Time evolution in an oscillating magnetic field for spin-1/2 particles

Question

This might be a rookie mistake. 
For a magnetic field oriented in the z-direction of form, $B = B_0 cos(omega t) hat{k}$. 
The Hamiltonian in this case will be $H = omega_0 cos(omega t) hat{S_z}$. 
For an initial state of,  $|{psi(0)} rangle  = |+z rangle $.
By solving the Schrodinger's equation, I obtain $$|psi(t) rangle = e^{-frac{i omega_0}{2} sin(omega t)} |+zrangle $$. 
But when I tried to find the time evolution of the state by using the relation  between 
the time evolution  operator and the Hamiltonian, of form $$hat{U} = e^{- frac{i hat{H}t}{hbar}}$$
I got the state to become into, $$|psi(t)rangle = e^{- frac{i omega_0 cos(omega t) t}{2}} |+zrangle $$
The second solution does not seem to agree with the Schrodinger's equation, is this because the Hamiltonian is explicitly dependent on time?

InertialObserver · Answer

Recall that the Schrodinger equation
$$
i frac{partial}{partial t} | psi rangle = H |psi rangle
$$
yields the time evolution
$$
|psi(t)rangle = e^{-i H t} | psi(0) rangle
$$
only when the Hamiltonian is time independent.
There are two other situations:
(1) If the Hamiltonian commutes with itself at all times, then the solution for the time evolution operator is given by
$$
|psi(t)rangle = e^{-i int_0^t H(t') dt'} | psi(0) rangle
$$
(2) If the Hamiltonian does not commute with itself at different times then the formal time evolution is a Dyson series
begin{align*}
|psi(t)rangle = left(mathbb{1} + sum_{n=1}^inftyint_0^t dt_1 int_0^{t_1} dt_2cdots int_0^{t_{n-1}} dt_n H(t_1) H(t_2)cdots H(t_n)right)|psi(0)rangle
end{align*}
For more on to handle these situation see pg. 72 of Sakurai.

Time evolution in an oscillating magnetic field for spin-1/2 particles

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