Eigenstateen

Eigenstateen, or eigenstates, are quantum states that remain proportional to themselves under the action of an observable operator. If A is an operator acting on a system’s state space, an eigenstate |ψ> satisfies A|ψ> = λ|ψ>, where λ is a scalar called the eigenvalue. For Hermitian (self-adjoint) operators, which correspond to physical observables, the eigenvalues λ are real, and eigenvectors associated with different eigenvalues are orthogonal.

When an operator A has a complete set of eigenstates, they form a basis for the Hilbert space. Any state |φ> can be written as a linear combination |φ> = ∑i c_i |λ_i>, or as an integral in the case of a continuous spectrum. The coefficients c_i = ⟨λ_i|φ⟩ give the amplitude for obtaining the eigenvalue λ_i upon measurement, with probability |c_i|^2.

The measurement postulate states that measuring A on a state |φ> yields an eigenvalue λ_i with probability |⟨λ_i|φ⟩|^2, and the system collapses to the corresponding eigenstate (normalized). If the initial state is already an eigenstate, the measurement yields the eigenvalue with certainty.

Time evolution: an energy eigenstate |E> of the Hamiltonian, satisfying H|E> = E|E>, evolves in time as |E,t> = e^{-iEt/ħ}|E>. Such states are stationary because the probabilities for measurement outcomes remain time-independent.

Degeneracy and generalized eigenstates: if several eigenvectors share the same eigenvalue, they span a degenerate subspace, and any normalized linear combination is also an eigenstate with that eigenvalue. In systems with a continuous spectrum, eigenvectors are not normalizable; spectral decomposition uses a suitable distribution of generalized eigenstates and projection-valued measures.