Guided reading
The section asks which observables describe a system with two independent angular momenta. The local basis is built first; the coupled basis appears when the total angular momentum is introduced.
Independent subspaces
For two angular momenta \(\vec{\hat L}_1\) and \(\vec{\hat L}_2\), each belongs to its own Hilbert subspace. Components belonging to different subspaces commute:
\[[\hat L_{1u},\hat L_{2v}]=0,\qquad (u,v)=(x,y,z).\]
\[[\hat L_1^2,\hat L_{2v}]=0,\qquad [\hat L_2^2,\hat L_{1v}]=0,\qquad [\hat L_1^2,\hat L_2^2]=0.\]
Each subspace still obeys the angular-momentum commutation rule.
\[[\hat L_{1u},\hat L_{1v}]=i\hbar\epsilon_{uvw}\hat L_{1w},\qquad [\hat L_{2u},\hat L_{2v}]=i\hbar\epsilon_{uvw}\hat L_{2w}.\]
Local basis
The algebra in each subspace gives
\[\hat L_{1z}|l_1,m_1\rangle=m_1\hbar|l_1,m_1\rangle,\qquad \hat L_1^2|l_1,m_1\rangle=l_1(l_1+1)\hbar^2|l_1,m_1\rangle,\]
\[\hat L_{2z}|l_2,m_2\rangle=m_2\hbar|l_2,m_2\rangle,\qquad \hat L_2^2|l_2,m_2\rangle=l_2(l_2+1)\hbar^2|l_2,m_2\rangle.\]
Therefore the commuting set
\[\{\hat L_1^2,\hat L_2^2,\hat L_{1z},\hat L_{2z}\}\]
is represented by \(|l_1,m_1\rangle\otimes |l_2,m_2\rangle=|l_1,l_2,m_1,m_2\rangle\). This is the local basis.
Total angular momentum
The total angular momentum is introduced as
\[\vec{\hat L}=\vec{\hat L}_1+\vec{\hat L}_2,\qquad \hat L^2=(\vec{\hat L}_1+\vec{\hat L}_2)^2=\hat L_1^2+\hat L_2^2+2\vec{\hat L}_1\cdot\vec{\hat L}_2.\]
\[\hat L_x=\hat L_{1x}+\hat L_{2x},\quad \hat L_y=\hat L_{1y}+\hat L_{2y},\quad \hat L_z=\hat L_{1z}+\hat L_{2z},\quad \hat L_\pm=\hat L_{1\pm}+\hat L_{2\pm}.\]
The new commuting set is
\[\{\hat L_1^2,\hat L_2^2,\hat L^2,\hat L_z\},\]
represented by \(|l_1,l_2,l,m\rangle\). This is the coupled basis.
Allowed values and projection condition
The total angular momentum follows the same algebra as one angular momentum:
\[[\hat L_u,\hat L_v]=i\hbar\epsilon_{uvw}\hat L_w.\]
\[\hat L_z|l,m\rangle=m\hbar|l,m\rangle,\qquad \hat L^2|l,m\rangle=l(l+1)\hbar^2|l,m\rangle,\qquad -l\le m\le +l.\]
Dimension preservation gives the possible values of \(l\):
\[|l_1-l_2|\le l\le l_1+l_2.\]
The projection condition is
\[m=m_1+m_2.\]
This condition is necessary, but not sufficient, for \(\langle m_1,m_2|l,m\rangle\) to be nonzero.
Example: two angular momenta l1 = l2 = 1/2
| Basis | Vectors in the short notation of the section | Dimension check |
|---|
| Local basis \(|m_1,m_2\rangle\) | \(|+,+\rangle,\ |+,-\rangle,\ |-,+\rangle,\ |-,-\rangle\) | \((2l_1+1)(2l_2+1)=4\) |
| Coupled basis \(|l,m\rangle\) | \(|1,1\rangle,\ |1,0\rangle,\ |1,-1\rangle,\ |0,0\rangle\) | \(\sum_{l=0}^{1}(2l+1)=4\) |
In this example \(+\) and \(-\) represent \(+1/2\) and \(-1/2\), respectively.
Exercise-ready boundary
This page supports compact guided exercises on: the two commuting sets that define local and coupled angular-momentum bases.
- 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: the two commuting sets that define local and coupled angular-momentum bases.
- Conceptual check: identify the basis or angular-momentum interaction before applying the equation.
- Key equation 1:
\[\{\hat L_1^2,\hat L_2^2,\hat L_{1z},\hat L_{2z}\}\]
- Key equation 2:
\[\{\hat L_1^2,\hat L_2^2,\hat L^2,\hat L_z\}\]
- 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.