Chapter 9 The Two-Body Problem

1.1. The Angular Momentum Operator
The theoretical framework for calculating the eigenvalues, specifically those of the orbital angular momentum using the algebraic technique and raising and lowering operators was fully developed in one of the very first papers on matrix mechanics [11]. The angular momentum which is defined by (9.27) obeys the commutation relation

To show this we write in terms of its components

and by direct calculation we find the following commutators;

We can write in a compact form as

where is the completely antisymmetric tensor defined by Eqs. (1.32) and (1.33). Equation is the vector version of .
Since and , we have

Similarly we find

The fact that and do not commute means that they are not simultaneously measurable. The only exception to this is when the eigenvalue of is zero, then all of the components have simultaneously zero eigenvalues.
We can choose one of the three components of , say , and measure it simultaneously with . Now we will show that commutes with . Consider the commutators

and

where in calculating (9.38) and (9.39) we have used the relation

By adding (9.38) to (9.39) and noting that we have

Next we want to show that any component of the angular momentum operator, , commutes with and (see also Sec. 3.3). For this we observe that the angular momentum commutator will not change if we make one of the two following replacements [2]:
(a) - If we replace by and by simultaneously.
(b) - If we replace by and at the same time by .
Now let us proceed with the proof that satisfies the following commutators (see also Sec. 3.5)

Using the fundamental commutation relations:

and with the other commutators for and being equal to zero we can calculate the following commutators:

By adding these three relations we get

Similarly we can show that

Now if we make the replacement given by (a) and use the same argument that we have just made, we reach the conclusion that

Finally if we make the substitution suggested in (b) we obtain

From this together with and , we get

We can summarize our result in this way: If is any scalar constructed from the operators shown in its argument, then every component of the angular momentum vector commutes with .