The Hydrogen Atom: Quantum Numbers and Orbital Shapes
One electron and one proton is the only atom the equation can be solved for exactly. Its solution produces the three spatial quantum numbers, and with them the shapes and sizes that orbital pictures have been standing in for.
What a learner can do afterwards
- Says which quantum number controls energy, which controls shape and which controls orientation, and gives the allowed values of each
- Reads a radial distribution plot and states where the electron is most likely to be found
- Counts radial and angular nodes for a named orbital and checks the total against the principal quantum number
- Explains why the hydrogen energy depends only on n while a many-electron atom breaks that pattern
1 · Read
Hydrogen is the one atom you can solve exactly, so it anchors everything after. Its solutions carry three labels. The number n sets size and energy, l sets shape, and ml sets orientation in space. The rules are strict: n runs 1, 2, 3 and up, l runs 0 to n minus 1, and ml runs minus l to plus l. Chemists nickname l = 0, 1, 2, 3 as s, p, d, f.
Each solution is a standing wave whose square is the electron cloud. A radial plot gives the chance of finding the electron at each distance, so its tallest peak marks the most likely distance from the nucleus. A 3s cloud has two spherical nodes, while a 2p cloud has one nodal plane through the nucleus.
Count nodes to check any label. The total is always n minus 1, split between radial shells and angular planes. A 3s orbital has 2 radial plus 0 angular, a 3p has 1 plus 1, and a 3d has 0 plus 2. If your two counts refuse to add to n minus 1, the label is wrong.
In hydrogen, energy depends only on n, so 2s and 2p sit at the same height. With more electrons, inner ones shield the nucleus, and s electrons slip through that shield more than p electrons do, so shape starts to matter for energy too.
Three numbers label each orbital, nodes check the label, and hydrogen energy needs only n.
Rewritten readings were written for this dataset.
2 · Watch
En anglaisTake it off screen
Where it sits
Optional connections
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.