Atomic Orbitals

Orbitals are wave patterns around a nucleus, not paths traced by tiny planets. Explore how angular nodes divide those patterns into the s, p, d, and f shapes, and how the patterns describe where an electron is likely to be found.

Source code

On this page · Compare the shapes

Start by comparing shapes

Choose an orbital family, then choose one of its orientations. Rotate the model to see the full three-dimensional pattern.

Choose an orbital family

Choose an orientation

Shell

n

: 2 · 0 radial nodes

Positive phase Negative phase

1a_0 \approx 52.9\,\mathrm{pm}

2p_y\;(\ell=1)

· 3 shapes

Rotate by dragging or with the arrow keys; zoom with pinch, scroll, or +/−. Denser points show where the electron is more likely to be found. Color shows the wave's sign, not a different charge. The shaded envelope is a guide to the shape, not a hard edge.

Model limits and physical scale

These are hydrogen-like models, not measured orbitals for neutral cerium. The scale bar uses Bohr radii ( $1a_0\approx 52.9\,\mathrm{pm}$ ); 90% of this orbital's radial probability lies within $8.0a_0\approx 423\,\mathrm{pm}$ of the nucleus. The camera fits each selection, so compare scale bars instead of the on-screen diameters.

Try subshells occupied in cerium

These buttons select representative hydrogen-like shapes for occupied subshells. They do not assign cerium's d or f electron to one unique orientation.

Notice where the pattern has lobes and where it has gaps. The gaps are places where the wave is zero; those surfaces help determine each orbital's shape.

Angular nodes make the shapes

The angular pattern describes how the wave changes with direction. Its zero-amplitude surfaces, called angular nodes, divide the surrounding space into lobes. Changing the angular quantum number $\ell$ changes this pattern.

Choose a pattern (angular quantum number

\ell

)

This cut shows how one angular pattern changes with direction. It is not the full orbital or a hard boundary. $\ell=1$ gives 1 angular node and 3 real-basis shapes.

The sequence is $s\;(\ell=0),\;p\;(\ell=1),\;d\;(\ell=2)$ , and $f\;(\ell=3)$ . Each family has $2\ell+1$ orientations: 1, 3, 5, and 7. The magnetic quantum number $m$ distinguishes angular patterns within a family; it does not change the family itself.

An orbital is a probability pattern

The wavefunction can have positive or negative values. Its sign is not electric charge. Squaring its magnitude, $|\psi|^2$ , gives the position probability density. Integrating that density over a region gives the probability of finding the electron there.

Show the same

2p_y

wave in two ways

Blue and orange mean opposite wave signs, not different charges.

The colored lobes are not solid walls. They summarize where the electron is more likely to be found; the color shows the wave's sign, not a different kind of charge.

The equation, drawn

The $p$ orbital is the probability pattern

For $p_z$ , the angular wave is proportional to $\cos\theta$ . It is positive above the nucleus, negative below, and zero all across the middle plane.

Squaring the wave removes the sign but keeps the zero. Probability is concentrated on either side of the nodal plane, creating two lobes.

The

p_z

state shown in an

\text{x--z}

slice; in 3D, the nodal plane extends around the nucleus.

Color = wave phase. The two sides have opposite signs; neither lobe is a different kind of charge.

Density = $|\psi|^2$ . Both lobes have positive probability density, with zero at the nodal plane.

Rotate the axis

Three orientations, same pattern

The three diagrams show the same two-lobed pattern aimed along different axes. For an isolated atom without an external field, these orientations have equal energy; the labels choose a coordinate direction.

p_x

p_y

p_z

The drawings are schematic cross-sections of probability density, not hard surfaces. An orbital is a quantum state; the electron does not trace out the pictured dumbbell.

Same shape, different shell

The shell number $n$ changes the radial pattern: how far from the nucleus the electron is likely to be found and how many spherical nodes appear. Changing $n$ can add radial structure while keeping the same angular family.

From the equation to an orbital shape

Follow the arrows

Worked example: where a p orbital gets its two lobes

We solve the stationary Schrödinger equation for an electron attracted to a nucleus. Each arrow shows the next mathematical choice and what it tells us about the shape.

01
Start with the governing equation
For hydrogen, treat the nucleus as fixed and use its Coulomb potential:
$\begin{aligned} -\frac{\hbar^2}{2m_e}\nabla^2\psi + V(r)\psi &= E\psi \\ V(r) &= -\frac{e^2}{4\pi\varepsilon_0 r} \end{aligned}$
This equation balances the wave's kinetic energy against its electric attraction to the nucleus. Solving it gives the allowed wavefunctions $\psi$ .
Try moving away from the nucleus
Distance from nucleus: $2.0a_0$
Only distance matters here. That spherical symmetry lets us split the equation into radial and angular parts.
The potential depends only on distance $r$ , so the problem has spherical symmetry.
02
Separate distance from direction
Spherical symmetry lets the solution split into a radial part and an angular part:
$\psi(r,\theta,\phi)=R_{n\ell}(r)Y_{\ell}^{m}(\theta,\phi)$
$R$ controls how the wave changes with distance. $Y$ , a spherical harmonic, describes how it changes from one direction to another.
The quantum numbers obey $n \ge 1$ , $0 \le \ell < n$ , and $-\ell \le m \le \ell$ . The real orbitals in the viewer combine complex magnetic-quantum-number states when $m$ is nonzero.
Probe a $2p_y$ wave at one position
Radius: $3.0a_0$
Angle from $+y$ : $30^{\circ}$
$\text{radial factor}=r e^{-r/2}=0.67$
$\text{angular factor}=\cos\theta=0.87$
$\text{wave}\propto\text{radial}\times\text{angular}=0.58$
These values omit a shared normalization constant. Crossing $90^{\circ}$ flips the wave’s sign; it does not make the electron’s charge change.
This probe uses $2p_y$ with its polar angle measured from the viewer's vertical $+y$ axis. The worked example below uses $2p_z$ and measures the polar angle from $+z$ . Rotating the coordinates gives the same two-lobed pattern.
The separated angular equation is an angular-momentum eigenvalue problem; its solutions are spherical harmonics.
03
Select the p pattern
The letter p means $\ell=1$ . The angular-momentum equation gives its allowed angular pattern; for the orientation called $p_z$ :
$\begin{aligned} \hat{L}^2Y_{\ell}^{m} &= \ell(\ell+1)\hbar^2Y_{\ell}^{m} \\ Y_1^0(\theta,\phi) &\propto \cos\theta \end{aligned}$
The cosine is positive on one side of the nucleus and negative on the other. At $\theta=\frac{\pi}{2}$ , it is zero: that is the plane $z=0$ .
For hydrogen's $2p$ state, the radial solution supplies the size and falloff with distance.
04
Combine the radial and angular waves
The hydrogen $2p$ radial solution is $R_{21}(r)\propto re^{-r/(2a_0)}$ . Multiplying by the angular solution gives:
$\psi_{2p_z}(r,\theta)\propto re^{-r/(2a_0)}\cos\theta$
The radial factor sets how far the cloud extends; the cosine factor sets its direction and the nodal plane.
The Born rule says measurement probability density is the absolute square of the wave.
05
Square the wave: the lobes appear
$|\psi_{2p_z}|^2\propto r^2e^{-r/a_0}\cos^2\theta$
At $\theta=\frac{\pi}{2}$ , $\cos\theta=0$ , so the probability density vanishes across the whole $z=0$ plane. At a given nonzero distance, $\cos^2\theta$ is largest along $+z$ and $-z$ . The high-probability regions on those two sides are the lobes of a $p$ orbital.

Symbol guide: what is inside Schrödinger's equation?

Symbols and SI units in the hydrogen orbital derivation
Symbol	Term	Meaning	SI unit
$\psi$	Wavefunction	The quantum state whose spatial shape we solve for.	$\mathrm{m}^{-3/2}$
$\hbar$	Reduced Planck constant	Sets the scale of quantum effects.	$\mathrm{J}\,\mathrm{s}$
$m_e$	Electron mass	Sets the kinetic-energy scale of the electron.	$\mathrm{kg}$
$\nabla^2$	Laplacian	Measures how the wave bends through space.	$\mathrm{m}^{-2}$
$V(r)$	Potential energy	The electron's electric potential energy, set by distance $r$ .	$\mathrm{J}$
$e$	Elementary charge	Sets the strength of the electron–nucleus attraction.	$\mathrm{C}$
$\varepsilon_0$	Vacuum permittivity	Relates electric charge to the Coulomb potential.	$\mathrm{F}/\mathrm{m}$
$r$	Radial distance	Distance from the nucleus to the electron.	$\mathrm{m}$
$E$	Energy eigenvalue	An allowed energy of the stationary quantum state.	$\mathrm{J}$
$a_0$	Bohr radius	Sets the radial scale in hydrogen.	$\mathrm{m}$

The wavefunction unit assumes a normalized three-dimensional spatial wavefunction. The Laplacian's unit describes the operator, before it acts on the wavefunction.

Sources and next steps

OpenStax: The Hydrogen Atom — equation, separation, and probability.
OpenStax: Development of Quantum Theory — orbital families and quantum numbers.
NIST: Cerium and NIST: Lanthanum — measured ground-state configurations.

Atomic Orbitals

Start by comparing shapes

Angular nodes make the shapes

An orbital is a probability pattern

The ppp orbital is the probability pattern

Three orientations, same pattern

Same shape, different shell

See the radial wave and node rule

From the equation to an orbital shape

Worked example: where a p orbital gets its two lobes

Start with the governing equation

Separate distance from direction

Select the p pattern

Combine the radial and angular waves

Square the wave: the lobes appear

Where do all four families appear in an atom?

Symbol guide: what is inside Schrödinger's equation?

Sources and next steps

The $p$ orbital is the probability pattern