Guided reading
The section uses the change-of-basis matrix obtained from Clebsch-Gordan coefficients to transform an operator. The purpose is practical: many operators are not diagonal in the local basis but become diagonal in the coupled basis.
Basis-change rule
The operator before the change of basis is \(\hat A_a\). After the change, it is
\[\hat A_d=\hat U\hat A_a\hat U^\dagger,\]
where \(\hat U\) is the change-of-basis matrix and \(\hat U^\dagger\) is its conjugate transpose.
Operator used in the example
The example returns to the scalar-product operator
\[\hat H=\vec{\hat S}_1\cdot\vec{\hat S}_2=\hat S_{1x}\hat S_{2x}+\hat S_{1y}\hat S_{2y}+\hat S_{1z}\hat S_{2z},\]
now with \(s_1=s_2=1/2\). In the local basis, after the Hilbert-space expansion, the operator is
\[\hat H_a=\frac{\hbar^2}{4}\begin{pmatrix}1&0&0&0\\0&-1&2&0\\0&2&-1&0\\0&0&0&1\end{pmatrix}.\]
Transform to the coupled basis
Using the matrix \(\hat U\) from the \(l_1=l_2=1/2\) coupling,
\[\hat H_d=\hat U\hat H_a\hat U^\dagger.\]
The result is diagonal:
\[\hat H_d=\frac{\hbar^2}{4}\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-3\end{pmatrix}.\]
Reading the result
The transformed operator has three degenerate states with eigenvalue \(\hbar^2/4\) and one non-degenerate state with eigenvalue \(-3\hbar^2/4\).
The section also emphasizes that changing from local to coupled basis does not necessarily diagonalize every operator.
Exercise-ready boundary
This page supports compact guided exercises on: operator basis change using the Clebsch-Gordan 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
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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.