Guided reading
Operators from different angular-momentum subspaces may have different matrix dimensions. The section solves this by expanding every operator and vector into the total Hilbert space.
Why expansion is needed
If a system has \(l_1=1/2\) and \(l_2=1\), the two Hilbert subspaces have dimensions \(2\times2\) and \(3\times3\). Operations between their matrices require a common total dimension.
\[\hbox{total dimension}=\prod_{i=1}^{N}(2l_i+1).\]
Operator expansion
The tensor products are
\[\begin{aligned}\hat L_{1u}&=\hat L_u^{(1)}\otimes 1^{(2)}\otimes 1^{(3)}\otimes\cdots\otimes 1^{(N)},\\ \hat L_{2u}&=1^{(1)}\otimes \hat L_u^{(2)}\otimes 1^{(3)}\otimes\cdots\otimes 1^{(N)},\\ \hat L_{Nu}&=1^{(1)}\otimes 1^{(2)}\otimes 1^{(3)}\otimes\cdots\otimes \hat L_u^{(N)}.\end{aligned}\]
Here \(u=(x,y,z)\). After the tensor products, all components have the correct common dimension.
Vector expansion
The vectors are expanded in the same way:
\[\begin{aligned}|l_1,l_2,l_3,\ldots,l_N,m_1,m_2,m_3,\ldots,m_N\rangle={}&|l_1,m_1\rangle\otimes |l_2,m_2\rangle\\ &\otimes |l_3,m_3\rangle\otimes\cdots\otimes |l_N,m_N\rangle.\end{aligned}\]
Example 7.4: spin product
The example considers
\[\hat H=\vec{\hat S}_1\cdot\vec{\hat S}_2=\hat S_{1x}\hat S_{2x}+\hat S_{1y}\hat S_{2y}+\hat S_{1z}\hat S_{2z},\]
with \(s_1=1/2\) and \(s_2=1\). One component is expanded as
\[\hat S_{1x}=\hat S_x^{(1)}\otimes 1^{(2)},\qquad \hat S_{2x}=1^{(1)}\otimes\hat S_x^{(2)}.\]
The same procedure is used for the other components before summing the Hamiltonian.
Example 7.4: expansion step by step
For \(s_1=1/2\) and \(s_2=1\), the Hilbert-space dimension is \((2s_1+1)(2s_2+1)=6\). The \(x\)-component of spin 1 is expanded with the identity in subspace 2:
\[\hat S_{1x}=\hat S_x^{(1)}\otimes 1^{(2)}=\frac{\hbar}{2}\begin{pmatrix}0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\\1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\end{pmatrix}.\]
The \(x\)-component of spin 2 is expanded with the identity in subspace 1:
\[\hat S_{2x}=1^{(1)}\otimes\hat S_x^{(2)}=\frac{\hbar}{\sqrt2}\begin{pmatrix}0&1&0&0&0&0\\1&0&1&0&0&0\\0&1&0&0&0&0\\0&0&0&0&1&0\\0&0&0&1&0&1\\0&0&0&0&1&0\end{pmatrix}.\]
The remaining components are obtained by the same tensor-product method before the products \(\hat S_{1x}\hat S_{2x}\), \(\hat S_{1y}\hat S_{2y}\) and \(\hat S_{1z}\hat S_{2z}\) are summed.
Matrix shown in the section
For \(s_1=1/2\) and \(s_2=1\), the resulting matrix is
\[\hat H=\frac{\hbar^2}{2}\begin{pmatrix}1&0&0&0&0&0\\0&0&0&\sqrt{2}&0&0\\0&0&-1&0&\sqrt{2}&0\\0&\sqrt{2}&0&-1&0&0\\0&0&\sqrt{2}&0&0&0\\0&0&0&0&0&1\end{pmatrix}.\]
The local basis vector example is \(|s_1,s_2,m_1,m_2\rangle=|s_1,m_1\rangle\otimes |s_2,m_2\rangle\).
Example local vector
The same expansion applies to vectors. The example state is
\[\left|\frac12,1,\frac12,0\right\rangle=\left|\frac12,\frac12\right\rangle\otimes |1,0\rangle.\]
With the column vectors used in the section, this gives
\[\left|\frac12,1,\frac12,0\right\rangle=\begin{pmatrix}1\\0\end{pmatrix}\otimes\begin{pmatrix}0\\1\\0\end{pmatrix}=\begin{pmatrix}0\\1\\0\\0\\0\\0\end{pmatrix}.\]
Exercise-ready boundary
This page supports compact guided exercises on: tensor-product expansion of operators and local basis vectors.
- 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: tensor-product expansion of operators and local basis vectors.
- Conceptual check: identify the basis or angular-momentum interaction before applying the equation.
- Key equation 1:
\[\hat L_{1u}=\hat L_u^{(1)}\otimes 1^{(2)}\otimes\cdots\otimes 1^{(N)}\]
- Key equation 2:
\[|l_1,\ldots,l_N,m_1,\ldots,m_N\rangle=|l_1,m_1\rangle\otimes\cdots\otimes |l_N,m_N\rangle\]
- 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.