Chapter 1
The Hydrogen Atom
One electron, one proton, and yet a whole zoo of shapes. Rotate, slice and superpose the orbitals of the simplest atom.
What is an orbital?
An orbital is not a path along which an electron circles. It is a wave function ψ (“psi”), the solution of the Schrödinger equation for an electron in the field of the nucleus. The square |ψ|² gives the probability of finding the electron at a given place. In the view above, brightness is exactly this probability density.
The hydrogen atom is the only atom that can be solved exactly. That is why its orbitals are the yardstick for all other atoms and molecules.
How to read the picture
- Brightness means probability: bright means the electron is often found here.
- Colour means the sign of ψ (blue positive, orange negative). The sign itself cannot be measured, but it decides how orbitals behave when they combine.
- Nodes are surfaces where ψ is zero. The electron is never there, although it occurs on both sides.
- “Bounded” shows the body in which the electron stays with 90% probability. “Soft” shows the whole cloud as a mist of dots.
The three quantum numbers
- n – the energy level
- In hydrogen it sets the energy (−13.6 eV / n²) and the size of the cloud, which grows roughly with n². At n = 1 the atom is about 0.1 nm across.
- l – the shape
- From 0 to n − 1. The letters s, p, d, f, g, h, i stand for l = 0, 1, 2, 3, 4, 5, 6 and date from early spectroscopy. l gives how many nodal surfaces pass through the nucleus.
- m – the orientation
- There are 2l + 1 ways to orient an orbital in space: one s, three p, five d and seven f orbitals. We show the real form, for which the names x, y, z, xy and so on can be used.
In hydrogen all orbitals with the same n have the same energy. In heavier atoms the interaction between electrons lifts this degeneracy, which is why 2s lies lower than 2p there.
Why do orbitals have these shapes?
An electron in an atom is a standing wave, much like the vibration of a guitar string or a drumhead. Only certain vibration patterns fit into the space around the nucleus. The more nodes a pattern has, the more energy it carries. The number of radial nodes is n − l − 1 and the number of nodal surfaces through the nucleus is l. Together they make n − 1.
Superposition: when an atom emits light
Choose “Superposition” above to mix two states. This is allowed because the Schrödinger equation is linear. The result is no longer a still orbital but a cloud that oscillates at the frequency ΔE/h. If the centre of charge moves while it does so (for example 1s + 2p), the atom radiates light of that frequency like a tiny antenna. This is how the spectral lines of hydrogen arise, for example Lyman-α at 121.6 nm or Balmer-α at 656 nm.
Frequently asked questions
- Where is the electron really?
- It has no definite position until you measure it. The orbital only tells you with what probability you will find it where.
- Doesn’t the electron orbit the nucleus like a planet?
- No. Bohr’s planetary model was an important stepping stone but is outdated. A 1s electron, for example, has no angular momentum, so it cannot “circle”.
- Why does the picture show signs if only |ψ|² can be measured?
- The sign decides how waves combine: equal signs reinforce, opposite signs cancel. This is the basis of the chemical bond (see the next chapter).
- How big is a hydrogen atom?
- The mean distance between electron and nucleus in the ground state is 1.5 Bohr radii, about 0.08 nm. The Bohr radius is 52.9 pm.
Think about it
Which orbital with n = 3 has two spherical shells as nodes?
The 3s orbital. The number of radial nodes is n − l − 1, so 3 − 0 − 1 = 2. Check it in the slice: you see two colour changes that appear as rings.
Why does 1s + 2p radiate light but 1s + 2s does not?
Only a cloud whose centre of charge moves back and forth can radiate. That is the case for 1s + 2p (Δl = ±1). For 1s + 2s the cloud just “breathes” spherically.
How many d orbitals are there on one level?
Five, because there are 2l + 1 orientations for l = 2. You see them under “Orientation” as soon as you choose n = 3 and l = d.
In the next chapter we combine hydrogen and oxygen into a water molecule.
A question or an idea?
Something unclear, an error you spotted, or a chapter you would like to see? Write to me. Good questions and their answers will appear here later as an FAQ.
Write a messageLast updated: 2026-10-03