Spherical Harmonics Explorer

This sim draws the spherical harmonics Y_ℓ^m(θ, φ), the angular wavefunctions of a particle with definite orbital angular momentum. Two integers, ℓ and m, choose the state, and the 3-D picture shows how the wavefunction varies with direction.

What's on screen

• Blue: Y > 0 in Real mode. Pale grey is Y ≈ 0.
• Red: Y < 0 in Real mode.
• Grey axes: x, y and z. The vertical z axis is the polar axis that θ is measured from.
• Legend (on the canvas): the colour scale for the current mode.
• Quantum Numbers: the label Y_ℓ^m and the two sliders.
• Nodal Structure: three node counts for the current state. The Guided Inquiry reveals it at card 4.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: only the angular factor Y is drawn. There is no radial part and no time dependence.

A hydrogen-like orbital factors as ψ = R(r) Y_ℓ^m(θ, φ). The sim draws only Y, which depends on direction alone, so no distance from a nucleus appears anywhere in the picture.

• Polar Surface. In each direction the surface sits at radius 1.2 |Y| / max|Y|. Where Y = 0 it pinches to the origin. In Real mode blue is Y > 0 and red is Y < 0.
• Projection. A sphere of fixed radius 1.5, coloured by Y. The nodes are where the colour crosses zero. There are no lobes.
• Point Cloud. Each direction is kept with probability proportional to |Y|², so the dots crowd where |Y|² is large. Each dot is still placed at a radius proportional to |Y|.

|Y|² is a probability density per unit solid angle, and the length you see is |Y|. The lobes are a plotting choice: Projection shows the same Y without them.

Y_ℓ^m is a simultaneous eigenfunction of L² (the squared magnitude of the orbital angular momentum) and L_z (its component along the grey z axis):

• ℓ sets the magnitude: |L| = √(ℓ(ℓ+1)) ħ.
• m sets the z component, L_z = mħ. One full turn round z must return e^imφ to itself, so m is an integer.
• The angular-momentum algebra allows only |m| ≤ ℓ. That leaves 2ℓ+1 states for each ℓ, the values the m slider steps through.
• ℓ = 0 has no preferred direction. Y₀⁰ = 1/√(4π) ≈ 0.282 is the same in every direction.

A node is a direction where Y = 0. Two factors of Y can vanish, and they give two families of node:

• Planes. In Real mode the azimuthal factor cos(mφ) or sin(|m|φ) gives |m| planes through the z axis.
• Cones. P_ℓ^|m|(cos θ) has ℓ−|m| zeros, each a cone about z. A cone at θ = 90° is the flat equatorial plane.
• Sign changes. In Real mode, going once round a circle of latitude that is not itself a node, the sign changes 2|m| times. The equator is a node when ℓ−|m| is odd.

A Complex harmonic is an L_z eigenstate, so L_z = mħ is sharp. But √(ℓ(ℓ+1)) > ℓ ≥ |m|, so L_z is always shorter than |L|. In the vector-model picture, L makes an angle θ_L with z, where cos θ_L = m/√(ℓ(ℓ+1)).

• Even at m = ℓ = 2 the best case is 2ħ out of 2.449ħ, a tilt of 35.3°.
• The missing length has no direction. ⟨L_x⟩ and ⟨L_y⟩ are zero, but their squares are not.
• This is forced by [L_x, L_y] = iħL_z: L² and one component can be sharp together, never all three components.

Both forms have the same ℓ and the same L². They differ in what they say about L_z.

• Complex is the L_z eigenstate. |Y| has no φ dependence, so the surface is symmetric about z, and the φ structure appears as hue: the phase winds |m| times round each circle of latitude.
• Real mixes +|m| and −|m| equally. It has lobes and sign changes, and ⟨L_z⟩ = 0.
• At m = 0 the two forms are identical, because there is no e^imφ factor.
• The sim plots Y(θ, φ) only. A stationary state's time factor e^−iEt/ħ is the same in every direction, so it would change no shape.

• Atomic orbitals. The angular part of every hydrogen-like orbital is a Y_ℓ^m. The letters s, p, d, f stand for ℓ = 0, 1, 2, 3, and the chemist's p_x, p_y, p_z are the Real forms. The radial part, which this sim does not draw, sets the size.
• Rotating molecules. The rotational states of a rigid diatomic molecule are Y_J^m, with energy ħ²J(J+1)/(2I) and 2J+1 states at each level.
• The cosmic microwave background. The temperature pattern over the whole sky is expanded in spherical harmonics. A multipole ℓ corresponds roughly to an angular scale of 180°/ℓ, and the first acoustic peak sits near ℓ ≈ 220.

Key points

• ℓ sets |L|, m sets L_z, and each ℓ has 2ℓ+1 values of m.
• The angular nodes are |m| planes plus ℓ−|m| cones.
• L_z is always shorter than |L|; the difference is spread over L_x and L_y.
• Complex harmonics are L_z eigenstates. Real harmonics mix +|m| and −|m|.

Common mix-ups

• "The lobes show how far the electron is from the nucleus." The radius is |Y| in that direction. Distance from the nucleus belongs to R(r), which is not drawn.
• "Red means negative probability." Red is Y < 0 in Real mode. The probability density |Y|² is never negative.
• "m is the angular momentum." m gives only the z component, L_z = mħ. The magnitude √(ℓ(ℓ+1)) ħ is always larger.

Things to try on your own

• Set ℓ = 4, m = −2 in Real mode. Work out the numbers of planes and cones, then check them against the Nodal Structure panel.
• At ℓ = 3, step m from 0 to 3 and note which values make the equator a node.
• Switch ℓ = 3, m = 1 to Complex and count how many times the hue cycles round a circle of latitude.