Addition of Angular Momenta

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Consider two sets of angular momentum operators $\vec{J}^{(1)}$ and  $\vec{J}^{(2)}$ such that every component of $\vec{J}^{(1)}$ commutes with every component of $\vec{J}^{(2)}$.
\begin{equation}
          [J_x^{(\alpha)}, J^{(\alpha)}_y] = i\hbar J_z^{(\alpha)} \qquad \alpha=1,2 \label{E1}
\end{equation}
and
\begin{equation}
[\vec{J}^{(1)}, \vec{J}^{(2)}] = 0\,\,\,. \label{E2}
\end{equation}
is obvious that $\vec{J}^{(1)^2}, \vec{J}^{(2)^2}, J_z^{(1)}$ and  $J_z^{(2)}$ from a commuting set. Let us now define total angular momentum $\vec{J}=\vec{J}^{(1)}+\vec{J}^{(2)}$ then

1. $\vec{J}$ satisfies angular momentum commutation relations
   \begin{equation}
   [J_x, J_y] = i\hbar J_z \qquad\qquad \mbox{etc.}   \label{E3} \end{equation}
2. All the three components of $\vec{J}$ commute with $\vec{J}^2_1$ and $\vec{J}_2^2$.
3. It follows from (b) that $\vec{J}^2$ commutes with $\vec{J}_1^2$ and $\vec{J}_2^2$.
   The angular momentum commutation relations for $\vec{J}$ imply that the possible  eigenvalues of $J^2$ are $j(j+)\hbar^2, j=0,1/2.1,3/2,\ldots$. For a fixed $j$, $J_z$ eigenvalues are $j\hbar, (j-1)\hbar, \ldots, -j\hbar$. 

Angular momentum eigenvalues 

In fact the component of $\vec{J}$ along every unit vector $\hat{n}$ has the same eigenvalues as $J_z$.

Given the angular momentum commutation relations it is straight forward to see that one can have two sets of commuting operators. Each of these two sets of operators can have  simultaneous eigenvectors. The two commuting sets of operators  and the notation for their simultaneous eigenvectors is shown in the table below.

The vectors $\ket{j_1,j_2m_1m_2}$ listed in the second column of the first row are  just the direct product of vectors $\ket{j_1m_1}$ and $\ket{j_2m_2}$ which are eigenvectors of  $\vec{J}^{(1)2}, J_z^{(1)}$ and of $\vec{J}^{(2)2}, J_z^{(2)}$, respectively.
\begin{equation}
\ket{j_1,j_2m_1m_2}\equiv \ket{j_1m_1}\ket{j_2m_2} \label{E4}
\end{equation}
The problem of addition of angular momenta consists of the following questions

•  For a state $\ket{j_1,j_2,m_1,m_2}$, which has definite values $j_1,m_1$ and $j_2,m_2$ for the individual values of the square, and the $z-$ component of the angular momenta, what values of square of total angular momentum $\vec{J}^2$ and  $J_z$ are allowed?
• The second part of the problem is to construct the transformation matrix connecting the two sets of simultaneous eigenvetors,  one set for each commuting  set.
  The states $\ket{j_1j_2JM}$ can be expressed as linear combination of states  $\ket{j_1j_2m_1m_2}$ and vice versa \begin{equation} \ket{j_1j_2m_1m_2} = \sum_{JM} C(JM; j_1j_2m_1m_2)\ket{j_1j_2JM} \label{E5} \end{equation} 

The coefficients  $C(JM; j_1j_2m_1m_2)$ are found by using orthogonality of  states $\ket{j_1j_2JM}$ for different $J,M$ giving
\begin{eqnarray}
C(JM; j_1j_2m_1m_2)&=& \innerproduct{j_1 j_2, JM}{ j_1j_2,m_1m_2}\label{E6} \\
\ket{j_1j_2m_1m_2} &=&\sum_{j=|j_1-j_2|}^{j_1+j_2}\innerproduct{j_1 j_2, JM}{ j_1j_2,m_1m_2}\ket{j_1j_2JM}\label{E7}
\end{eqnarray}
where $M=m_1+m_2$ and conversely
\begin{equation}
     \ket{j_1j_2JM} = \sum_{\scriptstyle m_1,m_2\atop m_1+m_2=M} 
     \innerproduct{j_1j_2,m_1m_2}{j_1j_2,JM}\ket{j_1 j_2m_1m_2} \label{E8}
\end{equation}
Since the values of $j_1, j_2$ are the common for all the states, we shall  frequently supress these labels and write the kets $\ket{j_1j_2JM}$ and $ \ket{j_1j_2m_1m_2}$ as $ \ket{JM}$ and as $\ket{j_1j_2m_1m_2}$ respectively. The coefficients $\innerproduct{j_1j_2,JM}{j_1j_2,m_1m_2}$, written in short as $\innerproduct{JM}{j_1j_2m_1m_2}$, are known as Clebsch Gordon coefficients and expressions are tabulated in books. The problem of addition of angular momenta consists of finding allowed range of values of $J,M$ and obtaining expressions for the Clebsch Gordon coefficients.