Hydrogen Radial Probability Density

Radial probability density P(r) = r²|Rnl(r)|² for hydrogen eigenstates (units: Bohr radius a₀ = 1).

Radial Probability Density vs r (in units of a₀)
n = ℓ =
P(r) = r²|Rnl(r)|² (radial probability density)
|Rnl(r)|² (wavefunction squared, no r² factor)
rmp — most probable radius (peak of P(r))
⟨r⟩ — mean radius
Running Integral: P(r < x)
Drag over main plot or use slider to explore cumulative probability.
—
P(r < — ) = —

State Info

Quantum numbers (n, ℓ) n=1, ℓ=0
Radial nodes (n−ℓ−1) 0
rmp (most probable) 1.000 a₀
⟨r⟩ (mean radius) 1.500 a₀
P(r < a₀) 32.3%
∫₀^∞ P(r) dr 1.0000

Wavefunction Rnl(r)

R₁₀ = 2·e^(−r)

Physics Notes

P(r) ≠ |ψ|²: The radial probability density P(r) = r²|Rnl|² includes the r² Jacobian factor from the spherical volume element dV = r² sin θ dr dθ dφ.

Most probable vs mean: For n=1, rmp = a₀ but ⟨r⟩ = 3a₀/2. Mean > most probable because P(r) is skewed rightward.

Nodes: The number of radial nodes is n−ℓ−1. Nodes are r values (excluding r=0) where Rnl = 0.