eigenstate

An eigenstate of a linear operator is a state that, when the operator acts on it, yields a scalar multiple of itself. In mathematical terms, for an operator A on a vector space, an eigenvector or eigenstate |ψ> satisfies A|ψ> = λ|ψ>, where λ is the eigenvalue.

In quantum mechanics, observables are represented by Hermitian operators on a Hilbert space. An eigenstate of an observable corresponds to a state with a definite value of that observable. If you measure the observable corresponding to A in the state |ψ>, the possible outcomes are the eigenvalues λ, and the system collapses to an eigenstate (or to a state within the corresponding eigenspace) with probability given by the projection of |ψ> onto that eigenspace, p = ||P_λ|ψ>||^2.

If A has a discrete spectrum and a complete set of eigenstates, any state can be expanded as a linear combination of eigenstates: |ψ> = ∑_λ c_λ |λ>, with c_λ = ⟨λ|ψ⟩. For continuous spectra, the expansion uses an integral over eigenstates. For Hermitian A, eigenstates with distinct eigenvalues are orthogonal and can be normalized to form an orthonormal basis.

The time evolution of an eigenstate of the Hamiltonian H is simple: if H|E> = E|E>, then |E>(t) = e^{-iEt/ħ}|E>; such states are called stationary. In many systems, the full Hamiltonian has a complete set of eigenstates that provides the spectral decomposition of any state.

Examples include energy eigenstates of the hydrogen atom and spin eigenstates of a component of spin, such as Sz eigenstates |↑> and |↓> for a spin-1/2 particle.