# Quantum Numbers Calculator

URL: https://speedycalculator.com/calculators/physics/quantum-number/
Last updated: 2026-10-30

> The quantum number calculator lists the allowed azimuthal (l) and magnetic (m_l) quantum numbers for a principal quantum number n, and the number of subshells, orbitals and electrons in that shell.

## Inputs

- Principal quantum number n

## Results

- Allowed l values
- m_l range for the highest l
- Number of subshells
- Number of orbitals n²
- Maximum electrons 2n²

## Formula

```
l = 0 … n − 1,  m_l = −l … +l,  m_s = ±½,  orbitals = n²,  electrons = 2n²
```

## Explanation

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 nNumber of orbitals n²1.522.255393.75144.520636

## 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.

## FAQ

**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.


## Sources

- [OpenStax – Chemistry 2e](https://openstax.org/details/books/chemistry-2e)
- [OpenStax – University Physics Volume 3](https://openstax.org/details/books/university-physics-volume-3)
- [NIST – Fundamental physical constants](https://physics.nist.gov/cuu/Constants/)