Quantum Number Definition

A quantum number is a value that is used when describing the energy levels available to atoms and molecules. An electron in an atom or ion has four quantum numbers to describe its state and yield solutions to the Schrödinger wave equation for the hydrogen atom.

There are four quantum numbers:

• n - principal quantum number: describes the energy level
• ℓ - azimuthal or angular momentum quantum number: describes the subshell
• m_ℓ or m - magnetic quantum number: describes the orbital of the subshell
• m_s or s - spin quantum number: describes the spin

Quantum Number Values

According to the Pauli exclusion principle, no two electrons in an atom can have the same set of quantum numbers. Each quantum number is represented by either a half-integer or integer value.

• The principal quantum number is an integer that is the number of the electron's shell. The value is 1 or higher (never 0 or negative).
• The angular momentum quantum number is an integer that is the value of the electron's orbital (for example, s=0, p=1). ℓ is greater than or equal to zero and less than or equal to n-1.
• The magnetic quantum number is the orientation of the orbital with integer values ranging from -ℓ to ℓ. So, for the p orbital, where ℓ=1, m could have values of -1, 0, 1.
• The spin quantum number is a half-integer value that is either -1/2 (called "spin down") or 1/2 (called "spin up").

Quantum Number Example

For the outer valence electrons of a carbon atom, the electrons are found in the 2p orbital. The four quantum numbers used to describe the electrons are n=2, ℓ=1, m=1, 0, or -1, and s=1/2 (the electrons have parallel spins).

Not Just for Electrons

While quantum numbers are commonly used to describe electrons, they may be used to describe the nucleons (protons and neutrons) of an atom or elementary particles.