 Energy eigenvector

An energy eigenvector is a nonzero vector $|\psi ⟩$ in the operator domain of a Quantum Hamiltonian satisfying $H|\psi ⟩=E|\psi ⟩$. Normalizing it gives a pure energy eigenstate, for which an ideal energy measurement yields $E$ with probability one.

The eigenvector can be multiplied by any nonzero scalar without changing its eigenvalue. Several independent eigenvectors can also share $E$ when the energy eigenspace is degenerate. The zero vector belongs to every eigenspace algebraically but represents no physical state and is excluded by the word eigenvector. A free particle plane wave is instead a generalized eigenfunction on unbounded space, since it is not a normalizable vector in the physical Hilbert space.