Commutation Rules

Consider the most general case. Suppose that we have two sets of angular momentum operators, ${\bf J}_1$ and ${\bf J}_2$ . By definition, these operators are Hermitian, and obey the fundamental commutation relations

$${\bf J}_1\times {\bf J}_1 = {\rm i}\,\hbar\,{\bf J}_1,$$ (6.1)

$${\bf J}_2\times {\bf J}_2 = {\rm i}\,\hbar\,{\bf J}_2.$$ (6.2)

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

$$[J_{1\,i}, J_{2\,j}] = 0,$$ (6.3)

where $i, j$ stand for either $x$ , $y$ , or $z$ . (See Section 2.2.) For instance, ${\bf J}_1$ could be an orbital angular momentum operator, and ${\bf J}_2$ a spin angular momentum operator. Alternatively, ${\bf J}_1$ and ${\bf J}_2$ 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 $J_1^{\,2}$ and $J_2^{\,2}$ can be written $j_1\,(j_1+1)\,\hbar^{\,2}$ and $j_2\,(j_2+1)\, \hbar^{\,2}$ , respectively, where $j_1$ and $j_2$ are either integers, or half-integers. We also know that the eigenvalues of $J_{1\,z}$ and $J_{2\,z}$ take the form $m_1\,\hbar$ and $m_2\,\hbar$ , respectively, where $m_1$ and $m_2$ are numbers lying in the ranges $j_1, j_1-1,\cdots, -j_1+1, -j_1$ and $j_2, j_2-1,\cdots, -j_2+1, -j_2$ , respectively.

Let us define the total angular momentum operator

$${\bf J} = {\bf J}_1 + {\bf J}_2.$$ (6.4)

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

$${\bf J} \times {\bf J} = {\rm i}\,\hbar\, {\bf J}.$$ (6.5)

Thus, ${\bf J}$ possesses all of the expected properties of an angular momentum operator. It follows that the eigenvalue of $J^{\,2}$ can be written $j\,(j+1)\,\hbar^{\,2}$ , where $j$ is an integer, or a half-integer. Moreover, the eigenvalue of $J_z$ takes the form $m\,\hbar$ , where $m$ lies in the range $j, j-1,\cdots, -j+1, -j$ . At this stage, however, we do not know the relationship between the quantum numbers of the total angular momentum, $j$ and $m$ , and those of the individual angular momenta, $j_1$ , $j_2$ , $m_1$ , and $m_2$ .

Now,

$$J^{\,2} = J_1^{\,2} + J_2^{\,2} + 2\,{\bf J}_1 \cdot {\bf J}_2.$$ (6.6)

Furthermore, we know that

$$[J_1^{\,2}, J_{1\,i} ] =0,$$ (6.7)

$$[J_2^{\,2}, J_{2\,i} ] =0,$$ (6.8)

and also that all of the $J_{1\,i}$ , $J_1^{\,2}$ operators commute with the $J_{2\,i}$ , $J_2^{\,2}$ operators. It follows from Equation (6.6) that

$$[J^{\,2}, J_1^{\,2}] = [J^{\,2}, J_2^{\,2}] = 0.$$ (6.9)

This implies that the quantum numbers $j_1$ , $j_2$ , and $j$ 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

$$[J^{\,2}, J_{1\,z}] \neq 0,$$ (6.10)

$$[J^{\,2}, J_{2\,z}] \neq 0.$$ (6.11)

This suggests that it is not possible to measure the quantum numbers $m_1$ and $m_2$ simultaneously with the quantum number $j$ . Thus, we cannot determine the projections of the individual angular momenta along the $z$ -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 $J_1^{\,2}, J_2^{\,2}, J_{1\,z}$ , and $J_{2\,z}$ . The second group is $J_1^{\,2}, J_2^{\,2}, J^{\,2},$ and $J_z$ . These two groups of operators are incompatible with one another. We can define simultaneous eigenkets of each operator group. The simultaneous eigenkets of $J_1^{\,2}, J_2^{\,2}, J_{1z}$ , and $J_{2z}$ are denoted $\vert j_1,j_2; m_1,m_2\rangle$ , where

$$J_1^{\,2}\, \vert j_1,j_2; m_1,m_2\rangle = j_1\,(j_1+1)\,\hbar^{\,2}\,\vert j_1,j_2; m_1,m_2\rangle,$$ (6.12)

$$J_2^{\,2} \,\vert j_1,j_2; m_1,m_2\rangle = j_2\,(j_2+1)\,\hbar^{\,2}\,\vert j_1,j_2; m_1,m_2\rangle,$$ (6.13)

$$J_{1z}\, \vert j_1,j_2; m_1,m_2\rangle = m_1\,\hbar\,\vert j_1,j_2; m_1,m_2\rangle,$$ (6.14)

$$J_{2z}\, \vert j_1,j_2; m_1,m_2\rangle = m_2\,\hbar\,\vert j_1,j_2; m_1,m_2\rangle.$$ (6.15)

The simultaneous eigenkets of $J_1^{\,2}, J_2^{\,2}, J^{\,2}$ and $J_z$ are denoted $\vert j_1, j_2; j, m\rangle$ , where

$$J_1^{\,2}\, \vert j_1,j_2; j,m\rangle = j_1\,(j_1+1)\,\hbar^{\,2}\,\vert j_1,j_2; j,m\rangle,$$ (6.16)

$$J_2^{\,2} \,\vert j_1,j_2; j,m\rangle = j_2\,(j_2+1)\,\hbar^{\,2}\,\vert j_1,j_2; j,m\rangle,$$ (6.17)

$$J^{\,2} \,\vert j_1,j_2; j,m\rangle =j\,(j+1)\,\hbar^{\,2}\,\vert j_1,j_2; j,m\rangle,$$ (6.18)

$$J_{z}\, \vert j_1,j_2; j,m\rangle = m\,\hbar\,\vert j_1,j_2; j,m\rangle.$$ (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 $J_1^{\,2}$ and $J_2^{\,2}$ are common to both operator groups, we can assume that the quantum numbers $j_1$ and $j_2$ 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 $m_1$ and $m_2$ , or the quantum numbers $j$ and $m$ , 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:

$$\sum_{m_1}\sum_{m_2 }\vert j_1,j_2; m_1, m_2\rangle \langle j_1,j_2; m_1, m_2\vert =1,$$ (6.20)

$$\sum_{j}\sum_{m} \vert j_1,j_2; j, m\rangle \langle j_1,j_2; j, m\vert =1,$$ (6.21)

where the right-hand sides denote the identity operator in the ket space corresponding to states of given $j_1$ and $j_2$ . The summation is over all allowed values of $m_1$ , $m_2$ , $j$ , and $m$ .