In the Heisenberg picture, the quantum state is assumed to remain constant at its initial value | ψ ψ ( 0 ) ⟩ ⟩ , whereas operators evolve with time according to the definition A H ( t ) := U † † ( t ) A S ( t ) U ( t ) . This readily implies ⟨ ⟨ A ⟩ ⟩ t = ⟨ ⟨ ψ ψ ( 0 ) | A H ( t ) | ψ ψ ( 0 ) ⟩ ⟩ , so the same expectation value can be obtained by working in either picture. The Schrödinger equation for the time-evolution operator is d d t U ( t ) = − − i ℏ ℏ H S ( t ) U ( t ) . It follows that d d t A H ( t ) = ( d d t U † † ( t ) ) A S ( t ) U ( t ) + U † † ( t ) A S ( t ) ( d d t U ( t ) ) + U † † ( t ) ( ∂ ∂ A S ∂ ∂ t ) U ( t ) = i ℏ ℏ U † † ( t ) H S ( t ) A S ( t ) U ( t ) − − i ℏ ℏ U † † ( t ) A S ( t ) H S ( t ) U ( t ) + U † † ( t ) ( ∂ ∂ A S ∂ ∂ t ) U ( t ) = i ℏ ℏ U † † ( t ) H S ( t ) U ( t ) U † † ( t ) A S ( t ) U ( t ) − − i ℏ ℏ U † † ( t ) A S ( t ) U ( t ) U † † ( t ) H S ( t ) U ( t ) + ( ∂ ∂ A S ∂ ∂ t ) H = i ℏ ℏ [ H H ( t ) , A H ( t ) ] + ( ∂ ∂ A S ∂ ∂ t ) H , where differentiation was carried out according to the product rule. This is Heisenberg's equation of motion. Note that the Hamiltonian that appears in the final line above is the Heisenberg Hamiltonian H H ( t ) , which may differ from the Schrödinger Hamiltonian H S ( t ) .