Each electron in an atom is described by four quantum numbers. The principal number n sets the shell; l the subshell shape (s, p, d, f…); m_l the orbital orientation; m_s the spin. Together they explain the periodic table.
How to use the Quantum Numbers Calculator
- Principal quantum number n
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the principal quantum number n (1–7 for known atoms).
Common mistakes to avoid
- Mixing units between fields.
- Using a rule outside the range where it applies.
- Expecting fun estimates to be exact predictions.
What each result means
- Allowed l values = [Principal quantum number n] – 1
- m_l range for the highest l = [Principal quantum number n] – 1
- Number of orbitals n² = [Principal quantum number n]^2
- Maximum electrons 2n² = 2×[Principal quantum number n]^2
Example values
How the main result changes when Principal quantum number n changes and everything else stays at the example values above:
| Principal quantum number n | Number of orbitals n² |
|---|---|
| 1.5 | 2 |
| 2.25 | 5 |
| 3 | 9 |
| 3.75 | 14 |
| 4.5 | 20 |
| 6 | 36 |
Formula
l = 0 … n − 1, m_l = −l … +l, m_s = ±½, orbitals = n², electrons = 2n²
Worked example
Inputs — Principal quantum number n: 3.
Results — Allowed l values = 0, 1, 2 (s, p, d); m_l range for the highest l = −2 … +2; Number of subshells = 3; Number of orbitals n² = 9; Maximum electrons 2n² = 18.
Key terms
- Orbital
- Region where an electron with given n, l, m_l is likely found.
Frequently asked questions
Why can l only go up to n − 1?
It follows from solving Schrödinger's equation for the atom.
What do s, p, d and f mean?
Historical spectroscopic labels: sharp, principal, diffuse, fundamental.
Why 2 electrons per orbital?
The Pauli exclusion principle allows two opposite spins.
How many electrons fit in shell 3?
18.