Wigner d-Matrix Calculator

Inputs

j (spin)

Use integer or half-integer values, e.g., 0.5, 1, 1.5.

m

Must satisfy −j ≤ m ≤ j.

m'

Must satisfy −j ≤ m' ≤ j.

β (rotation angle)

This calculator uses the reduced rotation about the y-axis.

Angle unit

Internally converted to radians.

Decimal precision

Higher precision helps for larger j.

Also compute the full d-matrix for this j and β

Example data table

Example for j = 1, β = 60° (values shown for illustration).

j	m	m'	β (deg)	d^j_m,m'(β)
1	1	1	60	0.7500000000
1	1	0	60	-0.6123724357
1	1	-1	60	0.2500000000

Formula used

The reduced Wigner rotation element (small-d) for a rotation by β about the y-axis is evaluated using the standard finite sum:

d^j_{m,m'}(β) = \sum_k (-1)^{k-m+m'} \frac{\sqrt{(j+m)!(j-m)!(j+m')!(j-m')!}}{(j+m-k)!\,k!\,(m'-m+k)!\,(j-m'-k)!} \\ \times (\cos(β/2))^{2j+m-m'-2k} (\sin(β/2))^{m'-m+2k}

The sum runs over integer k values that keep every factorial argument non-negative. Factorials for half-integers are computed via the Gamma function: x! = Γ(x+1).

How to use this calculator

Enter the spin value j (integer or half-integer).
Set m and m' within the range −j to j.
Provide the rotation angle β and select degrees or radians.
Choose a decimal precision suitable for your j value.
Optionally enable “full matrix” to compute all m,m' entries.
Press Compute to display results, then export to CSV or PDF.

Notes on conventions

This tool computes the reduced rotation matrix d(β) associated with a y-axis rotation.
m and m' advance in steps of 1, matching the spin ladder spacing.
Results depend on phase conventions; keep consistent conventions across calculations.
If you need the full D-matrix, combine with Euler phases: D = e^{-imα} d(β) e^{-im'γ}.

Professional article

1) Wigner d-matrix in rotation theory

The reduced Wigner rotation matrix d^j_m,m′(β) describes how angular-momentum eigenstates transform under a pure rotation about the y-axis. It is a central building block for the full Wigner D-matrix and appears whenever rotational symmetry is expressed in the |j,m⟩ basis.

2) Quantum indices and allowed ranges

The calculator accepts integer or half-integer j, with m and m′ running from −j to +j in unit steps. For example, j = 3/2 yields m ∈ {3/2, 1/2, −1/2, −3/2}. These quantized labels encode the dimensionality (2j+1) and selection constraints used in the summation.

3) Meaning of the angle β

β is the polar rotation angle about the y-axis and is often paired with Euler angles (α,β,γ) for general rotations. In spectroscopy and scattering, β connects laboratory and molecular frames; in spin dynamics, it describes the tilt between quantization axes. Internally the algorithm uses sin(β/2) and cos(β/2).

4) Stable summation with gamma scaling

Direct factorials overflow quickly, especially when j grows. This implementation evaluates factorial terms through the log-gamma function, using x! = Γ(x+1), and forms each term in log space before exponentiation. That approach improves stability for moderate-to-large j and helps maintain useful precision when many k-terms contribute with alternating signs.

5) Useful numerical checks

For any fixed j and β, the matrix d(β) is orthogonal: rows and columns are normalized and mutually orthogonal (within rounding). Special angles provide quick validation: at β = 0, d^j_m,m′(0) equals 1 when m=m′ and 0 otherwise. At β = π, values follow known sign and permutation patterns.

6) Building full matrices for spin systems

When you enable the full-matrix option, the tool computes every entry for m and m′ across the (2j+1)×(2j+1) grid. This is practical for constructing rotation operators in finite-dimensional Hilbert spaces, rotating density matrices, or generating basis-change tables used in coupled-spin Hamiltonians and polarization-transfer simulations.

7) Where the d-matrix is applied

Common applications include spherical-harmonic rotations (j=ℓ), rigid-rotor spectroscopy, quantum scattering amplitudes, and spinor rotations in magnetic resonance. In molecular physics, d-elements appear in direction-cosine expansions and in transforming transition moments. In quantum information, they provide exact single-qudit rotation blocks for spin-j encodings.

8) Recommended settings and interpretation

For larger j, increase decimal precision and prefer radians for reproducible inputs. If results seem noisy, compare symmetry relations (such as m↔m′ patterns under β→−β) and verify orthogonality using the exported table. Remember that phase conventions matter when combining with α and γ to form the full D-matrix.

FAQs

1) What does this calculator compute?

It computes the reduced Wigner rotation element d^j_m,m′(β) for a y-axis rotation, and can also generate the full (2j+1)×(2j+1) matrix for the selected j and angle.

2) Can j, m, and m′ be half-integers?

Yes. Enter values in steps of 0.5. The tool validates that m and m′ remain within −j to +j and lie on the same integer/half-integer grid as j.

3) Why do some entries change sign with β?

The sum includes alternating-sign terms and powers of sin(β/2) and cos(β/2). These factors encode the rotation geometry and phase convention, producing sign changes that are physically meaningful.

4) What precision should I choose?

Use higher precision for larger j or when β is near angles that cause cancellation. Ten to sixteen decimals is typical for j above about 10, especially when exporting full matrices for downstream calculations.

5) How is the full Wigner D-matrix obtained?

Combine the reduced element with Euler phase factors: D^j_m,m′(α,β,γ) = e^−imα d^j_m,m′(β) e^−im′γ. This tool supplies the d-part.

6) Are there quick sanity checks for my inputs?

Yes. At β=0 the matrix should be the identity. Also, rows and columns should be approximately orthonormal for any β. Export a full matrix and verify dot products if needed.

7) Why is there a suggested limit on j?

Very large j can amplify cancellation and rounding errors in finite-precision arithmetic. The log-gamma approach improves stability, but practical limits keep results dependable without requiring arbitrary-precision libraries.