Index
Chapter 7 · Item 7.4
Vectors in the coupled basis
Sequential and non-sequential construction
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Guided reading

The section extends the two-angular-momentum example to \(N\) angular momenta. The construction always adds angular momenta in pairs and checks the allowed interval \(|l_a-l_b|\le l\le l_a+l_b\) at every step.

Sequentially coupled basis

For four angular momenta \(l_1=l_2=l_3=l_4=1/2\), the sequential construction is

\[\vec{\hat L}_{12}=\vec{\hat L}_1+\vec{\hat L}_2,\qquad \vec{\hat L}_{13}=\vec{\hat L}_{12}+\vec{\hat L}_3,\qquad \vec{\hat L}=\vec{\hat L}_{14}=\vec{\hat L}_{13}+\vec{\hat L}_4.\]

The allowed intermediate values are obtained from

\[|l_1-l_2|\le l_{12}\le l_1+l_2,\qquad |l_{12}-l_3|\le l_{13}\le l_{12}+l_3,\qquad |l_{13}-l_4|\le l\le l_{13}+l_4.\]
Sequential construction of the coupled basis
Sequential construction for four \(1/2\) angular momenta: first \(l_{12}\), then \(l_{13}\), then \(l\).
Sequential vectors

The simplified sequential coupled basis omits the constant \(l_1,l_2,l_3,l_4\) and is written as \(|l_{12},l_{13},l,m\rangle\):

\(|l_{12},l_{13},l,m\rangle\)
\(|1,3/2,2,m\rangle\)
\(|1,3/2,1,m\rangle\)
\(|1,1/2,1,m\rangle\)
\(|1,1/2,0,m\rangle\)
\(|0,1/2,1,m\rangle\)
\(|0,1/2,0,m\rangle\)

The coupled basis has one quintuplet, three triplets and two singlets, preserving the Hilbert-space dimension \(16\).

Non-sequentially coupled basis

The angular momenta do not need to be added sequentially. For the same example of four spins \(1/2\), with \(l_1=l_2=l_3=l_4=1/2\), the section also uses

\[\vec{\hat L}_{12}=\vec{\hat L}_1+\vec{\hat L}_2,\qquad \vec{\hat L}_{34}=\vec{\hat L}_3+\vec{\hat L}_4,\qquad \vec{\hat L}=\vec{\hat L}_{12}+\vec{\hat L}_{34}.\]

The allowed intermediate values are obtained from

\[|l_1-l_2|\le l_{12}\le l_1+l_2,\qquad |l_3-l_4|\le l_{34}\le l_3+l_4,\qquad |l_{12}-l_{34}|\le l\le l_{12}+l_{34}.\]
Non-sequential construction of the coupled basis
Non-sequential construction: first \(l_{12}\) and \(l_{34}\), then the total \(l\).

This choice depends on the system under study and still uses the same allowed-value condition at each pairwise addition. Other non-sequential choices and the extension to three or more angular momenta remain in Chapter 7.

Non-sequential vectors

The corresponding basis is written as \(|l_{12},l_{34},l,m\rangle\):

\(|l_{12},l_{34},l,m\rangle\)
\(|1,1,2,m\rangle\)
\(|1,1,1,m\rangle\)
\(|1,1,0,m\rangle\)
\(|1,0,1,m\rangle\)
\(|0,1,1,m\rangle\)
\(|0,0,0,m\rangle\)

As expected, the dimension is preserved again: one quintuplet, three triplets and two singlets.

Exercise-ready boundary

This page supports compact guided exercises on: construction of coupled-basis vectors for several angular momenta.

  • 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: construction of coupled-basis vectors for several angular momenta.
  • Conceptual check: identify the basis or angular-momentum interaction before applying the equation.
  • Key equation 1:
    \[|l_{12},l_{13},l,m\rangle\]
  • Key equation 2:
    \[|l_{12},l_{34},l,m\rangle\]
  • Typical task: apply one central equation and state what basis, quantum numbers or physical consequence it uses.
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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.