On this page
Every electron in an atom can be described by a unique set of four numbers, like an address: which floor, which apartment, which room, and which of two beds. These quantum numbers come out of the mathematics of quantum mechanics, and they explain the shape of the periodic table itself.
1. Principal quantum number, n (the shell)
- What it tells you: the main energy level (shell) and roughly how far the electron is from the nucleus.
- Allowed values: 1, 2, 3, 4 … (positive whole numbers)
- Larger n means higher energy and a larger orbital.
The row (period) an element is in matches the highest n of its electrons. Sodium’s outer electron has n = 3, and sodium is in period 3.
2. Angular momentum quantum number, l (the subshell)
- What it tells you: the shape of the orbital — which subshell.
- Allowed values: 0 up to n − 1.
| l | Subshell | Shape |
|---|---|---|
| 0 | s | sphere |
| 1 | p | dumbbell |
| 2 | d | mostly four-lobed (“cloverleaf”) |
| 3 | f | more complex |
Because l can’t exceed n − 1, the first shell (n = 1) only has an s subshell; the second (n = 2) has s and p; the third has s, p and d. That’s why there’s no “1p” or “2d”. (The letters come from old descriptions of spectral lines: sharp, principal, diffuse, fundamental.)
3. Magnetic quantum number, mₗ (the orbital)
- What it tells you: the orientation of the orbital in space — which specific orbital within a subshell.
- Allowed values: whole numbers from −l to +l.
| Subshell | l | mₗ values | Number of orbitals |
|---|---|---|---|
| s | 0 | 0 | 1 |
| p | 1 | −1, 0, +1 | 3 |
| d | 2 | −2, −1, 0, +1, +2 | 5 |
| f | 3 | −3 … +3 | 7 |
The three p orbitals point along three perpendicular directions (often labelled pₓ, p_y, p_z).
4. Spin quantum number, mₛ
- What it tells you: the electron’s intrinsic spin — drawn as an up or down arrow.
- Allowed values: +½ or −½.
Spin isn’t the electron literally rotating; it’s an intrinsic quantum property that makes each electron behave like a tiny magnet.
The Pauli exclusion principle
No two electrons in the same atom can have the same set of all four quantum numbers. Since n, l and mₗ together identify an orbital, and mₛ has only two possible values, each orbital holds at most two electrons, with opposite spins. See electron configuration rules.
How it builds the periodic table
From those rules:
- Each orbital: 2 electrons
- s subshell (1 orbital): 2 electrons
- p subshell (3 orbitals): 6
- d subshell (5 orbitals): 10
- f subshell (7 orbitals): 14
- A whole shell n holds 2n² electrons: 2, 8, 18, 32…
Those numbers are the widths of the s, p, d and f blocks, which is why the periodic table is shaped the way it is.
Worked examples
Q: Give a possible set of quantum numbers for the outer electron of potassium. Potassium is [Ar] 4s¹. The electron is in 4s: n = 4, l = 0, mₗ = 0, mₛ = +½ (or −½).
Q: How many electrons can have n = 3 and l = 2? That’s the 3d subshell: 5 orbitals × 2 = 10 electrons.
Q: Is n = 2, l = 2 allowed? No — l must be less than n. There is no 2d subshell.
Q: How many orbitals are in the n = 4 shell? n² = 16 (one 4s, three 4p, five 4d, seven 4f).
Quick answers
What are the four quantum numbers? Principal (n), angular momentum (l), magnetic (mₗ) and spin (mₛ).
Which quantum number describes orbital shape? l, the angular momentum (or azimuthal) quantum number.
Why can’t two electrons have the same four quantum numbers? It’s the Pauli exclusion principle, a fundamental property of electrons and similar particles (fermions).
Related
See how the model developed in the history of atomic models, and check any element’s configuration with the electron configuration generator.
Advertisement