Chapter 9 The Two-Body Problem

1.3. Matrix Elements of Scalars and Vectors and the Selection Rules
For a two-particle system interacting with a central potential, the angular momentum vector which we define by Eqs. (9.32)-(9.34) commutes with any scalar function of the operators and . This is because any rotation of the coordinate system leaves scalar quantities unchanged. If we denote the scalar quantity by , we have

Since commutes with and , therefore the matrix form of will be diagonal in a representation where and are diagonal matrices. If is an eigenstate of and , and denotes all the remaining quantum numbers which define the state of the system, then we want to show that the matrix elements

are independent of . To prove this result we first note that from we get

Now we find the matrix element of this operator with the states and

where we have used the fact that has nonzero elements only when . We observe that the matrix elements involving do not depend on or , therefore (9.90) yields the result that

or that the above matrix element will be the same for all allowed values [7]. From this it follows that the nonzero elements of are

and are independent of . Noting that the Hamiltonian is a scalar we can draw an interesting conclusion from (9.92), that the energy of stationary states

Now we consider the same two-body system and we assume that there is a real vector which corresponds to a physical quantity. An infinitesimal rotation about the -th axis will change the components of and the new components will be linear combinations of the old ones. Thus the commutator of with the operator will produce components of . For instance if we choose to be the radius vector , then

On the other hand if , then

In general we have the commutator

a result that we will use later to solve the problem of the hydrogen atom [7].
It is convenient to set in the following derivation, and keeping this in mind from we find

and

By adding these three relations we obtain

Having found the commutator we calculate the double commutator in two different ways :
(1) - We can write and for each term calculate

using the commutation relations for different components of and and then adding up the results we find

(2) - We can also write the double commutator as

By equating (9.101) and (9.102) we obtain

The matrix element corresponding to the transition of Eq. (9.103) is

We arrive at this result by noting that

This follows from the fact that the matrix representing the scalar is diagonal with respect to and , and the matrix is diagonal with respect to and . By rearranging the terms in (9.104) we find

The first bracket in (9.106) cannot be zero, since and are equal or greater than zero and . Therefore we have the condition

which makes the second bracket in (9.106) zero or otherwise must vanish. In addition to (9.107) the case where

also allows for the nonzero matrix element of . If we replace either by and we find the same results.
The rules (9.107) and (9.108) are called selection rules. In addition to these rules any transition between two states with is forbidden. This can easily be shown by taking the matrix elements of

The left-hand side of (9.109) is zero since is diagonal with the eigenvalue . Thus we have

Similar selection rules can be found with respect to the quantum number .