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Quantum Mechanics

Clebsch–Gordan Coefficients

This page gives low-spin Clebsch–Gordan coefficients for common quantum-mechanics problems. It is not a full generated atlas.

For table-reading rules, phase conventions, ordering checks, and Wigner 3j3j conversion factors, see Clebsch–Gordan Tables and Conventions.

Convention

Section titled “Convention”

The product basis is

∣j1m1⟩∣j2m2⟩,\lvert j_1m_1\rangle\lvert j_2m_2\rangle,

and coupled states are expanded as

∣JM⟩=∑m1,m2⟨j1m1,j2m2∣JM⟩∣j1m1⟩∣j2m2⟩.\lvert JM\rangle =\sum_{m_1,m_2} \langle j_1m_1,j_2m_2\vert JM\rangle \lvert j_1m_1\rangle\lvert j_2m_2\rangle.

The phase convention is Condon–Shortley.

Two Spin-One-Half Particles

Section titled “Two Spin-One-Half Particles”

Write ∣+⟩=∣1/2,1/2⟩\lvert+\rangle=\lvert1/2,1/2\rangle and ∣−⟩=∣1/2,−1/2⟩\lvert-\rangle=\lvert1/2,-1/2\rangle.

Coupled StateProduct-Basis Expansion
∣1,1⟩\lvert1,1\rangle∣++⟩\lvert++\rangle
∣1,0⟩\lvert1,0\rangle(∣+−⟩+∣−+⟩)/2(\lvert+-\rangle+\lvert-+\rangle)/\sqrt2
∣1,−1⟩\lvert1,-1\rangle∣−−⟩\lvert--\rangle
∣0,0⟩\lvert0,0\rangle(∣+−⟩−∣−+⟩)/2(\lvert+-\rangle-\lvert-+\rangle)/\sqrt2

One Plus One-Half

Section titled “One Plus One-Half”

For j1=1j_1=1 and j2=1/2j_2=1/2:

Coupled StateProduct-Basis Expansion
∣3/2,3/2⟩\lvert3/2,3/2\rangle∣1,1⟩∣1/2,1/2⟩\lvert1,1\rangle\lvert1/2,1/2\rangle
∣3/2,1/2⟩\lvert3/2,1/2\rangle2/3∣1,0⟩∣1/2,1/2⟩+1/3∣1,1⟩∣1/2,−1/2⟩\sqrt{2/3}\lvert1,0\rangle\lvert1/2,1/2\rangle+\sqrt{1/3}\lvert1,1\rangle\lvert1/2,-1/2\rangle
∣3/2,−1/2⟩\lvert3/2,-1/2\rangle1/3∣1,−1⟩∣1/2,1/2⟩+2/3∣1,0⟩∣1/2,−1/2⟩\sqrt{1/3}\lvert1,-1\rangle\lvert1/2,1/2\rangle+\sqrt{2/3}\lvert1,0\rangle\lvert1/2,-1/2\rangle
∣3/2,−3/2⟩\lvert3/2,-3/2\rangle∣1,−1⟩∣1/2,−1/2⟩\lvert1,-1\rangle\lvert1/2,-1/2\rangle
∣1/2,1/2⟩\lvert1/2,1/2\rangle1/3∣1,0⟩∣1/2,1/2⟩−2/3∣1,1⟩∣1/2,−1/2⟩\sqrt{1/3}\lvert1,0\rangle\lvert1/2,1/2\rangle-\sqrt{2/3}\lvert1,1\rangle\lvert1/2,-1/2\rangle
∣1/2,−1/2⟩\lvert1/2,-1/2\rangle2/3∣1,−1⟩∣1/2,1/2⟩−1/3∣1,0⟩∣1/2,−1/2⟩\sqrt{2/3}\lvert1,-1\rangle\lvert1/2,1/2\rangle-\sqrt{1/3}\lvert1,0\rangle\lvert1/2,-1/2\rangle

Selection Rules

Section titled “Selection Rules”
RuleMeaning
M=m1+m2M=m_1+m_2magnetic quantum numbers add
∣j1−j2∣≤J≤j1+j2\lvert j_1-j_2\rvert\le J\le j_1+j_2triangle condition
Coefficients are zero outside these rulesdo not compute forbidden entries

Common Mistakes

Section titled “Common Mistakes”
  • Comparing coefficients from different phase conventions.
  • Reversing the order of the two angular momenta.
  • Forgetting that coupled states form a complete basis of the same tensor-product space.
  • Treating the singlet state as symmetric; it is antisymmetric under exchange of the two spin labels.
Section titled “Canonical Links”

References

Section titled “References”
  • A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
  • J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
  • D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.