Atomic orbitals

The Hydrogen-like Atom

A single electron bound to a nucleus of charge $Z$ feels a Coulomb potential

V(r) = -\frac{Ze^2}{4\pi\varepsilon_0 r}

Because $V$ depends only on $r$ , the time-independent Schrödinger equation separates in spherical coordinates into a radial part and an angular part:

\psi_{nlm}(r,\theta,\phi) = R_{nl}(r)\, Y_l^m(\theta,\phi)

The three quantum numbers are coupled: the principal number $n = 1, 2, 3, \dots$ sets the energy, the orbital angular momentum $l$ satisfies $0 \le l < n$ , and the magnetic number $m$ satisfies $-l \le m \le l$ . The radial part

R_{nl}(\rho) \propto e^{-\rho/2}\, \rho^{\,l}\, L_{n-l-1}^{2l+1}(\rho), \qquad \rho = \frac{2Zr}{n}

involves an associated Laguerre polynomial, while $Y_l^m$ is the familiar spherical harmonic that gives each orbital its characteristic lobed shape.

2p orbital

$n = 2$ · $l = 1$ · $m = 0$ · $Re (ψ)$

+ ψ

- ψ

Quantum numbers

n

principal

l

angular momentum

m

magnetic

+ ψ

- ψ

Real part of

ψ

for a hydrogen-like orbital — blue where

ψ > 0

, red where

ψ < 0

. Drag to rotate, scroll to zoom.

Reading the shape

The rendered surfaces enclose the region where $|\psi_{nlm}|$ exceeds a chosen fraction of its peak value colored by the sign of $\psi_{nlm}$ itself rather than by $|\psi_{nlm}|^2$ : blue where the wavefunction is positive, red where it is negative. The boundary between the two colors is exactly a node, a surface where $\psi_{nlm} = 0$ . Push $n$ up to see the number of radial nodes grow, each one another red/blue boundary along the radius; push $l$ up at fixed $n$ to watch the shape flatten from a sphere ( $s$ ) into the familiar two-toned dumbbell ( $p$ ) and four-lobed cloverleaf ( $d$ ); sweep $m$ to see the lobes rotate around the $z$ -axis, the axis along which angular momentum is quantised.