The shapes of the simplest atom in the universe.

The hydrogen atom is the only atom with an exact analytical solution to the Schrödinger equation. Its wave functions $\psi_{nlm}(r, \theta, \phi)$ are the foundation of all atomic physics.

The solution separates into three pieces — one for each coordinate:

where $R_{nl}(r)$ is the radial wave function and $Y_l^m(\theta, \phi)$ is the spherical harmonic we explored in the previous article. The quantum numbers are:

• $n = 1, 2, 3, \ldots$ — the principal quantum number, setting the energy level $E_n = -13.6\text{ eV}/n^2$
• $l = 0, 1, \ldots, n-1$ — the orbital angular momentum quantum number
• $m = -l, \ldots, +l$ — the magnetic quantum number, the $z$-projection of angular momentum

The Radial Wave Function $R_{nl}(r)$

The radial part solves the radial Schrödinger equation with the Coulomb potential $V(r) = -e^2/(4\pi\epsilon_0 r)$. The solution involves the associated Laguerre polynomials $L_{n-l-1}^{2l+1}$:

where $a_0 \approx 0.529\text{ Å}$ is the Bohr radius. The first few radial functions (in units of $a_0$):

The number of radial nodes — spherical shells where $R = 0$ — is $n - l - 1$. Explore the radial wave functions below:

Notice how $R_{10}$ has no radial nodes (it just decays from the origin), while $R_{20}$ passes through zero once — creating a nodal sphere at $r = 2a_0$. The electron has zero probability of being found at that radius.

Radial Probability Distribution $r^2|R_{nl}|^2$

The probability of finding the electron between $r$ and $r + dr$ is not simply $|R|^2$. Because we're in 3D, the volume element is $r^2 \sin\theta \, dr \, d\theta \, d\phi$, so the radial probability density is $r^2|R_{nl}(r)|^2$.

This makes a large difference: even though $|R_{10}|^2$ is largest at $r = 0$, the probability of finding the electron near the origin is small because the volume shell $4\pi r^2 dr$ is tiny there. The peak of $r^2|R_{10}|^2$ occurs at $r = a_0$ — the Bohr radius, exactly where Bohr's classical model predicted.

For $n = 2$, $l = 0$ (the 2s orbital), the probability peaks twice — corresponding to the electron spending time both near the nucleus and further out, with a radial node in between. For $n = 2$, $l = 1$ (the 2p orbital), there is a single peak further from the nucleus. Higher $n$ pushes the electron further out and adds more peaks.

Counting Nodes

The wave function's nodes reveal deep structure. For hydrogen:

• Radial nodes: $n - l - 1$ — spherical shells where $R_{nl} = 0$
• Angular nodes in $\theta$: $l - |m|$ — conical surfaces about the $z$-axis where $Y_l^m = 0$
• Angular nodes in $\phi$: $|m|$ — planes through the $z$-axis where $\text{Re}(e^{im\phi}) = 0$ (for real harmonics)
• Total nodes: $n - 1$

The total is always $n - 1$, split between radial and angular. As $l$ increases (for fixed $n$), radial nodes convert into angular nodes: the wave function trades spherical shells for cones and planes.

3D Orbital Shapes: Isosurfaces of $|\psi|^2$

The most intuitive way to visualize an orbital is as an isosurface of constant probability density. We pick a threshold value and draw the surface where $|\psi_{nlm}|^2$ equals that value. The electron is most likely found inside this surface.

Blue lobes show regions where $\psi > 0$; pink lobes show $\psi < 0$. The sign has no physical meaning for probability (which depends on $|\psi|^2$), but it reveals the wave function's phase structure — crucial for understanding chemical bonding.

The 1s Orbital: Ground State

The simplest hydrogen orbital has $n = 1$, $l = 0$, $m = 0$:

This is spherically symmetric — no angular dependence at all. The electron cloud is a fuzzy sphere, densest at the nucleus and decaying exponentially. There are zero nodes ($n - 1 = 0$). The energy is $E_1 = -13.6\text{ eV}$.

The 2s Orbital: First Radial Node

For $n = 2$, $l = 0$:

Still spherically symmetric ($l = 0$ means $Y_0^0$ is constant), but now there is one radial node at $r = 2a_0$. The isosurface shows two concentric shells — an inner sphere and an outer shell, separated by a nodal surface where $\psi = 0$.

The 2p Orbitals: Angular Structure

The three $n = 2$, $l = 1$ orbitals ($m = -1, 0, +1$) introduce angular nodes. The $m = 0$ case:

This is the $p_z$ orbital: two lobes along the $z$-axis, separated by the $xy$-plane (where $\cos\theta = 0$). No radial nodes ($n - l - 1 = 0$), one angular node ($l - |m| = 1$, the equatorial plane which is a cone of half-angle $\pi/2$).

The $m = \pm 1$ cases give two more lobes oriented in the $x$ and $y$ directions. In the real-harmonic basis, these become the familiar $p_x$ and $p_y$ orbitals.

The d and f Orbitals

For $l = 2$, the five $d$ orbitals have intricate shapes: cloverleaf patterns ($d_{xy}$, $d_{x^2-y^2}$), figure-eights with a torus ($d_{z^2}$), and tilted double lobes ($d_{xz}$, $d_{yz}$). These are the orbitals responsible for the colors of transition metals and the geometry of coordination compounds.

The $l = 3$ $f$ orbitals ($n \ge 4$) have even more complex shapes with up to 6 angular nodes. They play a central role in the chemistry of rare earth elements. Select the 4f₀ preset in the explorer above to see them.

Energy Levels and Degeneracy

A remarkable feature of hydrogen: the energy depends only on $n$, not on $l$ or $m$:

This means all orbitals with the same $n$ are degenerate — they have the same energy despite very different shapes. The 2s and three 2p orbitals are all at $E_2 = -3.4\text{ eV}$. This accidental degeneracy is unique to the Coulomb potential and is broken in multi-electron atoms by electron-electron repulsion.

The total number of degenerate states at level $n$ is $n^2$ (ignoring spin) or $2n^2$ (with spin). This directly gives the structure of the periodic table.

Exercises

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