eigenvalue

An eigenvalue is the special number an operator hands back when it acts on one of its own eigenstates without changing the state's direction. For most states an operator scrambles them into something new, but for the privileged eigenstates it merely multiplies them by a single number — and that number is the eigenvalue. The word comes from the German eigen, meaning 'own' or 'characteristic'.

Eigenvalues matter so much in quantum mechanics because they are precisely the values you can measure. When an observable is measured, the result is always one of the eigenvalues of its operator, never anything in between. If those eigenvalues form a discrete, separated set — as they do for the energy of a bound particle — then the quantity itself is quantized, coming only in fixed allowed amounts.

Because observables are represented by Hermitian operators, their eigenvalues are guaranteed to be real numbers, which is exactly what an honest measurement must yield. The full list of an operator's eigenvalues is called its spectrum, and reading off that spectrum is often the central goal of solving a quantum problem — for example, finding the allowed energy levels of an atom.