Estimate interaction time in Relativistic quantum scattering

Question

Consider a scattering collision between a relativistic electron and a Hydrogen atom, which is assumed to be in the ground state. Assume that the electron velocity is comparable to the speed of light $v \thicksim c$. For the discussion purposes, denote by $D$ the diameter of a ball, within which the probability to find the atomic-electron (in the ground state) is at least 0.99. 

What is needed is some reasonable higher bound on the time duration of the interaction, during collision between the electron and the Hydrogen atom.
Intuitively the interaction time can be estimated as 
$\Delta t\sim \mu\frac{D}{v}$
where $\mu$  is some reasonable coefficient in the range of $0.333 < \mu < 3$. 

Apparently just invoking classical mechanics won't do, since the objects are described by quantum wavefunctions and their motion by special relativity.
Yet some justified argument must exist to estimate and justify that, $\it{without}$ invoking the whole formalism of relativistic quantum scattering, which is extremely heavy. 

From the principles of wavefunctions-overlap integrals, the fundamental definition of velocity the estimate above seems true. 
Looking for better justified argumentation and estimation for the estimate above.

Comments

Apparently just invoking classical mechanics won't do, since the objects are described by quantum wavefunctions and their motion by special relativity.

Yes, CM will do since fast projectiles resemble geometrical optics approximation. Then the impact parameter will tell you how long the intercation will last (assuming the elastic scattering from the Hydrogen).

In case of inelastic scattering, there will be two sizes: the initial size and the final one of the Hydrogen atom ;-)