Power of Schrödinger’s Equation

Solutions of Schrödinger’s equation explain the shapes of hydrogen orbitals.

Schrödinger Eq & Hydrogen Orbitals

Time-independent Schrödinger equation

Ĥψ = Eψ

[ -ħ²/(2μ) ∇² - e²/(4πϵ₀r) ] ψ(r, θ, φ) = E ψ(r, θ, φ)

Hydrogen-atom solution

ψ_nℓm(r, θ, φ) = R_nℓ(r) Y_ℓ^m(θ, φ)

R_nℓ(r): radial part → size and radial structure
Y_ℓ^m(θ, φ): angular part → shape and orientation

Quantum numbers

n = principal quantum number → energy level and size
ℓ = angular quantum number → orbital type and shape
m = magnetic quantum number → orientation in space

Hydrogen orbitals by n and ℓ

Quantum Controls

Principal quantum number n

Angular quantum number ℓ

Magnetic quantum number m

Quick preset

Show axes Show nucleus Auto rotate

positive phase negative phase nucleus

Model uses real hydrogen-like wavefunctions with real spherical harmonics and the exact radial structure for hydrogen orbitals.