Where is the electron, really? Spin the orbitals and see their shapes for yourself

How to use this simulation

1. The simulation opens with a 2pz orbital slowly turning. Every dot is one imagined snapshot of the electron, and thousands of snapshots build up the dumbbell. Dots in the theme colour mark the positive lobe and orange dots mark the negative lobe.
2. Pick 1s, 2s, 2p or 3d with the "Which orbital" buttons. The readings panel updates n, l and the mₗ values, the node counts, how many orbitals and electrons the subshell holds, and the average distance from the nucleus.
3. Drag the canvas with a mouse or finger to rotate it yourself, or focus it and use the arrow keys. Press Pause to freeze the turning and study a shape. Use the zoom and points sliders to move in or thicken the cloud.
4. Tick "Show nodes". Red surfaces mark places where the chance of finding the electron is exactly zero: a sphere in 2s, one plane in 2p, two planes in 3d (two cones in dz²). Check each against the node count in the panel.

You cannot catch an electron, only hunt for it

Picture a mosquito buzzing in a dark room. You fire a camera flash once a second and take a photo each time. Lay a thousand photos on top of each other and you do not get a path. You get a cloud of dots, thick in some corners of the room and thin in others. Where the cloud is thick, the mosquito spends most of its time.

An electron in an atom behaves the same way, with one twist: the flash needed to photograph an electron shoves it away. Heisenberg's uncertainty principle says you can never know both its exact position and its exact velocity at once. So instead of asking "where is the electron?" we ask "how likely is it to be here?".

That map of likelihood is the orbital. Each dot in the simulation is one imagined photo, and the thick and thin patches together draw the orbital. The dots are placed by mathematics from a fixed starting seed, so the picture is steady and does not flicker.

Below we build the idea from scratch: why an orbital is not an orbit, what the four quantum numbers say, how to count nodes, and why s, p and d have the shapes they do. You can check every claim in the simulation as you go.

Orbit and orbital: two different ideas

In 1913 Niels Bohr proposed that an electron circles the nucleus on a fixed circular path, like a planet around the sun. For hydrogen the first such path has radius 52.9 pm, a length called the Bohr radius, a₀. The picture is simple, and it explained the spectrum of hydrogen well.

But it does not survive. It fails for atoms with more than one electron, and it clashes head-on with the uncertainty principle, because a definite path means a definite position and velocity together. In 1926 Schrödinger wrote an equation whose solutions are wavefunctions, written ψ (psi).

ψ itself cannot be measured directly. But ψ² is the probability density: the chance of finding the electron in a tiny volume around a point is proportional to ψ² there. Where ψ² is large the electron spends more time. The three-dimensional shape of the region where ψ² is large is what we call the orbital.

The words you need, in one table

Definitions score marks only if the words are exact. These terms come back again and again.

The four quantum numbers: an electron's address

Think of a postal address: area, street, house number, floor. Every electron in an atom has an address of four numbers. The first three say which orbital it is in. The fourth says how the electron itself is spinning inside that orbital.

s, p, d: the three families of shapes

Now sit in front of the simulation. Each shape has a reason, and the reason comes straight from the value of l. Learning the reason beats memorising the picture.

A simple rule for counting nodes

There are two kinds of node. A radial node is a sphere around the nucleus. An angular node is a plane (or cone) through the nucleus. The counting rules are short: radial nodes = n − l − 1, angular nodes = l, so total nodes = n − 1.

At a node the probability is zero, so you will not find the electron there, yet it can still get from one side to the other. That is a strange feature of the quantum world, and it means a node must not be pictured as a wall.

What the + and − of a lobe mean

Did you notice two colours of dots? That is the sign of ψ. A wavefunction is a wave, so in places it is positive and in others negative, like the crest and trough of a water wave. Negative does not mean negative charge; it means the opposite phase of the wave.

Probability depends on ψ², so the sign does not change the density, and both colours are sampled equally thickly. Then why does it matter? When two atoms bond, their orbitals overlap. Lobes of the same sign add up, the wave strengthens, electrons gather between the nuclei and a bond forms.

Lobes of opposite sign cancel, a new node appears between the nuclei, and that is an antibonding combination. The sign is the root of how bond angles and molecule shapes work out. Two pz lobes meeting head on make a σ bond; side by side they make a π bond.

How many electrons shells and subshells hold

There is one master rule: an orbital holds at most two electrons. A subshell has (2l + 1) orbitals, so it holds 2(2l + 1): 2 in s, 6 in p, 10 in d and 14 in f. A whole n-th shell holds 2n²: 2 in the first, 8 in the second, 18 in the third and 32 in the fourth.

The filling order follows the (n + l) rule: the lower n + l fills first, and on a tie the lower n goes first. For 4s, n + l = 4; for 3d it is 5. So 4s fills first. The full order is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s.

Pour electrons in that order and the running totals are: 1s: 2 · 2s: 4 · 2p: 10 · 3s: 12 · 3p: 18 · 4s: 20 · 3d: 30 · 4p: 36 · 5s: 38. So filling 4s takes 20 electrons and finishing 3d takes 30, which is the atomic number of zinc.

Experiments to run in the simulation

Each takes a minute or two. The rules will click into place on their own as you go.

• Experiment 1 (size): choose 1s, 2s and 3d and read the average distance each time. The bigger the shell, the more spread out the orbital, so the electron sits further out on average.
• Experiment 2 (counting nodes): switch nodes on for 2s and for 2pz. Both have the same total number of nodes (n − 1) but one is a sphere and the other a plane. Watch where the dot colour flips.
• Experiment 3 (three p directions): pick 2px, 2py and 2pz in turn and watch the lobes change direction. The three shapes are identical and differ only in direction, so their energy is the same.
• Experiment 4 (d clovers): compare 3dxy with 3dx²−y². One is the other turned through 45°. Then open 3dz² and find its ring.
• Experiment 5 (viewpoint): press Pause and drag the cloud until you look straight down the z axis. A pz orbital then looks like a round blob, yet it is a dumbbell. One view is never enough to judge a 3D shape.
• Experiment 6 (density): drop the points to 500, then raise them to 4000. With few dots the shape is vague, with many it is sharp. That is exactly the character of a probability cloud.

Worked problems

Every number below is computed from the rules above. Follow the formula and check it yourself.

The most probable distance of the electron

The average distance and the most probable distance are not the same thing. For a subshell with l = n − 1 (such as 1s, 2p and 3d) the most probable distance is n² a₀. The values below are computed from that formula.

Mistakes almost everyone makes

These slips cost marks in exams. See whether you are making any of them.

• Calling an orbital an orbit. An orbit is a fixed path; an orbital is a region of probability. The two words are not interchangeable.
• Thinking the electron in a p orbital runs along a figure of eight. It has no definite path, it may be in either lobe, and it is simply never found on the nodal plane.
• Reading + and − as electric charge. They are the sign of the wavefunction; the electron's charge is always negative.
• Treating a node as a wall. A node only says the probability is zero there; the electron can still get from one side to the other.
• Writing l ≥ n. The largest l is n − 1, so there is no 2d and no 1p.
• Listing the wrong mₗ values: forgetting that they run from −l to +l, which gives the wrong orbital count.
• Misreading the ring of 3dz². It is part of a d orbital, not a separate s orbital, and its nodes are two cones.
• Mixing up the filling order of 4s and 3d. 4s fills first, but when a transition metal loses electrons it gives up the 4s ones first.

Orbitals in everyday life

Orbitals are not just a picture in a textbook. They explain a lot of what you can see around you.

• Coloured compounds: copper sulfate is blue because electrons in the 3d orbitals of Cu²⁺ absorb light and jump between d orbitals. With no d orbitals there would be no such colour.
• Firework colours: heat lifts electrons to higher orbitals, and they give out light of a fixed wavelength as they fall back. Sodium gives yellow, strontium red and copper blue-green.
• Molecule shapes: the four C–H bonds in methane are identical because one s and three p orbitals blend into four equal sp³ hybrids. Knowing orbital shapes tells you molecular shapes.
• Magnetism: iron has unpaired electrons in its 3d orbitals, so it is drawn toward a magnet. Counting unpaired electrons tells you whether a substance is paramagnetic.
• Semiconductors and LEDs: the s and p orbitals of silicon blend into energy bands. The gap between the bands decides what colour of light an LED gives.

Exam corner: CBSE and NCERT style questions

Orbitals and quantum numbers appear nearly every year, in one-mark, two-mark and long-answer form.

Revision in one glance

The night before an exam, reading this list once is enough.

• An orbital is the most probable three-dimensional region for an electron around a nucleus, the dense part of ψ².
• Four quantum numbers: n (shell and size), l (shape), mₗ (orientation), mₛ (spin).
• l runs from 0 to n − 1; mₗ from −l to +l; mₛ is ±½.
• Radial nodes = n − l − 1, angular nodes = l, total nodes = n − 1.
• s is a sphere, p a dumbbell (three, at 90°), d a clover (five, with dz² the exception).
• Subshell capacity 2(2l + 1); a shell has n² orbitals and 2n² electrons.
• Pauli principle: at most two electrons per orbital, with opposite spins.
• Filling order follows (n + l): 4s before 3d.
• The + and − of a lobe are the phase of the wave, not charge, and they decide bonding.

Frequently asked questions