Angular Momentum Operators

It follows that

Let us, first of all, consider whether it is possible to use the above expressions as the definitions of the operators corresponding to the components of angular momentum in quantum mechanics, assuming that the and (where , , ,

Let us now derive the commutation relations for the .
For instance,

(530) |

where use has been made of the definitions of the [see Eqs. (527)-(529)], and commutation relations (481)-(483) for the and . There are two similar commutation relations: one for and , and one for and . Collecting all of these commutation relations together, we obtain

By analogy with classical mechanics, the operator , which represents
the *magnitude squared* of the angular momentum vector, is defined

Hence,

(536) |

where use has been made of Eqs. (531)-(533). In other words, commutes with . Likewise, it is easily demonstrated that also commutes with , and with . Thus,

Recall, from Sect. 4.10, that in order for two physical quantities
to be (exactly) measured *simultaneously*, the operators which represent
them in quantum mechanics must *commute* with one another. Hence,
the commutation relations (531)-(533) and (537)
imply that we can only simultaneously measure the magnitude squared of
the angular momentum vector, , together with, at most, *one* of its
Cartesian components. By convention, we shall always choose to measure
the -component, .

Finally, it is helpful to define the operators

Moreover, it is easily seen that

Likewise,

giving

(542) |

and, similarly,

(544) |