Angular Momentum Addition (Clebsch-Gordan) Calculator
Find the exact Clebsch-Gordan coefficient ⟨j₁m₁j₂m₂|jm⟩ for combining two angular momenta using Racah's formula.
🔗 What is the Clebsch-Gordan Calculator?
This Clebsch-Gordan calculator finds the exact coefficient that appears when two separate quantum angular momenta are combined into a state of definite total angular momentum. Enter the two individual angular momentum quantum numbers and their z-components, plus the target total j, and it returns the exact Clebsch-Gordan coefficient, the implied total m, and the probability of measuring that total angular momentum.
Whenever two quantum systems with angular momentum combine, whether two electron spins, an electron's orbital motion and its spin, or two atomic nuclei, the combined system can be described in two equally valid ways: by the individual quantum numbers of each part, or by the total angular momentum of the whole. Clebsch-Gordan coefficients are the exact bridge between these two descriptions.
This calculator uses Racah's sum formula, a finite, closed-form expression involving factorials and an alternating sum, not a numerical approximation. Every value it returns is exact to floating-point precision, verified here against the standard reference tables for combining two spin-1/2 particles and combining spin-1 with spin-1/2.
This calculator is useful for students and researchers working with spin-orbit coupling, nuclear and atomic spectroscopy, entangled quantum states, and any system where two angular momenta need to be added together correctly.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Two spin-1/2 particles, stretched triplet state
Example 2 - Two spin-1/2 particles, the singlet state
Example 3 - Spin-1 combined with spin-1/2 (spin-orbit-type coupling)
❓ Frequently Asked Questions
🔗 Related Calculators
What is a Clebsch-Gordan coefficient?
A Clebsch-Gordan coefficient, written ⟨j1 m1 j2 m2 | j m⟩, is the expansion coefficient that appears when combining two separate angular momentum states |j1 m1⟩ and |j2 m2⟩ into a state of definite total angular momentum |j m⟩. It tells you exactly how much of each total-angular-momentum state is present in a given combination of the two individual states.
What is the formula for Clebsch-Gordan coefficients?
This calculator uses Racah's closed-form sum formula, which expresses the coefficient exactly as a product of factorials and a finite alternating sum over an integer index k. It is not an approximation, it produces the exact value (to floating-point precision) for any valid combination of j1, m1, j2, m2, j, and m.
Why must total m equal m1 plus m2?
The z-component of angular momentum is additive: Jz = J1z + J2z always, so any combined state with total magnetic quantum number m can only be built from individual states whose m1 and m2 add up to that same m. Any Clebsch-Gordan coefficient with m ≠ m1+m2 is automatically zero, this is a strict selection rule, not an approximation.
What is the triangle rule for combining angular momenta?
When combining two angular momenta j1 and j2, the total j can only take the values |j1−j2|, |j1−j2|+1, ..., j1+j2, stepping by whole integers. This mirrors the classical picture of adding two vectors of length j1 and j2, the resultant length ranges from their difference (anti-parallel) to their sum (parallel).
What does combining two spin-1/2 particles give?
Two spin-1/2 particles (like two electrons) combine into a spin-1 triplet (three states, j=1, m=1,0,−1) and a spin-0 singlet (one state, j=0, m=0). The Clebsch-Gordan coefficients for this case are especially famous: the m=0 triplet and singlet states are both equal superpositions of the two product states, differing only by a relative sign, forming the basis of entangled Bell states in quantum information.
What is the physical meaning of the coefficient squared?
If a system made of two subsystems is prepared in the definite product state |j1 m1⟩|j2 m2⟩, then |⟨j1 m1 j2 m2 | j m⟩|² is the probability of measuring the combined system to have total angular momentum quantum numbers j and m. Summing this probability over every allowed j for fixed m1, m2 always adds up to exactly 1.
Where are Clebsch-Gordan coefficients used in physics?
They appear whenever two angular momenta combine: adding electron orbital and spin angular momentum (spin-orbit coupling, fine structure), combining the spins of two electrons or two nucleons, coupling nuclear spins in NMR, and describing entangled states and selection rules in atomic, nuclear, and particle physics.
Why is the general formula so complicated?
The Racah sum formula involves six or more factorials and a finite sum, because it has to correctly encode both the overall normalization of the coupled states and the specific overlap between two different, non-commuting ways of describing the same physical system (individual vs. total angular momentum). Despite the complexity, every term is an exact rational or square-root expression, there is no numerical approximation involved.
Can Clebsch-Gordan coefficients be negative?
Yes. Following the standard Condon-Shortley sign convention (used by this calculator), coefficients can be positive, negative, or zero. The sign matters physically, it determines whether product states interfere constructively or destructively when combined into a total angular momentum eigenstate, as seen in the antisymmetric singlet state of two spin-1/2 particles.
What happens if I enter an invalid combination?
The calculator checks the triangle rule (j between |j1−j2| and j1+j2), that m1 and m2 fall within their allowed ranges, and that all quantum numbers are consistent multiples of 1/2 before computing anything. If m1+m2 does not equal j's implied m, or j falls outside the allowed range, the Clebsch-Gordan coefficient is genuinely zero by the underlying selection rules, not just unavailable.