Quantum Number
Quantum numbers are discrete integer or half-integer values that characterise the quantum state of an electron (or other quantum particle) in an atom, arising naturally as solutions to the Schrödinger equation. The four quantum numbers for electrons — principal (n), azimuthal (l), magnetic (mₗ), and spin (mₛ) — together uniquely specify the quantum state of each electron in accordance with the Pauli Exclusion Principle. They determine the energy, shape, spatial orientation, and spin of atomic orbitals, providing the foundation for the periodic table and chemical bonding.
The Four Quantum Numbers for Electrons in an Atom
| Quantum Number | Symbol | Allowed Values | Physical Meaning |
|---|---|---|---|
| Principal | n | 1, 2, 3, … (positive integers) | Energy level / shell size |
| Azimuthal (angular momentum) | l | 0 to n−1 | Orbital shape (s, p, d, f) |
| Magnetic | mₗ | −l to +l (integers) | Orbital orientation in space |
| Spin magnetic | mₛ | +½ or −½ | Intrinsic angular momentum (spin up/down) |
| Example: 2p electron | — | n=2, l=1, mₗ=−1,0,+1, mₛ=±½ | One of six 2p states |
Interactive Tools
Khan Academy – Quantum Numbers
Comprehensive lesson on the four quantum numbers and their allowed values
Open ToolPtable Interactive Periodic Table
Visualise electron configurations and quantum numbers for all elements
Open ToolWolframAlpha Quantum Numbers
Look up quantum number assignments for hydrogen and other atoms
Open Tool
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Related Terms
Atomic Orbital
An atomic orbital is a mathematical function describing the wave-like behaviour and probable location of an electron in an atom, representing a region of space where there is a high probability (typically 90–95%) of finding the electron. Orbitals are characterised by three quantum numbers (n, l, mₗ) and have distinct shapes: s-orbitals are spherical, p-orbitals are dumbbell-shaped, and d- and f-orbitals have more complex geometries. Atomic orbitals form the basis for understanding electron configurations, chemical bonding, and molecular orbital theory.
Electron Spin
Electron spin is an intrinsic quantum mechanical property of electrons (and other fermions) that represents a form of angular momentum with no classical analogue — the electron does not physically rotate, but behaves as if it does, with a fixed magnitude of angular momentum. Electrons have a spin quantum number s = 1/2, giving two possible spin states: spin-up (mₛ = +1/2) and spin-down (mₛ = −1/2). Electron spin is responsible for the Pauli Exclusion Principle, magnetic properties of atoms (paramagnetism and diamagnetism), and is the basis for technologies such as MRI and spintronics.
Wave Function
The wave function (denoted Ψ) is a mathematical function in quantum mechanics that completely describes the quantum state of a particle or system. Its squared modulus |Ψ|² gives the probability density for finding the particle at a given position and time, as interpreted by Max Born in 1926. The wave function must be continuous, single-valued, and square-integrable (normalised so that the total probability integrates to one), and it evolves deterministically according to the Schrödinger equation.
The term "quantum number" comes from the quantisation of atomic energy levels discovered through spectroscopy in the early 20th century. Principal quantum number n was introduced by Niels Bohr (1913); azimuthal and magnetic numbers were formalised by Arnold Sommerfeld (1916); spin quantum number was proposed by Goudsmit and Uhlenbeck (1925).