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Concept
Principal quantum number
Summary
In quantum mechanics, the principal quantum number (symbolized n) is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable.
Apart from the principal quantum number, the other quantum numbers for bound electrons are the azimuthal quantum number l, the magnetic quantum number ml, and the spin quantum number 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 n − 1 inclusively, hence higher-n electron states are more numerous. 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.
Hydrogen-like atom
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.
Official source
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