Answer:
The final size is approximately equal to the initial size due to a very small relative increase of
in its size
Solution:
As per the question:
The energy of the proton beam, E = 250 GeV =
Distance covered by photon, d = 1 km = 1000 m
Mass of proton, 
The initial size of the wave packet, 
Now,
This is relativistic in nature
The rest mass energy associated with the proton is given by:


This energy of proton is 
Thus the speed of the proton, v
Now, the time taken to cover 1 km = 1000 m of the distance:
T = 
T = 
Now, in accordance to the dispersion factor;


Thus the increase in wave packet's width is relatively quite small.
Hence, we can say that:

where
= final width