Space Quantization: Orbital Angular Momentum Vector Model

Orbital Quantum Number l

l 2

Magnetic Quantum Number m_l

Precession rate 1.0×

Live Readouts

|l, m_l⟩ = |2, 0⟩

|L| (ħ)

2.449

L_z (ħ)

0.000

Cone angle θ

90.0°

Azimuth φ

0.0°

How it works

Orbital angular momentum in quantum mechanics is quantized. Its magnitude and its projection on one chosen axis (conventionally z) can be known simultaneously and exactly, but the other two components (L_x, L_y) cannot — this is space quantization:

|L| = ħ√(l(l+1)),   l = 0, 1, 2, …
Lz  = m_l · ħ,       m_l = −l, …, −1, 0, +1, …, +l   (2l+1 values)
cone half-angle:  θ = arccos( m_l / √(l(l+1)) )

Because L_x and L_y are not simultaneously well-defined with L_z (they don't commute: [L_x, L_y] = iħL_z), the classical picture is a vector of fixed length |L| and fixed z-projection L_z, free to point anywhere on the cone of half-angle θ — its azimuth φ is completely undetermined. The "Precession" toggle animates a vector sweeping that cone to make this indeterminacy visible; it is a classical stand-in for a genuinely unknown azimuth, not literal orbital motion.

l slider — sets the orbital quantum number (0=s, 1=p, 2=d, 3=f, 4=g), which fixes |L| and the number of allowed cones (2l+1).
m_l pills — pick the projection quantum number; the highlighted cone updates θ and L_z instantly.
Show all cones — renders every allowed m_l cone for the current l at once, so you can see the full quantized fan from −l to +l.
Precession — animates the vector's azimuth on the selected cone at the chosen rate; toggle off to freeze it and inspect a single orientation.

This is the same quantization rule (with spin s = ½ instead of orbital l) behind the Stern–Gerlach experiment and the Zeeman splitting of atomic spectral lines, and it underlies why atomic orbitals (s, p, d, f) come in the specific shapes and counts they do.

Selected L vector Allowed m_l cones

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