Energy eigenvalue

An energy eigenvalue $E$ is an eigenvalue of the Hamiltonian: $H|\psi ⟩=E|\psi ⟩$ for some nonzero state $|\psi ⟩$.

An energy eigenstate is a nonzero quantum state represented by an eigenvector of the Hamiltonian operator. Its time evolution changes only its overall phase.

The energy eigenspace for $E$ is $\ker (H-EI)$.

An excited state is an energy eigenstate whose energy exceeds the ground state energy.

The first excited state has the smallest energy strictly above the ground-state energy.

An eigenstate of an observable is a quantum state represented by an eigenvector of that observable.

A momentum eigenstate is an eigenstate of the momentum operator. In one dimension it is a plane wave $e^{ikx}$ with momentum eigenvalue $\hslash k$.