Quantum Number Calculator

Enter the principal quantum number (n) to find all possible azimuthal (l), magnetic (mₗ), and spin (mₛ) quantum numbers for an electron. The Quantum Number Calculator shows every allowed value, the orbital subshell labels, the angular momentum magnitude, and the maximum electron count for that shell. Also try the Energy to Wavelength Calculator.

The shell number (1 = K, 2 = L, 3 = M, …). Allowed values: 1 to 7.

Results

Max Electrons in Shell

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Number of Subshells (l values)

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Total Orbitals in Shell

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Azimuthal Quantum Numbers (l)

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Magnetic Quantum Number Range (mₗ)

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Spin Quantum Numbers (mₛ)

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Results Table

Frequently Asked Questions

What is a quantum number?

A quantum number is a value that describes a specific property of an electron in an atom — such as its energy level, orbital shape, orbital orientation, or spin. Four quantum numbers together uniquely identify every electron in an atom, following the Pauli Exclusion Principle, which states no two electrons can share the same set of four quantum numbers. See also our Hydrogen Atom Energy Levels.

What does the principal quantum number (n) determine?

The principal quantum number n defines the electron's main energy level or shell and its average distance from the nucleus. It takes positive integer values (1, 2, 3, …). Higher n means higher energy and a larger orbital. Each shell can hold a maximum of 2n² electrons.

Which characteristic is given by the angular momentum (azimuthal) quantum number?

The azimuthal quantum number l describes the shape of the electron's orbital and its angular momentum. For a given n, l ranges from 0 to n−1. The values correspond to subshell types: l=0 (s), l=1 (p), l=2 (d), l=3 (f), and so on.

Which quantum number describes the orientation of an orbital?

The magnetic quantum number mₗ specifies the orientation of the orbital in space relative to an external magnetic field. For a given l, mₗ takes integer values from −l to +l, giving 2l+1 possible orientations (orbitals) per subshell. You might also find our use the Compton Wavelength Calculator useful.

How do I find the magnetic quantum number?

Once you know the azimuthal quantum number l, the magnetic quantum number mₗ can be any integer from −l to +l. For example, if l=2 (a d subshell), mₗ can be −2, −1, 0, +1, or +2, giving five d-orbitals.

What are the spin quantum numbers?

The spin quantum number mₛ describes the intrinsic angular momentum (spin) of an electron. It has only two possible values: +½ (spin-up, ↑) and −½ (spin-down, ↓). Because of the Pauli Exclusion Principle, each orbital can hold at most two electrons with opposite spins.

How many electrons can fit in a shell?

The maximum number of electrons in a shell with principal quantum number n is 2n². For example, the first shell (n=1) holds 2 electrons, the second (n=2) holds 8, the third (n=3) holds 18, and so on.

Can a quantum numbers calculator be used for educational purposes?

Absolutely. This calculator is ideal for students studying atomic structure in chemistry and physics. It automates the rules for deriving all allowed quantum numbers from a given shell, making it much easier to check homework, study for exams, or explore how electron configurations are built up.