Probability Densities and Orbitals

A hydrogen wavefunction $\psi_{n\ell m}$ is not directly observable, but $|\psi_{n\ell m}|^2$ is: by the Born rule it is the probability density for finding the electron at a given point. The familiar pictures of "orbitals" are visualizations of this density.

Density versus radial distribution

The probability density in space is

|\psi_{n\ell m}(r,\theta,\phi)|^2 = |R_{n\ell}(r)|^2\,|Y_\ell^m(\theta,\phi)|^2.

But the chance of finding the electron at a particular radius (in any direction) requires integrating over the spherical shell of radius $r$ , whose volume element is $r^2\,dr$ times the solid angle. Defining the radial distribution function

P(r) = r^2\,|R_{n\ell}(r)|^2,

the probability of finding the electron between $r$ and $r + dr$ is $P(r)\,dr$ (the angular part integrates to 1 because the spherical harmonics are normalized). The factor $r^2$ is essential and often surprising.

The ground state: density versus shell

For the $1s$ state, $|R_{10}|^2 \propto e^{-2r/a_0}$ is largest at $r = 0$ — the electron is most likely to be found at the nucleus per unit volume. Yet the radial distribution

P(r) = r^2\,|R_{10}|^2 \propto r^2 e^{-2r/a_0}

vanishes at $r = 0$ and peaks at $r = a_0$ , the Bohr radius. There is no contradiction: near the origin the density is high but the available volume $\propto r^2$ is tiny. The most probable radius is $a_0$ , even though the most probable point is the origin.

Angular shapes: s, p, d

The angular factor $|Y_\ell^m|^2$ sets the shape:

$\ell = 0$ (s orbitals): $Y_0^0$ is constant, so the density is spherically symmetric — a ball.
$\ell = 1$ (p orbitals): the densities have two lobes. The real combinations $p_x$ , $p_y$ , $p_z$ point along the coordinate axes, each with a nodal plane through the nucleus.
$\ell = 2$ (d orbitals): more complex four-lobed (and one ring-plus-lobes) shapes, with two angular nodes.

The number of angular nodes is exactly $\ell$ , and the number of radial nodes is $n-\ell-1$ , so the total node count is $n-1$ for every state — a unifying bookkeeping rule.

From orbitals to chemistry

These shapes are the foundation of chemical bonding. The directional lobes of $p$ and $d$ orbitals explain why molecules have specific geometries, and the spherical $s$ orbitals explain the inert, closed-shell behaviour of noble gases. The hydrogen solutions are the template from which multi-electron atoms are approximately built.

The takeaway

The orbital picture is a plot of $|\psi_{n\ell m}|^2$ . The radial distribution $P(r)=r^2|R_{n\ell}|^2$ peaks at $a_0$ for the ground state, while the angular factor $|Y_\ell^m|^2$ gives the spherical ( $s$ ), two-lobed ( $p$ ), and multi-lobed ( $d$ ) shapes, with $\ell$ angular nodes and $n-\ell-1$ radial nodes.