The physics behind these orbitals
Each cloud you see is a Monte-Carlo sampling of the position-probability density |ψnℓm(r,θ,φ)|² for a single electron bound to a proton. This page explains where every part of the picture comes from.
1. The hydrogen Schrödinger equation
The non-relativistic, time-independent Schrödinger equation for an electron in the Coulomb field of a proton is, in atomic units (ℏ = me = e = 4πε₀ = 1):
Because the potential V(r) = −1/r depends only on the distance to the nucleus, the equation is separable in spherical coordinates. We write ψ as a product:
The radial part Rnℓ(r) carries energy and shell information; the angular part Yℓm is a spherical harmonic that carries angular-momentum information.
2. Three quantum numbers, three meanings
| Symbol | Range | Physical meaning |
|---|---|---|
| n (principal) | 1, 2, 3, … | Energy En = −1/(2n²) and overall size of the orbital. |
| ℓ (azimuthal) | 0, 1, …, n−1 | Magnitude of orbital angular momentum: |L|² = ℏ²ℓ(ℓ+1). |
| m (magnetic) | −ℓ, …, +ℓ | Projection along z: Lz = mℏ. |
3. The radial part Rnℓ(r)
Solving the radial equation gives products of an exponential decay and an associated Laguerre polynomial:
The polynomial creates n − ℓ − 1 radial nodes — spherical shells where the probability density drops to zero. Those are the dark gaps you see between lobes inside the cross-section. For example:
- 3p R31(r) ∝ r(6−r)e−r/3 has a node at r = 6 a₀ — the small inner sphere visible in the slice.
- 3s R30(r) ∝ (27 − 18r + 2r²)e−r/3 has two radial nodes — three concentric shells.
- 3d R32(r) ∝ r²e−r/3 has no radial nodes — a single, thicker shell.
4. The angular part Yℓm(θ,φ)
Spherical harmonics in their complex form are simultaneous eigenfunctions of L² and Lz:
Because |eimφ|² = 1, the probability density |Yℓm|² depends only on θ — it is rotationally symmetric about the z-axis. That is why the m = ±1 clouds come out as tori around z, and the m = 0 cloud comes out as a vertical dumbbell.
Densities used in this app
| ℓ | m | |Yℓm(θ)|² ∝ | shape |
|---|---|---|---|
| 0 | 0 | 1 | spherical |
| 1 | 0 | cos²θ | dumbbell along z |
| 1 | ±1 | sin²θ | torus around z |
| 2 | 0 | (3cos²θ − 1)² | two lobes + waist (dz²) |
| 2 | ±1 | sin²θ cos²θ | four-leaf around z |
| 2 | ±2 | sin⁴θ | flatter torus around z |
5. Why the orbitals are shown rotating
Although |ψnℓm|² is time-independent and azimuthally symmetric, the wavefunction itself is not: ψ ∝ e−i En t/ℏ · eimφ. The probability current circulates around the z-axis whenever m ≠ 0:
The current's direction follows the sign of m and its
magnitude
scales with |m|. So in this visualizer the cloud spins about z with
ω ∝ sign(m)·|m|. For the m = 0 state, Lz
vanishes but |L|² = ℏ²ℓ(ℓ+1) does not — there is angular momentum,
just not in z. The dumbbell is shown tumbling about a horizontal
axis to convey that.
6. Why we slice the cloud
Without a slice, the dense outer shell hides the radial nodes and inner lobes inside it. The slice plane cuts away the front half so the interior structure — the small inner sphere of 3p, the ring-and-core of 3d, the layered shells of 3s — becomes visible. The plane is fixed in world space, so as the cloud rotates, you can see individual sample points pass through the cut and reappear on the other side.
7. How the points are drawn
Each frame uses ≈ 80 000 points. To place them, we sample independently:
- r from p(r) ∝ r²|Rnℓ(r)|² (the r² is the spherical-shell Jacobian)
- cos θ from p(cosθ) ∝ |Yℓm(θ)|²
- φ uniformly on [0, 2π)
Both 1-D distributions are sampled by rejection against their numerical maxima. The result is a faithful representative of |ψnℓm|² in 3-D, which is why dense regions of the cloud match where the electron is most likely to be found.
References
Griffiths, Introduction to Quantum Mechanics, ch. 4 (hydrogen atom). Sakurai & Napolitano, Modern Quantum Mechanics, ch. 3 (angular momentum) and ch. 4 (central potentials).