The three orbital angular momentum component operators do not commute with one another.

To show that , we substitute eq72 and eq73 in , giving , which when substituted with eq74, returns

Repeating the above procedure, we get

Hence, each of the three orbital angular momentum component operators do not commute with the other two. Next, to show that commutes with all 3 orbital angular momentum component operators, we begin with

Using the identity ,

Substitute eq99 and eq101 in the above equation, noting that , we have . Repeating the steps for and gives

As mentioned in an earlier article, a common complete set of eigenfunctions can be selected for two operators only if they commute. Therefore, shares a common set of eigenfunctions with each of , and , but we cannot select a common set of eigenfunctions for any pair of angular momentum component operators.

###### Question

Show that each of the three orbital angular momentum component operators commute with , , , and , where and .

Answer

Substituting eq74 in , , , , and (noting that , where ) and carrying out the derivatives, we have

Using the identities and ,

can be inferred from eq103. Repeating the same logic for and . we have

The commutation relations in the above Q&A are applicable to hydrogenic systems. For a system of 2-electrons, there are cross terms like:

which are useful in determining the commutation relations between and the multi-electron Hamiltonian, for example .