In terms of the [time evolution operators](time%20evolution%20operators.md) for an [isolated quantum system](Closed%20quantum%20systems.md) we may define its [Hamiltonian](Hamiltonian%20operator.md) as $\hat{H}(t) = i\hbar\frac{d\hat{U}(t,t_0)}{dt}\hat{U}^\dagger(t,t_1)=-i\hbar\hat{U}(t,t_0)\frac{d\hat{U}^{\dagger}(t,t_0)}{dt}$ The application of this operator gives the total energy of that quantum system at time $t.$ The Hamiltonian is always independent of $t_0,$ the point in time when the system is prepared. ^11f289 %%The above should be easy to derive from the Heisenberg picture%% # Hamiltonian operators under time dependent perturbation Consider a [isolated quantum system](Closed%20quantum%20systems.md) whose energy is described by a [time independent Hamiltonian operator](Hamiltonian%20operator%20(time%20independent).md) $\hat{H}_0.$ If it is then subject to any external forces that change with time, we want to write its [Hamiltonian](Hamiltonians.md) such that we define the energy contribution due to those external forces as a potential, $\hat{V}(t)$. Thus we express the _time dependent Hamiltonian_ as $\hat{H}(t)=\hat{H}_0+\hat{V}(t)$This Hamiltonian is then required in order to construct the [time evolution operators](time%20evolution%20operators%20with%20time%20dependent%20Hamiltonians.md) for this system. ^f68af8 # Constructing time dependent Hamiltonians ## Time dependent Hamiltonians of finite dimensional systems ## 2nd quantized Hamiltonian operators # Solving for energies of quantum systems with time dependence --- # Recommended reading The quantum Hamiltonian operator is defined in terms of the [time evolution operators](time%20evolution%20operators.md) in the following text: * [Peres, A., _Quantum Theory Concepts and Methods_, Kluwer Academic Publishers, 2002](Peres,%20A.,%20Quantum%20Theory%20Concepts%20and%20Methods,%20Kluwer%20Academic%20Publishers,%202002.md) pg. 239. %%Definition from Peres pg. 239%% ^0290d2 #QuantumMechanics/QuantumDynamics #QuantumMechanics/QuantumMeasurement/QuantumObservables