04 Jun 2026
Addition of Angular Momentum and Clebsch-Gordan Coefficients
Addition theorem, coupled and uncoupled bases, Clebsch-Gordan coefficients, and simple examples.
Many quantum systems contain more than one angular momentum. An electron in an atom has orbital angular momentum and spin. Two particles may each carry spin. In such cases the total angular momentum is obtained by adding angular momentum operators.
The main question is how to describe the same state in two languages: individual angular momenta, or total angular momentum. Clebsch-Gordan coefficients provide the conversion between these two descriptions.
Total angular momentum
Let two angular momenta be $\mathbf J_1$ and $\mathbf J_2$. The total angular momentum is
\[\mathbf J=\mathbf J_1+\mathbf J_2.\]The corresponding operators satisfy
\[J_z=J_{1z}+J_{2z}\]and
\[J^2=(\mathbf J_1+\mathbf J_2)^2 =J_1^2+J_2^2+2\mathbf J_1\cdot\mathbf J_2.\]Uncoupled basis
The uncoupled basis is
\[\lvert j_1m_1\rangle\lvert j_2m_2\rangle \equiv \lvert j_1m_1;j_2m_2\rangle.\]It diagonalizes
\[J_1^2,\quad J_2^2,\quad J_{1z},\quad J_{2z}.\]In this basis, the total magnetic quantum number is
\[m=m_1+m_2.\]This basis is natural when the two angular momenta are considered separately.
Coupled basis
The coupled basis is
\[\lvert j_1j_2;JM\rangle.\]It diagonalizes
\[J_1^2,\quad J_2^2,\quad J^2,\quad J_z.\]The allowed total angular momentum values are
\[\boxed{ J=\lvert j_1-j_2\rvert,\ \lvert j_1-j_2\rvert+1,\dots,j_1+j_2. }\]For each $J$,
\[M=-J,-J+1,\dots,J.\]This basis is natural when the Hamiltonian depends on the total angular momentum, for example through spin-orbit or spin-spin coupling.
Clebsch-Gordan expansion
The coupled and uncoupled bases are related by
\[\boxed{ \lvert j_1j_2;JM\rangle = \sum_{m_1,m_2} C^{JM}_{j_1m_1\,j_2m_2} \lvert j_1m_1;j_2m_2\rangle. }\]The numbers
\[C^{JM}_{j_1m_1\,j_2m_2} = \langle j_1m_1;j_2m_2\mid j_1j_2;JM\rangle\]are called Clebsch-Gordan coefficients.
They are nonzero only when
\[\boxed{ M=m_1+m_2. }\]Example: two spin-one-half particles
Let
\[j_1=j_2=\frac12.\]The allowed total angular momenta are
\[J=1,\ 0.\]The triplet states are
\[\lvert 1,1\rangle=\lvert +\rangle\lvert +\rangle,\] \[\lvert 1,0\rangle= \frac{1}{\sqrt2} \left( \lvert +\rangle\lvert -\rangle+ \lvert -\rangle\lvert +\rangle \right),\] \[\lvert 1,-1\rangle=\lvert -\rangle\lvert -\rangle.\]The singlet state is
\[\lvert 0,0\rangle= \frac{1}{\sqrt2} \left( \lvert +\rangle\lvert -\rangle- \lvert -\rangle\lvert +\rangle \right).\]The triplet states are symmetric under particle interchange, while the singlet state is antisymmetric.
Physical importance
Angular momentum addition is used in:
- spin-orbit coupling,
- fine structure of atoms,
- addition of orbital and spin angular momentum,
- coupling of nuclear spins,
- selection rules in atomic transitions.
Example with $j_1=1$ and $j_2=1/2$
If
\[j_1=1,\qquad j_2=\frac12,\]then
\[J=|1-\tfrac12|,\ |1-\tfrac12|+1,\dots,1+\tfrac12.\]Therefore
\[\boxed{ J=\frac12,\ \frac32. }\]The number of coupled states is
\[(2\times\tfrac12+1)+(2\times\tfrac32+1) =2+4=6.\]This agrees with the number of uncoupled product states:
\[(2j_1+1)(2j_2+1)=3\times2=6.\]Selection rule for coefficients
A Clebsch-Gordan coefficient
\[C^{JM}_{j_1m_1\,j_2m_2}\]can be non-zero only if
\[M=m_1+m_2.\]For example, if $m_1=1$ and $m_2=-1/2$, then
\[M=\frac12.\]Any coefficient with the same $m_1,m_2$ but a different $M$ is zero.
Main points
- Angular momenta add as operators: $\mathbf J=\mathbf J_1+\mathbf J_2$.
- The allowed total values are $J=\lvert j_1-j_2\rvert,\dots,j_1+j_2$.
- Clebsch-Gordan coefficients connect coupled and uncoupled bases.
- The condition $M=m_1+m_2$ must hold.
- Two spin-one-half particles give one triplet and one singlet multiplet.
Practice questions
- Find the possible $J$ values when $j_1=1$ and $j_2=1/2$.
- How many total states are obtained by adding $j_1=1$ and $j_2=1$?
- Write the singlet state for two spin-one-half particles.
- Why must $M=m_1+m_2$ for a nonzero Clebsch-Gordan coefficient?
- Distinguish coupled and uncoupled bases.
Discussion