Spherical Harmonics Yₗᵐ

Angular functions on S², with integer degree 0≤ℓ≤20 and |m|≤ℓ. θ is measured from +z; φ runs from +x toward +y. The complex Y uses the orthonormal Condon–Shortley convention. Re Y and Im Y have squared norm 1/2 for nonzero m; for m=0 they have norms 1 and 0.

The real orthonormal basis uses Tℓ0=Yℓ0, Tℓm=√2(−1)^m Re Yℓm for m>0, and Tℓm=√2(−1)^|m| Im Yℓ|m| for m<0. Positive orders are cosine modes; negative orders are sine modes. Every T has squared norm 1. Changing m genuinely changes the angular function.

Color maps the selected value S. Signed modes use a symmetric scale [−Bℓ,Bℓ], with Bℓ=√((2ℓ+1)/(4π)); magnitude uses [0,Bℓ]. This addition-theorem bound is independent of a sampled maximum. Amplitude view uses r=|S|/Bℓ, with no positive baseline: nodes meet the origin. Sphere view keeps r=1. These are angular diagrams, not atomic electron-density orbitals or physical distances.

Drag to inspect; plain wheel scrolls the page, Ctrl/Meta+wheel zooms. Auto-rotation is optional and pauses in hidden tools. Angle inputs move the white probe without rebuilding the mesh; exact values are in the expandable readings. The squared norm shown there is analytical, not a measured grid integral. A finite triangle mesh approximates the surface, especially at high ℓ. Input changes discard stale surfaces; repeated renders dispose old GPU resources. PNG includes parameters and current camera view.

NIST DLMF 14.30: normalization, symmetry and addition theorem.

Examples choose degree and order in the real basis. Changing a component, surface or palette updates the plot immediately; valid degree/order edits update after a short pause or Enter. Fit view centers and sizes the surface. The shape explanation is available in the color legend tooltip. A completed surface is reconstructed from its exact parameters, not a screenshot, using the bounded mesh calculation. No surface is generated for a fresh or unfinished session.

Only input values are remembered in this browser. Calculated results and experiment histories are not saved.