eigenkets

An eigenket is a vector in a Hilbert space that satisfies A|ψ⟩ = λ|ψ⟩ for some linear operator A and scalar λ called an eigenvalue. The ket |ψ⟩ is an eigenstate of A associated with the eigenvalue λ.

In quantum mechanics, observables are represented by Hermitian operators. The eigenkets of an observable correspond to possible measurement outcomes, and the eigenvalues are the values that may be obtained when the observable is measured. If the eigenvalue is non-degenerate, the corresponding eigenket is unique up to a phase; if the eigenvalue is degenerate, the set of eigenkets forms a subspace, and one can choose an orthonormal basis within that subspace.

Spectral decomposition expresses the operator in terms of its eigenvalues and projectors. For a discrete spectrum, A = ∑_λ λ P_λ, where P_λ = ∑_i |λ,i⟩⟨λ,i| projects onto the eigen-subspace of λ. For continuous spectra, the decomposition uses an integral A = ∫ λ dP_λ. These projectors satisfy P_λ P_μ = δ_λμ P_λ and ∑_λ P_λ = I.

A general state |Ψ⟩ can be expanded in the eigenkets of A: |Ψ⟩ = ∑_λ c_λ |λ⟩, with probabilities P(λ) = |c_λ|^2 for measuring the eigenvalue λ (assuming normalized eigenkets). Upon measurement, the state collapses to the corresponding eigenket |λ⟩.

In linear algebra, eigenkets and eigenvectors are the same mathematical objects, with Dirac notation emphasizing quantum states and measurement statistics. Degeneracy adds structure through subspaces of eigenkets, enriching spectral decompositions and state representations.