Clebsch-Gordan tables tell you how to combine two angular momenta in quantum mechanics, but their compact notation makes them notoriously hard to read at first glance. Each table is essentially a lookup chart: you go in with quantum numbers for two individual particles or systems, and you come out with the coefficients needed to express combined states as sums of product states (or vice versa). Once you understand the layout, reading them becomes mechanical.
How the Table Is Organized
There is one table (or block) for each pair of angular momentum values j₁ and j₂. You’ll see this pair labeled at the top, written as something like “1 × 1/2” or “1 × 1.” Everything inside that block only applies to that specific combination.
Within each block, the columns are labeled by J and M, the quantum numbers for the total (coupled) angular momentum. The rows are labeled by m₁ and m₂, the quantum numbers for each individual angular momentum. The entry sitting at the intersection of a row and column is the Clebsch-Gordan coefficient connecting those particular quantum numbers. The whole table is really just a change-of-basis matrix: it translates between the “individual” description (m₁, m₂) and the “combined” description (J, M).
The Square Root Convention
This trips up almost everyone the first time. The numbers printed in the table are not the coefficients themselves. Every entry is understood to be under a square root sign. So if you see “2/3” in the table, the actual coefficient is √(2/3). If you see “1,” the coefficient is √1 = 1.
When a minus sign appears, it sits outside the square root. So “−1/3” in the table means −√(1/3), not √(−1/3). Some tables print this slightly differently, but the Particle Data Group tables and most textbook versions follow this rule consistently.
Finding a Specific Coefficient
Suppose you need the Clebsch-Gordan coefficient for combining j₁ = 1 and j₂ = 1/2 into a total state with J = 3/2, M = −1/2. Here’s the process step by step:
- Locate the right block. Find the table labeled “1 × 1/2.”
- Find the column. Look along the top row for the column headed J = 3/2, M = −1/2.
- Find the row. Look down the left side for the row with your chosen m₁ and m₂ values. Since M = m₁ + m₂ must equal −1/2, the possible rows are (m₁ = 1, m₂ = −1/2), (m₁ = 0, m₂ = 1/2), and (m₁ = −1, m₂ = 1/2) if it satisfies the constraint.
- Read the entry. The number at that intersection, placed under a square root (with any minus sign outside), is your coefficient.
For this particular example, reading down the J = 3/2, M = −1/2 column gives two nonzero entries: √(1/3) for the row (m₁ = 1, m₂ = −1/2) and −√(2/3) for the row (m₁ = 0, m₂ = 1/2). That tells you the combined state is:
|J = 3/2, M = −1/2⟩ = √(1/3) |m₁ = 1, m₂ = −1/2⟩ − √(2/3) |m₁ = 0, m₂ = 1/2⟩
Reading Columns vs. Reading Rows
The table works in both directions, and understanding this is the key to using it flexibly.
Reading down a column gives you a coupled state (|J, M⟩) expressed as a sum of product states (|m₁, m₂⟩). Each nonzero entry in that column is one term in the sum, with the coefficient telling you how much of that particular product state contributes. This is the direction you use when you know the total angular momentum and want to decompose it.
Reading across a row gives you the reverse: a product state (|m₁, m₂⟩) expressed as a sum of coupled states (|J, M⟩). For instance, in the 1/2 × 1/2 table, reading across the row for (m₁ = 1/2, m₂ = −1/2) tells you that the “particle 1 spin-up, particle 2 spin-down” state equals (1/√2)|J = 1, M = 0⟩ + (1/√2)|J = 0, M = 0⟩. You use this direction when you know the state of each particle individually and want to figure out the probability of measuring a particular total angular momentum.
Rules That Make Entries Zero
Before you even look at the table, you can rule out many coefficients as zero. A Clebsch-Gordan coefficient is automatically zero unless two conditions hold:
- m₁ + m₂ = M. The individual magnetic quantum numbers must add up to the total magnetic quantum number. This is why, for a given column (fixed M), only certain rows have nonzero entries.
- The triangle inequality is satisfied. The total angular momentum J must fall in the range |j₁ − j₂| ≤ J ≤ j₁ + j₂. For j₁ = 1 and j₂ = 1/2, J can only be 3/2 or 1/2. You will never find a J = 5/2 column in that block because it violates this rule.
These two rules dramatically reduce the size of each table. Most entries that could exist in principle are zero, so the printed tables only include the nonzero entries and the few zeros needed to fill out the grid.
The Sign Convention
Clebsch-Gordan coefficients are not uniquely determined by the physics alone. There is a freedom in choosing the overall sign (phase) of each coupled state, and different choices would give different tables. Nearly all published tables, including those from the Particle Data Group, use the Condon-Shortley phase convention. Under this convention, the coefficient ⟨j₁, j₂, j₁, J − j₁ | J, J⟩ is always positive. In plain terms: the “stretched” state where m₁ takes its maximum value j₁ always has a positive coefficient.
This matters when you compare results from different sources. As long as both use the Condon-Shortley convention (and almost all do), the tables will agree. One practical consequence of the phase convention is the symmetry relation: swapping the two particles (exchanging j₁ with j₂ and m₁ with m₂) multiplies the coefficient by (−1) raised to the power J − j₁ − j₂. So the order of coupling matters for the sign, even though the magnitude stays the same.
Checking Your Reading With Orthogonality
If you’re unsure whether you’ve read the table correctly, there’s a built-in consistency check. The columns of a Clebsch-Gordan table are orthonormal: if you take two different columns (different J or M values) and multiply them entry by entry, then sum those products, you get zero. If you do the same with a column against itself, the sum equals one. Rows have the same property.
This is worth doing the first time you use a new table. Pick any column, square each entry, and add them up. If you get 1, you’re reading the coefficients correctly (remembering to take the square root of the printed values first). If you get something else, you’ve likely misidentified which number sits under the radical or missed a minus sign.
A Worked Example: 1/2 × 1/2
The simplest nontrivial table combines two spin-1/2 particles. Here j₁ = 1/2 and j₂ = 1/2, so J can be 1 (the triplet) or 0 (the singlet). There are four product states: (m₁ = +1/2, m₂ = +1/2), (+1/2, −1/2), (−1/2, +1/2), and (−1/2, −1/2).
The table has four columns for the coupled states: |1, 1⟩, |1, 0⟩, |1, −1⟩, and |0, 0⟩. Reading each column:
- |1, 1⟩ = |+1/2, +1/2⟩. Just one term with coefficient 1.
- |1, 0⟩ = (1/√2)|+1/2, −1/2⟩ + (1/√2)|−1/2, +1/2⟩. An equal mix of both “one up, one down” arrangements.
- |1, −1⟩ = |−1/2, −1/2⟩. Again a single term.
- |0, 0⟩ = (1/√2)|+1/2, −1/2⟩ − (1/√2)|−1/2, +1/2⟩. Same magnitudes as |1, 0⟩, but with a crucial minus sign making it antisymmetric.
Notice that the |1, 0⟩ and |0, 0⟩ columns both involve the same two product states but with different signs. That minus sign is the entire difference between the triplet and singlet states, and it comes straight from the orthogonality requirement. You can verify it: multiply the |1, 0⟩ column by the |0, 0⟩ column entry by entry and sum the results. You get (1/√2)(1/√2) + (1/√2)(−1/√2) = 1/2 − 1/2 = 0, confirming the columns are orthogonal.