What are the four quantum numbers?

The four quantum numbers are: n (principal, n = 1, 2, 3, …), l (orbital angular momentum, 0 ≤ l ≤ n−1), ml (magnetic, −l ≤ ml ≤ l), and ms (spin, +½ or −½). Together they uniquely identify each electron in an atom.

How does the Quantum Number Calculator validate combinations?

The Quantum Number Calculator checks all four quantum-number selection rules simultaneously. It flags exactly which rule is violated (e.g. l ≥ n, or |ml| > l) so you can identify and correct the error immediately.

How many electrons can occupy each shell?

Shell n can hold a maximum of 2n² electrons. Shell 1 holds 2, shell 2 holds 8, shell 3 holds 18, and shell 4 holds 32.

What does the ‘Show All Combinations’ feature do?

It generates every valid (n, l, ml, ms) combination for the chosen principal quantum number n, showing all allowed electron states in that shell. This is useful for drawing orbital diagrams and writing electron configurations.

No. All calculations run entirely in your browser. Nothing is transmitted to a server.

Quantum Number Calculator

How to Use Quantum Number Calculator

The Quantum Number Calculator validates quantum number sets and lists allowed orbital states.

Enter a Principal Quantum Number n — Type any positive integer (1, 2, 3, …). The Quantum Number Calculator immediately shows the allowed range for l.
Set l, ml, and ms — Enter values within the displayed constraints. The calculator highlights in red any value that violates the selection rules and shows the specific rule violated.
Read the Validation Result — A valid set shows the subshell name (s, p, d, f), the orbital label, and the spin label. An invalid set shows which rule is broken.
Click Show All Combinations — The Quantum Number Calculator lists every valid (n, l, ml, ms) set for the chosen n, with subshell groupings, so you can fill an orbital diagram step by step.

Formula & Theory — Quantum Number Calculator

The Quantum Number Calculator enforces the four quantum-number selection rules from quantum mechanics:

n  = 1, 2, 3, …              (principal quantum number)
l  = 0, 1, …, n−1            (orbital angular momentum quantum number)
ml = −l, −l+1, …, 0, …, l   (magnetic quantum number)
ms = +½ or −½               (spin quantum number)

Symbol	Name	Allowed Values	Physical Meaning
n	Principal	1, 2, 3, …	Shell / energy level
l	Orbital (azimuthal)	0 to n−1	Subshell shape
ml	Magnetic	−l to +l	Orbital orientation
ms	Spin	+½ or −½	Electron spin direction

Subshell Names

l value	Subshell name	Max electrons (2(2l+1))
0	s	2
1	p	6
2	d	10
3	f	14

Total Electrons per Shell

Maximum electrons in shell n:

N_max = 2n²

For n=1: 2 electrons; n=2: 8; n=3: 18; n=4: 32.

Use Cases for Quantum Number Calculator

The Quantum Number Calculator supports students and researchers across chemistry and physics:

General chemistry homework — Students verify quantum number sets and fill in electron configurations without guessing at which values are allowed.
AP Chemistry and IB Chemistry — Exam preparation questions on orbital diagrams and electron spin are answered and explained instantly.
Physical chemistry courses — Undergraduate students exploring wave mechanics and the Schrödinger equation use the Quantum Number Calculator to cross-check hand-derived solutions.
Materials science — Predicting magnetic properties (ferromagnetism, paramagnetism) from the number of unpaired d or f electrons starts with validating quantum number sets.
Spectroscopy — Interpreting atomic emission and absorption spectra requires knowing which transitions are allowed, which in turn depends on valid quantum number differences.
Nuclear and quantum physics — Researchers cross-check shell-model quantum number assignments for nucleons using the same selection-rule framework.