Hydrogen orbitals

Two-dimensional plot of the probability density of hydrogen orbitals (the squared modulus of the wave function), based on the stationary states of the hydrogen atom:
$$\psi_{nlm}(r, \theta, \phi) = C_{nlm}\, e^{-r/na_0} \left( \dfrac{2r}{na_0} \right)^l \times$$ $$\times\, L_{n-l-1}^{2l+1} \left( \dfrac{2r}{na_0} \right) \times Y_l^m(\theta, \phi)$$ Only the three principal planes (x = 0, y = 0, and z = 0) are available for plotting.

Adjust the quantum numbers n, ℓ, and m in the "Quantum numbers" panel—the valid range of ℓ depends on n, and the range of m depends on ℓ. In the "Graph" panel, change the plotted region size (Span) and resolution (pixels). Choose the cross-sectional plane in the "Plane" panel, the plotted quantity (|Ψ|² or |Ψ|) in the "Plot" panel, and the color map in the "Scale" panel. On mobile devices, touch and drag with one finger to zoom into the selected region; double-tap to return to the original plot.

The quantum numbers n, l, and m determine, respectively, the energy, orbital angular momentum, and its projection onto the z-axis—and directly shape the displayed probability cloud, from spherically symmetric orbitals (l = 0) to the characteristic lobes of p, d, and f orbitals.

Marco P. M. de Souza