calculating quantum wavelengths

[nf405]

1. The problem statement, all variables and given/known data
Ok,as part of my dissertation I'm trying to explain about how quarks have a bigger quantum wavelength than protons so when you localise a quark inside a proton, the uncertainty in energy of the system becomes really big so there's a load of creation/anhilitation of other particles going on inside the proton.
This is how my supervisor explained it but I'm not sure I get it...



3. The attempt at a solution
This is what my supervisor said;
(I'm using D as 'uncertainty in', and h means h bar)

to work out quantum wavelength of a proton;
Start with Heisenberg

1)DpDx>h (for a start I think this should be h/2)

2)then use E=pc ------I don't understand why you can use this- it's not a realtivistic case. My supervisor said it works if you say that the proton is gaining energy by being hit with photons... but then

3) sub equation (2) into(1)

DE>hc/Dx

but the momentum in 1) is the momentum of the proton and in 2) its the momentum of the photon so how does that work?

4) then you sub in E=mc^2 to get; Dx>hc/mc^2

and the quantum wavelength is h/mc

I don't think this is right does anyone have a better argument?

[Gokul43201]

That's a poor argument, though the end result is correct (to within some factors of pi and 2). The correct argument involves the limitation that dE be less than mc^2, in order that you do not create two particles out of one.

[nf405]

I still don't quite understand- do you still start from heisenberg?

Ok is it something like;

DpDx>h/2

then I'm thinking maybe use kinetic energy/ momentum relation to get to energy?
(p=(2mE)^-1/2)

and when you say dE<mc^2 is that the mass of the proton? cause surely it should be the mass of any other possible particle...?