Let us assume that the two groups of operators correspond to different degrees of freedom of the system, so that

(6.3) |

where stand for either , , or . (See Section 2.2.) For instance, could be an orbital angular momentum operator, and a spin angular momentum operator. Alternatively, and could be the orbital angular momentum operators of two different particles in a multi-particle system. We know, from the general properties of angular momentum outlined in the previous two chapters, that the eigenvalues of and can be written and , respectively, where and are either integers, or half-integers. We also know that the eigenvalues of and take the form and , respectively, where and are numbers lying in the ranges and , respectively.

Let us define the total angular momentum operator

(6.4) |

Now, is an Hermitian operator, because it is the sum of Hermitian operators. Moreover, according to Equation (4.14), satisfies the fundamental commutation relation

(6.5) |

Thus, possesses all of the expected properties of an angular momentum operator. It follows that the eigenvalue of can be written , where is an integer, or a half-integer. Moreover, the eigenvalue of takes the form , where lies in the range . At this stage, however, we do not know the relationship between the quantum numbers of the total angular momentum, and , and those of the individual angular momenta, , , , and .

Now,

Furthermore, we know that

(6.7) | ||

(6.8) |

and also that all of the , operators commute with the , operators. It follows from Equation (6.6) that

(6.9) |

This implies that the quantum numbers , , and can all be measured simultaneously. In other words, it is possible to determine the magnitude of the total angular momentum together with the magnitudes of the component angular momenta. However, it is apparent from Equations (6.1), (6.2), and (6.6) that

(6.10) | ||

(6.11) |

This suggests that it is not possible to measure the quantum numbers and simultaneously with the quantum number . Thus, we cannot determine the projections of the individual angular momenta along the -axis together with the magnitude of the total angular momentum.

It is clear, from the preceding discussion, that we can form two alternate groups of mutually commuting operators. The first group is , and . The second group is and . These two groups of operators are incompatible with one another. We can define simultaneous eigenkets of each operator group. The simultaneous eigenkets of , and are denoted , where

(6.12) | ||

(6.13) | ||

(6.14) | ||

(6.15) |

The simultaneous eigenkets of and are denoted , where

(6.16) | ||

(6.17) | ||

(6.18) | ||

(6.19) |

Each set of eigenkets are complete, mutually orthogonal (for eigenkets corresponding to different sets of eigenvalues), and have unit norms. Because the operators and are common to both operator groups, we can assume that the quantum numbers and are known. In other words, we can always determine the magnitudes of the individual angular momenta. In addition, we can either know the quantum numbers and , or the quantum numbers and , but we cannot know both pairs of quantum numbers at the same time. Finally, we can write a conventional completeness relation for both sets of eigenkets:

(6.20) | ||

(6.21) |

where the right-hand sides denote the identity operator in the ket space corresponding to states of given and . The summation is over all allowed values of , , , and .