Clebsch–Gordan Computation

Computing Clebsch–Gordan coefficients is a structured change-of-basis problem. Reliable output requires exact admissibility tests, a declared phase convention, and checks of orthogonality and angular-momentum eigenvalues. A list of plausible square roots is insufficient: inconsistent signs can change interference terms while preserving individual probabilities.

Required background. Clebsch–Gordan coefficients defines the coupled basis; tables and conventions fixes the Condon–Shortley convention used here.

Exact quantum numbers

Use

$$C^{JM}_{j_1m_1,j_2m_2} =\langle j_1m_1;j_2m_2\vert JM\rangle.$$

Store doubled quantum numbers $2j_1,2m_1,2j_2,2m_2,2J,2M$ as integers. Then half-integer parity checks and triangle rules become exact integer tests, avoiding comparisons of rounded decimal inputs.

A coefficient vanishes unless $m_1+m_2=M$, $|m_i|\le j_i$, $|M|\le J$, $|j_1-j_2|\le J\le j_1+j_2$, with $j_i-m_i$, $J-M$, and $j_1+j_2-J$ nonnegative integers. Check these conditions before evaluating a factorial expression. They eliminate invalid requests and prevent negative factorial arguments from masquerading as arithmetic failures.

For software returning Wigner symbols, the convention conversion is

$$C^{JM}_{j_1m_1,j_2m_2} =(-1)^{j_1-j_2+M}\sqrt{2J+1} \begin{pmatrix}j_1&j_2&J\\m_1&m_2&-M\end{pmatrix}.$$

The argument $-M$ and the phase both matter. Verify the software’s argument ordering rather than assuming every interface takes $j$ and $m$ values in the same order.

Constructing a coupling matrix

Build dimensionless single-spin matrices with

$$(J_z/\hbar)_{mm}=m, \qquad J_+\lvert j,m\rangle =\hbar\sqrt{(j-m)(j+m+1)}\lvert j,m+1\rangle.$$

In the ordered product basis form $\mathbf J=\mathbf J_1\otimes I+I\otimes\mathbf J_2$. Work in fixed-$M$ blocks of $J_z$ and diagonalize $J^2/\hbar^2$. Its eigenvalues are $J(J+1)$. The normalized eigenvectors supply coupling coefficients, but their arbitrary signs still need a convention.

For each $J$, choose the highest-weight state with the standard positive coefficient on its largest allowed $m_1$ component. Generate lower $M$ states with

$$\lvert J,M-1\rangle =\frac{J_-\lvert J,M\rangle} {\hbar\sqrt{(J+M)(J-M+1)}}.$$

This ties the phases within a multiplet together. Independently diagonalizing every $M$ block and accepting arbitrary signs does not do so. For coupling more than two spins, multiplicity spaces introduce further basis choices that must also be declared.

Two spin-one-half factors

Order the product states as $\lvert\uparrow\uparrow\rangle$, $\lvert\uparrow\downarrow\rangle$, $\lvert\downarrow\uparrow\rangle$, $\lvert\downarrow\downarrow\rangle$. In the $M=0$ block,

$$J^2/\hbar^2=\begin{pmatrix}1&1\\1&1\end{pmatrix}.$$

The symmetric vector has eigenvalue two and gives $\lvert1,0\rangle=(\lvert\uparrow\downarrow\rangle+\lvert\downarrow\uparrow\rangle)/\sqrt2$. The antisymmetric vector has eigenvalue zero and, with the stated phase, gives $\lvert0,0\rangle=(\lvert\uparrow\downarrow\rangle-\lvert\downarrow\uparrow\rangle)/\sqrt2$.

Place coupled states in columns of $U$. Then $U^\dagger U=I$ and $U^\dagger J^2U$ is diagonal. Together with the lowering relation, these tests check magnitudes, phases between multiplet states, and the interpretation of rows and columns.

Large angular momenta and cancellation

Direct factorial formulas can overflow even when the final coefficient has magnitude at most one. Log-factorials control range but do not remove cancellation between alternating terms. Use exact arithmetic for moderate quantum numbers or algorithms designed for stable recurrences or exact integer accumulation at larger values. Increasing floating-point precision should be a diagnostic with a demonstrated plateau, not an untested cure.

For a full fixed-$M$ block, check column orthogonality, row completeness, and the $J^2$ residual. Compare selected low-spin values with an independent convention reference. Exchange symmetry gives an additional test:

$$C^{JM}_{j_2m_2,j_1m_1} =(-1)^{j_1+j_2-J}C^{JM}_{j_1m_1,j_2m_2}.$$

Phase-insensitive tests alone cannot establish convention agreement. Conversely, a uniform sign change of an entire multiplet may be a convention difference rather than incorrect physics if all related quantities are transformed consistently.

Exercises

A routine returns the correct normalized triplet and singlet vectors but negates only $\lvert1,0\rangle$. Which check detects the inconsistency?

Solution

Orthogonality and the $J^2$ eigenvalue test still pass. The ladder test fails: $J_-\lvert1,1\rangle=\sqrt2\hbar\lvert1,0\rangle$ requires the phase fixed by the positive ladder coefficient. Negating every state in the multiplet instead would preserve that relation.

References

• Edmonds, A. R., Angular Momentum in Quantum Mechanics, Princeton University Press, 1957, reprinted 1996 — coupling phases and recurrence relations.
• Johansson, H. T., and C. Forssén, “Fast and Accurate Evaluation of Wigner 3j, 6j, and 9j Symbols Using Prime Factorisation and Multi-Word Integer Arithmetic,” SIAM Journal on Scientific Computing 38, A376–A384 (2016), author manuscript — exact accumulation avoids cancellation in factorial sums.
• National Institute of Standards and Technology, Digital Library of Mathematical Functions, §34.3, Basic Properties of the 3j Symbol — symmetry and orthogonality checks.