In quantum mechanics, the principal quantum number (n) of an electron in an atom indicates which electron shell or energy level it is in. Its values are natural numbers (1, 2, 3, ...).

Hydrogen and Helium, at their lowest energies, have just one electron shell. Lithium through Neon (see periodic table) have two shells: two electrons in the first shell, and up to 8 in the second shell. Larger atoms have more shells.

The principal quantum number is one of four quantum numbers assigned to each electron in an atom to describe the quantum state of the electron. The other quantum numbers for bound electrons are the total angular momentum of the orbit ℓ, the angular momentum in the z direction ℓ_z, and the spin of the electron s.

As n increases, the electron is also at a higher energy and is, therefore, less tightly bound to the nucleus. For higher n, the electron is farther from the nucleus, on average. For each value of n, there are n accepted ℓ (azimuthal) values ranging from 0 to ${\displaystyle n-1}$ inclusively; hence, higher-n electron states are more numerous.^{[citation needed]} Accounting for two states of spin, each n-shell can accommodate up to 2n² electrons.

In a simplistic one-electron model described below, the total energy of an electron is a negative inverse quadratic function of the principal quantum number n, leading to degenerate energy levels for each n > 1. In more complex systems—those having forces other than the nucleus–electron Coulomb force—these levels split. For multielectron atoms this splitting results in "subshells" parametrized by ℓ. Description of energy levels based on n alone gradually becomes inadequate for atomic numbers starting from 5 (boron) and fails completely on potassium (Z = 19) and afterwards.

The principal quantum number was first created for use in the semiclassical Bohr model of the atom, distinguishing between different energy levels. With the development of modern quantum mechanics, the simple Bohr model was replaced with a more complex theory of atomic orbitals. However, the modern theory still requires the principal quantum number.

There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers n, ℓ, m, and s specify the complete and unique quantum state of a single electron in an atom, called its wave function or orbital. Two electrons belonging to the same atom cannot have the same values for all four quantum numbers, due to the Pauli exclusion principle. The Schrödinger wave equation reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The principal quantum number arose in the solution of the radial part of the wave equation as shown below.

The Schrödinger wave equation describes energy eigenstates with corresponding real numbers E_n and a definite total energy, the value of E_n. The bound state energies of the electron in the hydrogen atom are given by: $${\displaystyle E_{n}={\frac {E_{1}}{n^{2}}}={\frac {-13.6{\text{ eV}}}{n^{2}}},\quad n=1,2,3,\ldots }$$The parameter n can take only positive integer values. The concept of energy levels and notation were taken from the earlier Bohr model of the atom. Schrödinger's equation developed the idea from a flat two-dimensional Bohr atom to the three-dimensional wavefunction model.