Evolution operator Please, what are the requirements, the Hamiltonian should satisfy before we can use the evolution operator: 
$$U(t) =\exp\left[-~\mathrm i \int_0^t \mathcal{H}(t^\prime)~\mathrm dt^\prime\right]$$ 
Can it be applied with the following? 
$$\mathcal {H}(t^\prime) =\epsilon (t^\prime) \sigma_z +\Delta(t^\prime) \sigma_x.$$
$\sigma_x $ and $\sigma_z$ are Pauli matrices ;  $\epsilon (t) $ and $\Delta(t) $ are real time-dependent parameters.  
 A: This formula can be used whenever you have a time dependent Hamiltonian provided the H's at different times commute.
For a particle with a magnetic moment moving in a magnetic field:


*

*If the magnetic field intensity is varying with time but is in a constant direction then the Hamiltonian will be proportional to a fixed Pauli matrix, so H's at different times will commute.

*If the magnetic field intensity is varying with time and the magnetic field direction is also varying with time then the Hamiltonian at different times will not commute.


If the Hamilonian at different times do not commute, you need to use the Dyson Series.
A: Yes if the exponential can be time ordered. ie
$$U(t) = \mathcal{T}\exp\left[-~\mathrm i\int_{0}^{t}\mathrm{d}t^\prime ~\mathcal{H}(t')\right]$$
or equivalently stated
$$\partial_t U(t) = - ~\mathrm i~ \mathcal{H}(t)U(t)\,.$$
As noted by Bruce Greetham, the time ordering has no effect, and can be dropped, only if $[\mathcal{H}(t),\mathcal{H}(t^\prime)]=0$, only in this (quite common) case is the form given by he OP correct.
