A250 gev beam of protons is fired over a distance of 1 km. if the initial size of the wave packet is 1 mm, find its final size

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