Wigner 6j Symbol Calculator

Formula used

The Wigner 6j symbol { j₁ j₂ j₃ ; j₄ j₅ j₆ } is evaluated with the Racah summation formula, using four triangle factors Δ(j₁,j₂,j₃), Δ(j₁,j₅,j₆), Δ(j₄,j₂,j₆), Δ(j₄,j₅,j₃). The symbol is zero unless these triads satisfy triangle and parity rules.

The triangle factor is Δ(a,b,c) = √[(a+b−c)!(a−b+c)!(−a+b+c)! / (a+b+c+1)!] (with factorial arguments becoming ordinary integers for valid half‑integer sets).

The remaining part is a finite sum over an integer index z, whose limits are determined by the same selection rules. This implementation uses log‑factorials (lgamma) for numerical stability.

How to use this calculator

1. Enter six non‑negative j values as integers or half‑integers.
2. Use fractions like 3/2 or decimals like 1.5.
3. Choose an output format and a display precision.
4. Click Calculate to see the value above the form.
5. Use Download CSV or Download PDF to export.

Example data table

These examples use the special case with j₆ = 0 and matching deltas.

Notes for advanced use

• The calculator checks each required triangle triad before summing.
• If the z-range is empty, the result is zero.
• Very large j values can cause loss of precision in floating arithmetic.

Wigner 6j symbol in practice

1) Why the 6j symbol matters

The Wigner 6j symbol encodes recoupling of three angular momenta. It links two equivalent coupling orders, such as (j1+j2)=j12 versus (j2+j3)=j23. In practice it appears in reduced matrix elements, hyperfine structure, line strengths, and tensor‑operator algebra. It is also a building block for 9j and higher 3nj symbols.

2) Inputs and accepted forms

Each j is a non‑negative integer or half‑integer. Enter values like 2, 3/2, or 1.5. The calculator validates “multiple of 0.5” inputs and rejects malformed entries before computing.

3) Triangle conditions as selection rules

The symbol is zero unless four triangle triads hold: (j1,j2,j3), (j1,j5,j6), (j4,j2,j6), and (j4,j5,j3). Each must satisfy |a-b| ≤ c ≤ a+b and have integer a+b+c. These constraints mirror angular‑momentum addition rules and act as fast filters.

4) Racah sum and the z‑range

With valid triangles, the Racah formula becomes a finite sum over an integer index z. The lower and upper bounds come from simple max/min combinations of the inputs, so the sum length is predictable. Typical cases sum over a few to a few dozen terms. If the bounds cross, the z-range is empty and the result is exactly zero.

5) Stable evaluation with log‑factorials

Factorials overflow quickly, so the computation uses log‑factorials via lgamma(n+1). This keeps terms like (z+1)! and denominator factorials well‑behaved and reduces underflow/overflow. The final value is reconstructed in floating arithmetic; for very large quantum numbers, cancellation can still limit digits.

6) Symmetries for verification

A 6j symbol is invariant under any permutation of its three columns and under swapping top/bottom entries in any two columns. These symmetries are practical checks: apply a valid symmetry transformation and confirm the same value is returned.

7) Interpreting sign and magnitude

Values are real for physical inputs and often small due to alternating signs and cancellation in the sum. The sign comes from the phase factor in the Racah expression, while the magnitude reflects overlap between coupling schemes. Near‑zero outputs can be meaningful and may indicate weak recoupling.

8) Using exports in real workflows

In atomic, nuclear, and molecular calculations, 6j symbols are evaluated repeatedly across quantum‑number grids. Use CSV to store batch inputs and results for reproducibility or regression tests. Use PDF to attach a clean record to lab notes and reports when you need an auditable trail.

FAQs

1) What formats can I use for half‑integers?

You can enter half‑integers as fractions like 3/2 or decimals like 1.5. The calculator verifies the value is a multiple of 0.5 before computing.

2) Why does the calculator sometimes return exactly zero?

If any required triangle condition fails, or if the computed z-sum has no valid integer range, the Wigner 6j symbol is mathematically zero, so the output is zero.

3) Does input order matter?

The numeric value respects 6j symmetry rules, but arbitrary reshuffling is not always a symmetry. If you permute according to column permutations or valid swaps, you should get the same result.

4) How accurate is the result for larger j values?

Log‑factorials improve stability, but very large quantum numbers can still accumulate floating‑point error due to cancellation. Use higher precision in external libraries if you need many significant digits.

5) What does the “precision” setting change?

It controls rounding of the displayed decimal value. The internal computation is the same, but the formatted output and PDF/CSV numeric fields follow the selected number of decimal places.

6) Can the 6j symbol be complex?

For standard angular momentum quantum numbers (integers and half‑integers) the Wigner 6j symbol is real. Complex results would indicate non‑physical inputs or a different generalized definition.

7) How do I cite or record my calculation?

Use the PDF export for a report‑friendly record including inputs and the computed value. The CSV export is better for datasets, auditing, or repeating the same parameter grid later.