energy eigenstate

An energy eigenstate is a quantum state that has one definite energy and nothing else. If you measure the energy of a system prepared in such a state, you always get the same value with no spread at all. Mathematically it is a wavefunction that the Hamiltonian merely scales: Ĥψ = Eψ, where the number E is its energy eigenvalue.

Energy eigenstates are precisely the stationary states, the special solutions of the time-independent Schrödinger equation. Each one is a particular spatial shape — a standing wave with its own number of bumps and nodes — and each carries its own energy. They are the natural 'pure tones' of a quantum system, just as a vibrating string has its fundamental and its overtones.

Their power lies in completeness. The energy eigenstates of a given Hamiltonian form a basis, meaning any state whatsoever can be written as a weighted sum of them. Because each one evolves in time so simply, expressing a complicated state in this language turns the hard problem of time evolution into easy bookkeeping of rotating phases.