eigenvalue equation

The eigenvalue equation is the simple but powerful statement that an operator acting on a state gives back the same state times a number. Written compactly it reads Â|ψ⟩ = a|ψ⟩, where  is the operator, |ψ⟩ is the eigenstate, and a is the eigenvalue. Solving this equation means finding all the special states that survive the operator unscathed and the numbers that go with them.

This single relation ties together the three central ideas of the subject. The states that satisfy it are the eigenstates; the numbers that appear are the eigenvalues; and for an observable those eigenvalues are precisely the values a measurement can return. So when you write down an operator and ask the eigenvalue equation, you are really asking nature what outcomes it permits.

An enormous amount of quantum mechanics is, in practice, the work of solving eigenvalue equations. The time-independent Schrödinger equation is exactly the eigenvalue equation for the energy operator, the Hamiltonian, and solving it yields the allowed energy levels of an atom or a well. The same template — set up the operator, solve its eigenvalue equation, read off the spectrum — recurs throughout the theory.