According to Schrödinger's equation, the quantum state at time t is | ψ ψ ( t ) ⟩ ⟩ = U ( t ) | ψ ψ ( 0 ) ⟩ ⟩ , where U ( t ) = T e − − i ℏ ℏ ∫ ∫ 0 t d s H S ( s ) is the time-evolution operator induced by a Hamiltonian H S ( t ) that could depend on time, and | ψ ψ ( 0 ) ⟩ ⟩ is the initial state. T refers to time-ordering, ħ is the reduced Planck constant, and i is the imaginary unit. The expectation value of an observable A S ( t ) in the Schrödinger picture, which is a Hermitian linear operator that could also be time-dependent, in the state | ψ ψ ( t ) ⟩ ⟩ is given by ⟨ ⟨ A ⟩ ⟩ t = ⟨ ⟨ ψ ψ ( t ) | A S ( t ) | ψ ψ ( t ) ⟩ ⟩ .