Spherical Harmonics Calculator

Calculator

Enter indices and angles

Degree l (0–20)

Higher l adds finer angular detail.

Order m (−l to +l)

Controls azimuthal oscillation in φ.

Angle unit

θ is polar, φ is azimuthal.

Polar angle θ

Valid range: 0 to π radians.

Azimuth angle φ

Common range: 0 to 2π radians.

Output mode

Choose what you want to display and export.

After calculation, results appear above this form.

Formula used

Complex spherical harmonic

This calculator uses the standard normalized complex form.

Y_l^m(θ, φ) = N_l^|m| P_l^m(cos θ) e^{i m φ}

N_l^|m| = √((2l+1)/(4π) · (l-|m|)!/(l+|m|)!)

The associated Legendre term P_l^m is computed via stable recurrences, including the Condon–Shortley phase.

Real-form convention

A common orthonormal real basis is reported as Y_real.

m > 0: Y_real = √2 · (−1)^m Re(Y_l^m)

m < 0: Y_real = √2 · (−1)^|m| Im(Y_l^|m|)

m = 0: Y_real = Y_l⁰

How to use this calculator

Choose degree l and order m, keeping −l ≤ m ≤ l.
Select degrees or radians, then enter θ (polar) and φ (azimuth).
Pick an output mode: complex, real, or both.
Click Calculate to display results above the form.
Use CSV or PDF buttons to export the displayed results.

Example data table

l	m	θ (deg)	φ (deg)	Notes
0	0	60	10	Constant over the sphere.
1	0	45	30	Dipole pattern aligned with z-axis.
2	1	45	30	Quadrupole with azimuthal variation.
3	-2	90	120	Higher detail; real form is often plotted.

Use the “Load example” button to auto-fill one set.

Article

1) What this calculator evaluates

Spherical harmonics are orthonormal functions defined on the unit sphere. This calculator evaluates Y_l^m(θ, φ) for chosen indices and angles, returning both complex components and a common real-form value. These outputs support angular basis expansions used in fields, waves, and potential theory.

2) Interpreting the indices l and m

The degree l controls angular detail: larger l creates finer features. For each l, the order m spans −l to +l, giving 2l+1 modes at that degree. If you sum all modes up to L, the total count is (L+1)², a useful rule for bandwidth and model size planning.

3) Angle conventions on the sphere

The polar angle θ runs from 0 to π, measured from the positive z-axis. The azimuth φ wraps around the z-axis and is periodic with 2π. You may enter degrees or radians; internally, the computation uses radians. Keeping θ within range avoids nonphysical inputs and improves numerical stability.

4) The role of the associated Legendre term

The angular shape in θ is carried by the associated Legendre function P_l^m(cos θ). Near the poles (θ≈0 or π), powers of sin θ affect magnitude strongly, especially for |m| > 0. The calculator uses recurrence relations to compute P_l^m efficiently for practical l values.

5) Normalization and orthogonality

The normalization factor N ensures an orthonormal basis: the integral of Y_l^m times the complex conjugate of Y_l′^m′ over the sphere equals 1 when (l,m)=(l′,m′), otherwise 0. This property makes coefficients comparable across modes and preserves energy in expansions.

6) Complex value and phase behavior

The φ dependence appears as e^{i m φ}, which rotates the complex value as φ changes. The calculator reports Re, Im, magnitude |Y|, and phase. Magnitude is helpful for amplitude maps, while phase captures directional structure that matters in interference, modal coupling, and harmonic analysis workflows.

7) Real-form harmonics for plotting

Many visualization pipelines use real-valued basis functions. The reported Y_real combines the real or imaginary parts of the m>0 modes with √2 scaling, while keeping m=0 unchanged. This yields an orthonormal real basis that is convenient for heatmaps, mesh shading, and coefficient fitting.

8) Practical workflow and exporting results

A typical workflow is to select l and m, scan θ and φ to sample the function, then build a table of outputs for plotting or regression. Use the export buttons to capture the computed terms and results. For high-resolution studies, increase l gradually and confirm trends before using larger mode counts.

FAQs

1) What values of l and m are valid?

Use l ≥ 0 and ensure −l ≤ m ≤ l. This calculator allows l from 0 to 20 to keep computations fast and numerically stable for typical educational and laboratory use.

2) What is the correct range for θ and φ?

θ should be between 0 and π. φ is periodic with 2π and can be any real value, but many users choose 0 to 2π for consistent sweeps.

3) Why do I see both complex and real outputs?

The complex form is the standard orthonormal definition. The real-form is a widely used real basis derived from the complex modes, which is often easier for plotting and fitting real-valued data.

4) What does the reported phase mean?

Phase is atan2(Im, Re) in radians. It tracks the angular rotation contributed mainly by the e^{i m φ} factor and is useful when comparing modes or analyzing interference patterns.

5) Why can Y become very small near the poles?

For |m| > 0, the associated Legendre term includes factors related to sin^|m|(θ), which tends to zero as θ approaches 0 or π, suppressing the amplitude near the poles.

6) Which output should I use for surface shading?

Use Y_real for direct shading, because it is a real value at each (θ, φ). If you need complex amplitude information, use magnitude and phase from the complex output.

7) Can I use these results for expansions on a sphere?

Yes. With consistent normalization, coefficients from inner products are comparable across modes. This calculator gives you normalized Y values suitable for building and checking harmonic series on angular grids.