D.39 Selection rules

This note derives the selection rules for electric dipole transitions
between two hydrogen states

According to chapter 4.3, the hydrogen states take the form

As noted in the text, allowed electric dipole transitions must respond
to at least one component of a constant ambient electric field. That
means that they must have a nonzero value for at least one electrical
dipole moment,

where

The trick in identifying when these inner products are zero is based
on taking inner products with cleverly chosen commutators. Since the
hydrogen states are eigenfunctions of

For the

The final inner product is the dipole moment of interest. Therefore, if a suitable expression for the commutator in the left hand side can be found, it will fix the dipole moment.

In particular, according to chapter 4.5.4

For the

Plugging that into (D.24) produces

From these equations it is seen that the

and if the

To derive selection rules involving the azimuthal quantum numbers

where by the definition of the commutator

Evaluating

For

The right hand side is obviously zero for

and the left hand side can be written in terms of these same factors as

Equating the two results and simplifying gives

The second factor is only zero if

The spin is not affected by the perturbation Hamiltonian, so the
dipole moment inner products are still zero unless the spin magnetic
quantum numbers

Now consider the effect of the magnetic field on transitions. For
such transitions to be possible, the matrix element formed with the
magnetic field must be nonzero. Like the electric field, the magnetic
field can be approximated as spatially constant and quasi-steady. The
perturbation Hamiltonian of a constant magnetic field is according to
chapter 13.4

Note that now electron spin must be included in the discussion.

According to this perturbation Hamiltonian, the perturbation
coefficient

and that is zero because

However, the

According to chapter 12.11, the effect of

The magnetic field simply wants to rotate the orbital angular momentum
vector in the hydrogen atom. That does not change the energy, in the
absence of an average ambient magnetic field. For the second inner
product, the spin magnetic quantum numbers have to be different by one
unit, while the orbital magnetic quantum numbers must now be equal.
So, all together

and either the orbital or the spin magnetic quantum numbers must be unequal. That are the selection rules as given in chapter 7.4.4 for magnetic dipole transitions. Since the energy does not change in these transitions, Fermi’s golden rule would have the decay rate zero. Fermi’s analysis is not exact, but such transitions should be very rare.

The logical way to proceed to electric quadrupole transitions would be
to expand the electric field in a Taylor series in terms of

The first term is the constant electric field of the electric dipole approximation, and the second would then give the electric quadrupole approximation. However, an electric field in which

It is necessary to retreat to the so-called vector potential

In terms of the vector potential, the perturbation Hamiltonian is,
chapter 13.1 and 13.4, and assuming a
weak field,

Ignoring the spatial variation of

That should be same as for the electric dipole approximation, since the field is now completely described by

Now consider the second term in the Taylor series of

The factor

The first term has already been accounted for in the magnetic dipole transitions discussed above, because the factor within parentheses is

As second terms in the Taylor series, both Hamiltonians will be much
smaller than the electric dipole one. The factor that they are
smaller can be estimated from comparing the first and second term in
the Taylor series. Note that

The selection rules for the electric quadrupole Hamiltonian can be
narrowed down with a bit of simple reasoning. First, since the
hydrogen eigenfunctions are complete, applying any operator on an
eigenfunction will always produce a linear combination of
eigenfunctions. Now reconsider the derivation of the electric dipole
selection rules above from that point of view. It is then seen that

That are the selection rules as given in chapter 7.4.4 for electric quadrupole transitions. These arguments apply equally well to the magnetic dipole transition, but there the possibilities are narrowed down much further because the angular momentum operators only produce a couple of eigenfunctions. It may be noted that in addition, electric quadrupole transitions from