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Hydrogen Atom Orbital Viewer

Visualize hydrogen atom electron orbitals by selecting quantum numbers (n, l, m). See the probability density as a 2D cross-section heatmap in the xz-plane, explore the radial probability distribution, and learn how quantum numbers determine orbital shape, energy, and node structure.

Probability Density Heatmap

Controls

Principal Quantum Number (n)1
12345
Angular Momentum (l)0 (s)
Magnetic Quantum Number (m)0
Resolution150 px

Results

Orbital:1s
Energy Level
-13.60 eV
Bohr Radius (n²a₀)
1.000 a₀
Radial Nodes
0
Angular Nodes
0
Total Nodes
0
n
1
l
0 (s)
m
0
Key Formulas
ψ1,0,0(1s orbital)\psi_{1,0,0} \quad (1s \text{ orbital})
E1=−13.6 eV12=−13.60 eVE_{1} = -\frac{13.6 \text{ eV}}{1^2} = -13.60 \text{ eV}
Radial nodes=n−l−1=0\text{Radial nodes} = n - l - 1 = 0
Angular nodes=l=0\text{Angular nodes} = l = 0
Total nodes=n−1=0\text{Total nodes} = n - 1 = 0

Radial Probability Distribution

The radial probability r²|R(r)|² shows the probability of finding the electron at distance r from the nucleus. Red dashed lines mark radial nodes where the wave function is zero.

Reference Guide

Quantum Numbers

Every hydrogen orbital is described by three quantum numbers that determine its size, shape, and orientation.

n (principal)

Ranges from 1, 2, 3, ... and determines the energy level and overall size of the orbital. Higher n means larger orbital and higher energy (less tightly bound).

l (angular momentum)

Ranges from 0 to n-1 and determines the shape. l = 0 (s) is spherical, l = 1 (p) is dumbbell, l = 2 (d) is cloverleaf, l = 3 (f) is more complex.

m (magnetic)

Ranges from -l to +l and determines the orientation of the orbital in space. Different m values give the same energy in the absence of a magnetic field (degeneracy).

Orbital Shapes

The wave function ψnlm(r,θ,ϕ)=Rnl(r)⋅Ylm(θ,ϕ)\psi_{nlm}(r,\theta,\phi) = R_{nl}(r) \cdot Y_l^m(\theta,\phi) separates into a radial part and an angular part.

s orbitals (l = 0)

Spherically symmetric. 1s has no nodes, 2s has one radial node (a spherical shell of zero probability), 3s has two.

p orbitals (l = 1)

Dumbbell-shaped with one angular node (a nodal plane). Three orientations (m = -1, 0, +1) correspond to different spatial directions.

d orbitals (l = 2)

Cloverleaf patterns with two angular nodes. Five orientations (m = -2 to +2) give the familiar d-orbital shapes used in chemistry.

Radial Probability

The radial probability distribution P(r)=r2∣Rnl(r)∣2P(r) = r^2 |R_{nl}(r)|^2 gives the probability of finding the electron at distance r from the nucleus, regardless of direction.

Rnl(r)=Nnl⋅e−r/(na0)⋅(2rna0)l⋅Ln−l−12l+1(2rna0)R_{nl}(r) = N_{nl} \cdot e^{-r/(na_0)} \cdot \left(\frac{2r}{na_0}\right)^l \cdot L_{n-l-1}^{2l+1}\left(\frac{2r}{na_0}\right)

For the 1s orbital, the most probable radius is exactly one Bohr radius a0≈0.529a_0 \approx 0.529 angstroms. Radial nodes occur where Rnl(r)=0R_{nl}(r) = 0, and there are n - l - 1 of them.

Energy Levels

The energy of a hydrogen atom depends only on the principal quantum number n (in the non-relativistic approximation).

En=−13.6 eVn2E_n = -\frac{13.6 \text{ eV}}{n^2}

The ground state (n = 1) has E = -13.6 eV. All orbitals with the same n have the same energy, giving n2n^2-fold degeneracy. Transitions between levels produce the characteristic hydrogen spectrum (Lyman, Balmer, Paschen series).

Total nodes = n - 1

The total number of nodes (radial + angular) is always n - 1. This is why higher n orbitals have more complex structures with more regions of zero probability.