Eigenstates

An eigenstate of a linear operator A on a vector space is a nonzero vector v that satisfies Av = λv for some scalar λ, called the eigenvalue. In finite-dimensional algebra, eigenstates are simply eigenvectors; in functional analysis and quantum mechanics, the concept extends to vectors in a Hilbert space and operators acting on them.

In quantum mechanics, eigenstates refer to the eigenvectors of observables, represented by Hermitian operators. If A|ψ> = a|ψ>, then the system prepared in |ψ> has a definite value a for the corresponding measurement. Upon measurement, outcomes are described by the spectral decomposition, and probabilities are determined by the projection of the state onto the eigenbasis, with probability |⟨ψ_i|Φ⟩|^2 for the eigenstate |ψ_i> with eigenvalue a_i. If the eigenvalue a is degenerate, the eigenspace contains multiple orthonormal eigenstates, and any orthonormal basis of that subspace is a valid eigenbasis.

Eigenstates of a Hermitian operator corresponding to distinct eigenvalues are orthogonal. If the operator is diagonalizable, its eigenstates form a complete basis; any state can be expressed as a superposition of eigenstates, and the operator admits a spectral decomposition A = ∑_i a_i P_i, where P_i are projectors onto the eigen-subspaces. Observables that commute share a common eigenbasis, allowing simultaneous definite values for those observables.

Stationary states in quantum mechanics are energy eigenstates of the Hamiltonian, evolving in time by a phase factor e^{-iEt/ħ} while preserving measurement statistics. Examples include position eigenstates |x> with X|x> = x|x>, momentum eigenstates |p> with P|p> = p|p>, and spin-1/2 eigenstates of σ_z. Normalization and orthogonality are standard: ⟨ψ_i|ψ_j⟩ = δ_ij, and completeness ∑_i |ψ_i⟩⟨ψ_i| = I.