Kets and bras Ket: ∣ψ⟩|\psi\rangle∣ψ⟩ Bra: ⟨ψ∣=(∣ψ⟩)†\langle\psi| = (|\psi\rangle)^\dagger⟨ψ∣=(∣ψ⟩)† Inner product ⟨ϕ∣ψ⟩\langle\phi|\psi\rangle⟨ϕ∣ψ⟩ Operator action ∣ψ′⟩=U∣ψ⟩|\psi'\rangle = U|\psi\rangle∣ψ′⟩=U∣ψ⟩ Expectation value (common form) ⟨A⟩=⟨ψ∣A∣ψ⟩\langle A \rangle = \langle\psi|A|\psi\rangle⟨A⟩=⟨ψ∣A∣ψ⟩
