Validity of the sudden/diabatic approximation The Schrodinger equation is given by
$$i\hbar\ \frac{\partial}{\partial t}\ \mathcal{U}(t,t_{0})=H\ \mathcal{U}(t,t_{0}),$$
where $\mathcal{U}(t,t_{0})$ is the time evolution operator for evolution of some physical state $|\psi\rangle$ from $t_0$ to $t$.

Rewriting time $t$ as $t=s\ T$, where $s$ is a dimensionless parameter and $T$ is a time scale, the Schrodinger equation becomes as
$$i\ \frac{\partial}{\partial s}\ \mathcal{U}(t,t_{0})=\frac{H}{\hbar/T}\ \mathcal{U}(t,t_{0})=\frac{H}{\hbar\ \Omega}\ \mathcal{U}(t,t_{0}),$$
where $\Omega \equiv 1/T$.
In the sudden/diabatic approximation, $T \rightarrow 0$, which means that $\hbar\ \Omega \gg H$.



*

*Are we allowed to redefine $H$ by adding or subtracting an arbitrary constant?

*How does this introduce some overall phase factor in the state vectors?

*Why does this imply that $\mathcal{U}(t,t_{0})\rightarrow 1$ as $t\rightarrow 0$?

*How does this prove the validity of the sudden approximation?

 A: Note: This is not a complete answer. However the essence of sudden approximation is as follows.
The sudden approximation is valid if the Hamiltonian varies drastically in a time interval that is very small compared to that required for the system to have a transition in between the corresponding eigen states.  
The above time evolution equation of the time evolution operator has a solution that can be approximated as  
$$\mathcal{U}(t,t_o)=1-\frac{i}{h}\int_{t_0}^{t}H(t')dt'$$  
When you parametrize it in terms of $s$, 
$$\mathcal{U}(s)=1-\frac{i}{h}T\int_{0}^{s}H(s')ds'$$  
In the limit $T\longrightarrow 0$, we have $\mathcal{U}(s)=1$.  Hence for $T\longrightarrow 0$ 
$$\vert \alpha,t_o;t\rangle=\lim_{T\to 0} \mathcal{U}(t,t_o)\vert\alpha,t_o\rangle=1\vert\alpha,t_o\rangle=\vert\alpha,t_o\rangle$$  
i.e., the system remains in the previous eigen ket. The system do not have enough time to adjust to the change as the Hamiltonian suffers a very rapid change. So, in the very small interval of time, $H(t_0)$ changes to $H(t)$. But the system remains in the eigen state of $H(t_0)$, which, in general, need not be the eigen state of the evolved Hamiltonian. Now, how to define the "rapidness" (or, as you asked, when this approximation is valid)?  
If $E_a$ and $E_b$ represent the energy eigen values of the Hamiltonian at the instants $t_0$ and $t$ respectively, then the change in the energy levels, $E_{ab}=\hbar\omega_{ab}$. If $T<<2\pi/\omega_{ab}$, then this approximation is valid.  
