In the Schrödinger picture, the states are time dependent and the operators are time independent. States obey the Schrödinger equation

\[\begin{equation} \label{eq:schr_equation} i\hbar\frac{\partial}{\partial t}\ket{\Psi_S(t)} = \hat{H}(t)\ket{\Psi_S(t)}, \end{equation}\]

where the Hamiltonian \(\hat{H}(t)\) is potentially time dependent. Whether or not the Hamiltonian is time dependent, we can write the evolution of a state at time \(t\) to a time \(t’\) in terms of the time evolution operator \(\hat{U}(t’,t)\).

\[\begin{equation} \label{eq:time_evolution} \ket{\Psi_S(t')} = \hat{U}(t',t)\ket{\Psi_S(t)} \end{equation}\]

If we plug \eqref{eq:time_evolution} into \eqref{eq:schr_equation}, then we find that

\[\begin{equation} i\hbar\frac{\partial}{\partial t'}\ket{\Psi_S(t')} = \frac{\partial}{\partial t'}\hat{U}(t',t) \ket{\Psi_S(t)} \end{equation}\]

Then, we use the Schrödinger equation to rewrite the left hand side as

\[\begin{equation} \hat{H}(t')\ket{\Psi_S(t')} = \frac{\partial}{\partial t'}\hat{U}(t',t) \ket{\Psi_S(t)} \end{equation}\]

Now, we write the \(\ket{\Psi_S(t’)} = \hat{U}(t’,t)\ket{\Psi(t)}\), and get

\[\begin{equation} \hat{H}(t') \hat{U}(t',t)\ket{\Psi_S(t)} = i\hbar\frac{\partial}{\partial t'}\hat{U}(t',t) \ket{\Psi_S(t)} \end{equation}\]

Since this equation is to be true for any state \(\ket{\Psi}\), we conclude that

\[\begin{equation} i\hbar\frac{\partial}{\partial t'}\hat{U}(t',t) = \hat{H}(t') \hat{U}(t',t) \end{equation}\]

which is the differential equation for the time evolution operator. In general, to solve this equation is extremely difficult, and it must be expressed as a time-ordered exponential. We will come to this detail later when we develop perturbation theory. For now however, we can discuss some simple cases where we can exactly solve for \(\hat{U}\).

If \(\hat{H}\) is a time independent Hamiltonian, then we can solve explicitly for this time evolution operator \(\hat{U}\). For a time independent Hamiltonian, we get

\[\begin{equation} \hat{U}(t',t) = e^{-i\hat{H}(t'-t)/\hbar}. \end{equation}\]

This allows us to define the Schrödinger states in terms of states at \(t = 0\). We can write a state at a general time \(t\) by

\[\begin{equation} \ket{\psi_S(t)} = \hat{U}(t,0)\ket{\psi_S(0)} = e^{-i\hat{H}t/\hbar} \ket{\psi_S(0)} \end{equation}\]

We may also write \(\hat{U}(t,0) = \hat{U}(t)\).