For any given atom, there are quantum numbers that can specify the wavefunction of that atom. Using the hydrogen atom as an example, four quantum numbers are required to fully describe the state of the system. Quantum numbers that are eigenvalues of the operators that commute with the wavefunction to describe physical aspects of the system, and are called “good” numbers because of this. Once good quantum numbers have been found for a given atomic transition, the selection rules determine what changes in quantum numbers are allowed.
The electric dipole (E1) transition of a hydrogen atom can be described with the quantum numbers l (orbital angular momentum quantum number), m (magnetic quantum number), m (electron spin quantum number), and n (principal quantum number). When evaluating the effect of the electric dipole moment operator μ on the wavefunction of the system, we see that all values of the eigenvalue are 0, except for when the changes in the quantum numbers follow a specific pattern.
For example in the E1 transition, unless Δ l = ± 1, Δ m = 0 or ± 1, Δ m = 0, and Δ n = any integer, the equation above will yield a value equal to zero and the transition would be known as a “forbidden transition”. For example, this would occur for certain cases like when Δ l = 2. In this case, the transition would not be allowed and therefore would be much weaker than an allowed transition. These specific values for the changes in quantum numbers are known as the selection rules for the allowed transitions and are shown for common transitions in the table below: