Quantum Numbers

Quantum Numbers

Quantum numbers describe the state of electrons in an atom. Learn about the principal quantum number, orbital quantum number, magnetic quantum number, and spin quantum number in this interactive tutorial.

Introduction to Quantum Numbers

Scientists from Bohr to de Broglie to Heisenberg have attempted to describe the nature of electrons. Erwin Schrödinger and Pauld Dirac developed equations that for now best describe the various states that electrons can assume. These states are known as quantum states and are described by four quantum numbers.

Principal Quantum Number (n)

The principal quantum number describes the energy level (or shell) that an electron occupies. It is shown as a positive integer n={1,2,3,4,…}

Oribtal Quantum Number (L)

The orbital quantum number identifies the subshell — the shape of the space the electron occupies. It is shown as a positive integer L={0,1,2,3,…(n-1)}, which corresponds to the subshell letter names s=0, p=1, d=2, f=3, etc.

Magnetic Quantum Number (mL)

The magnetic quantum number indicates a specific orbital within the subshell (since many subshells consist of multiple orbitals). It is shown as an integer mL={_l,…,0,…,+L}.

The maximum value of the magnetic quantum number depends on the orbital quantum number — the s subshell has one orbital, the p subshell has three, the d subshell has five, and the f subshell has seven orbitals.

Spin Quantum Number (ms)

Electrons spin within their orbitals, producing magnetic fields. These spins and field orientations are described by the spin quantum number.

For every combination of principal, orbital, and magnetic quantum numbers, the spin quantum number (ms) can only be +1/2 or -1/2.

Pauli Exclusion Principle

According to the Pauli Exclusion Principle, no two electrons in an atom can have the same set of quantum numbers. Thus, the two electrons in a helium atom may have the same principal (n=1), orbital (L=0), and magnetic (mL=0) quantum numbers, but the spin quantum numbers must be different – one must be ms=+1/2 and the other ms=-1/2.