Guided reading
After the Hilbert space is expanded, the next question is how to change from the local basis to the coupled basis. The answer is the Clebsch-Gordan matrix.
Starting point
The total angular momentum is
\[\vec{\hat L}=\vec{\hat L}_1+\vec{\hat L}_2.\]
The local basis is \(|l_1,l_2,m_1,m_2\rangle\), while the coupled basis is \(|l_1,l_2,l,m\rangle\). For the maximum projection, there is a univocal relation:
\[|l_1,l_2,l_{\max},m_{\max}\rangle=|l_1,l_2,m_{1\max},m_{2\max}\rangle.\]
Lowering operators and orthogonality then generate the remaining coupled vectors.
Example 7.5 route: lowering and orthogonality
For \(l_1=l_2=1/2\), the maximum projection fixes the first vector:
\[|1,+1\rangle=|+,+\rangle.\]
Then apply \(\hat L_-=\hat L_{1-}+\hat L_{2-}\):
\[\hat L_-|1,+1\rangle=(\hat L_{1-}+\hat L_{2-})|+,+\rangle.\]
This gives the \(m=0\) triplet vector; applying \(\hat L_-\) once more gives the \(m=-1\) triplet vector:
\[\begin{aligned}|1,0\rangle&=\frac{1}{\sqrt2}(|+,-\rangle+|-,+\rangle),\\ |1,-1\rangle&=|-,-\rangle.\end{aligned}\]
The missing singlet is found by writing \(|0,0\rangle=a|+,-\rangle+b|-,+\rangle\), using \(\langle1,0|0,0\rangle=0\), and normalizing:
\[|0,0\rangle=\frac{1}{\sqrt2}(|+,-\rangle-|-,+\rangle).\]
Completeness and coefficients
Using the completeness relation of the local basis, the coupled vector is written as
\[|l_1,l_2,l,m\rangle=\sum_{m_1}\sum_{m_2}\langle l_1,l_2,m_1,m_2|l_1,l_2,l,m\rangle |l_1,l_2,m_1,m_2\rangle.\]
The coefficients \(\langle l_1,l_2,m_1,m_2|l_1,l_2,l,m\rangle\) are the Clebsch-Gordan coefficients. They measure the relation between a local vector and a coupled vector. They are real and form a unitary matrix.
Selection conditions
The Wigner \(3j\) symbol and the Racah formula give the coefficients. The formula is zero unless the conditions below are satisfied:
\[m=m_1+m_2,\qquad |l_1-l_2|\le l\le l_1+l_2.\]
These are the same conditions obtained from the coupled-basis construction.
Example 7.6: change-of-basis matrix for l1 = l2 = 1/2
The local basis is ordered as \(|+,+\rangle, |+,-\rangle, |-,+\rangle, |-,-\rangle\). The coupled basis is ordered as \(|1,1\rangle, |1,0\rangle, |1,-1\rangle, |0,0\rangle\).
The Clebsch-Gordan coefficients form the matrix in
\[|l,m\rangle=\hat U|m_1,m_2\rangle.\]
With coupled vectors as rows and local vectors as columns, the matrix of coefficients is
\[\hat U=\begin{pmatrix}\langle +,+|1,1\rangle&\langle +,-|1,1\rangle&\langle -, +|1,1\rangle&\langle -,-|1,1\rangle\\ \langle +,+|1,0\rangle&\langle +,-|1,0\rangle&\langle -, +|1,0\rangle&\langle -,-|1,0\rangle\\ \langle +,+|1,-1\rangle&\langle +,-|1,-1\rangle&\langle -, +|1,-1\rangle&\langle -,-|1,-1\rangle\\ \langle +,+|0,0\rangle&\langle +,-|0,0\rangle&\langle -, +|0,0\rangle&\langle -,-|0,0\rangle\end{pmatrix}.\]
The zero entries are selected by \(m=m_1+m_2\); the nonzero entries are the Clebsch-Gordan coefficients. For this example:
\[\hat U=\begin{pmatrix}1&0&0&0\\0&1/\sqrt2&1/\sqrt2&0\\0&0&0&1\\0&1/\sqrt2&-1/\sqrt2&0\end{pmatrix}.\]
Reading U and U dagger
Each row of \(\hat U\) gives one coupled vector expanded in the local basis. This reproduces the vectors obtained in Example 7.5 by successive application of \(\hat L_-\).
Because \(\hat U\) is unitary, the inverse change of basis is
\[|m_1,m_2\rangle=\hat U^{-1}|l,m\rangle=\hat U^\dagger|l,m\rangle.\]
For the same ordering of vectors,
\[\hat U^\dagger=\begin{pmatrix}1&0&0&0\\0&1/\sqrt2&0&1/\sqrt2\\0&1/\sqrt2&0&-1/\sqrt2\\0&0&1&0\end{pmatrix}.\]
Reading boundary for 3+ angular momenta
For three or more angular momenta, the coupled basis can be sequential or non-sequential, and the change-of-basis matrix is built recursively from Clebsch-Gordan coefficients.
This app keeps the two-angular-momentum case as the worked example. The detailed construction for three and \(N\) angular momenta remains in Chapter 7.
Exercise-ready boundary
This page supports compact guided exercises on: Clebsch-Gordan coefficients and the local-to-coupled basis matrix.
- Use from this page: the definitions, physical setup, highlighted equations and quantum-number restrictions shown here.
- Keep in Chapter 7: the complete demonstrations, long examples, historical notes and extended exercise solutions.
- Boundary: do not introduce notation, Hamiltonians, coupling schemes or selection rules outside the sequence presented here.
Practice anchors
Use these anchors only within the material introduced on this page.
- Focus: Clebsch-Gordan coefficients and the local-to-coupled basis matrix.
- Conceptual check: identify the basis or angular-momentum interaction before applying the equation.
- Key equation 1:
\[|l_1,l_2,l,m\rangle=\sum_{m_1}\sum_{m_2}\langle l_1,l_2,m_1,m_2|l_1,l_2,l,m\rangle |l_1,l_2,m_1,m_2\rangle\]
- Key equation 2:
\[|l,m\rangle=\hat U|m_1,m_2\rangle\]
- Typical task: apply one central equation and state what basis, quantum numbers or physical consequence it uses.
Original book and previews: Source note: Original auxiliary summary for this book-app, based on Chapter 7 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc.; consult the original book for complete demonstrations, examples and exercises.