenstates ψ and ψ can alternatively be written in terms of the momentum operator, using the relationship
This relationship can be proven starting from the commutation relation between position x and the Hamiltonian :
Then
However, assuming that ψ and ψ are energy eigenstates with energy E and E, we can also write
Similar relations hold for y and z, which together give the relationship above.