Differential equation for time evolution operator
We drop the t0 index in the time evolution operator with the convention that t0 = 0 and write it as U(t). The Schrödinger equation is
where H is the Hamiltonian. Now using the time-evolution operator U to write ,
Since is a constant ket (the state ket at t = 0), and since the above equation is true for any constant ket in the Hilbert space, the time evolution operator must obey the equation
If the Hamiltonian is independent of time, the solution to the above equation is[note 1]
Since H is an operator, this exponential expression is to be evaluated via its Taylor series:
Therefore,
Note that is an arbitrary ket. However, if the initial ket is an eigenstate of the Hamiltonian, with eigenvalue E:
The eigenstates of the Hamiltonian are stationary states: they only pick up an overall phase factor as they evolve with time.
If the Hamiltonian is dependent on time, but the Hamiltonians at different times commute, then the time evolution operator can be written as
If the Hamiltonian is dependent on time, but the Hamiltonians at different times do not commute, then the time evolution operator can be written as
where T is time-ordering operator, which is sometimes known as the Dyson series, after Freeman Dyson.
The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. This is the Heisenberg picture.