Quantum Number Calculator (check n, l, mₗ, mₛ and count orbitals and electrons)

Understanding the quantum number calculator — concept and principle

This calculator checks whether a set of four quantum numbers (n, l, mₗ, mₛ) describes an allowed electron state, names the subshell, and counts the orbitals and electrons in the shell and subshell. It also gives the orbital angular momentum and the hydrogen energy level. It is used in chemistry and physics courses on atomic structure.

How it is calculated

• n = 1, 2, 3 …; l = 0 … n − 1; mₗ = −l … +l; mₛ = +½ or −½.
• Subshell letters: l = 0, 1, 2, 3 → s, p, d, f.
• Orbitals in a subshell = 2l + 1; electrons = 4l + 2. Orbitals in a shell = n²; electrons = 2n².
• |L| = √(l(l + 1)) ħ; hydrogen energy Eₙ = −13.6 eV ÷ n².

The principle

Solving the Schrödinger equation for an electron in an atom gives states labelled by whole numbers that can only take certain combinations. n sets the size and energy shell; l sets the shape and orbital angular momentum; mₗ sets the orientation; and the spin mₛ adds two choices for each orbital. The Pauli exclusion principle says no two electrons in an atom share all four numbers, which is why each orbital holds two electrons and a shell holds 2n².

Worked example

Default inputs: n = 3, l = 2, mₗ = −1, mₛ = +½.

1. l = 2 is within 0 … 2 and mₗ = −1 is within −2 … +2, so: Yes — this electron state is allowed.
2. Subshell 3d: 5 orbitals holding 10 electrons.
3. Shell n = 3: 9 orbitals, 18 electrons.
4. |L| = √(2 × 3) = 2.4495 ħ; hydrogen E₃ = −13.6 ÷ 9 = −1.511 eV.

How to read the result

A "No" verdict shows which rule is broken, such as l = n or |mₗ| > l. The energy row applies exactly only to hydrogen; in many-electron atoms the energy also depends on l, which is why 4s fills before 3d.

Common mistakes

• Allowing l to equal n.
• Letting mₗ exceed l in size.
• Assuming energies depend only on n in many-electron atoms.

Frequently asked questions

How many electrons fit in the n = 2 shell?

Eight: 2n² = 8.

Why does l go only up to n − 1?

The solutions of the Schrödinger equation allow only those values.

What does mₛ = +½ mean?

The electron's spin component is "up" along the chosen axis.

Why is the hydrogen energy negative?

The electron is bound; zero energy corresponds to a free electron at rest.

Sources: OpenStax, University Physics Volume 3 (2016), section 8.1 "The Hydrogen Atom" and chapter 8 key equations