Hydrogen Atom Wave Function Visualization

Quantum Theory of Hydrogen Atom

The Wave Function ψ_nlm(r,θ,φ)

The hydrogen atom's wave functions are solutions to the Schrödinger equation for a Coulomb potential. Each solution is described by three quantum numbers:

n - Principal quantum number (energy level, n ≥ 1)
l - Angular momentum quantum number (0 ≤ l < n)
m - Magnetic quantum number (-l ≤ m ≤ l)

The wave function separates into radial and angular components:

ψ_nlm(r,θ,φ) = R_nl(r) Y_l^m(θ,φ)

Radial Part R_nl(r)

Describes electron distribution as function of distance from nucleus:

R_nl(r) ∝ e^-r/(na₀) (2r/(na₀))^l L_n-l-1^2l+1(2r/(na₀))

Where L are associated Laguerre polynomials and a₀ is the Bohr radius.

Angular Part Y_l^m(θ,φ)

Spherical harmonics describing angular distribution:

Y_l^m(θ,φ) ∝ P_l^|m|(cosθ) e^imφ

Wave Function Visualizer

Principal Quantum Number (n)

123456

1

Angular Quantum Number (l)

012345

0

Magnetic Quantum Number (m)

-5-3-10135

0

Probability Density |ψ(r,θ,φ)|²

Cross-Section View

Interactive Explanation

What You're Seeing

3D Plot: Shows the probability density (darker regions indicate higher probability of finding the electron)
2D Slice: Displays a cross-section through the center of the atom (x-z plane)
Quantum Numbers: Change n, l, m to see different orbitals (1s, 2p, 3d, etc.)

Key Observations

n determines: Number of radial nodes (n-l-1) and size of orbital

l determines: Shape of orbital (s=sphere, p=dumbbell, d=cloverleaf, etc.)

m determines: Orientation of orbital in space

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