I'm trying to do a question which is as follows:
"Applying the uncertainty principle on the smallest scales, estimate the characteristic energy of the three relativistic quarks confined within the neutron. In the light of your answer comment on the mass of the neutron."
I'm able to write down that $\Delta x \Delta p \geq \hbar/2$ and perhaps that $\Delta E \Delta t \geq \sim \hbar$ but apart from that I'm not sure. Are there some other important relations that I'm supposed to remember?
Could I perhaps estimate $\Delta x$ as being on the scale of $h$, Planck's constant? Even then, how do I calculate the energy of the quarks if I don't know their mass?
This is a past exam question for a 3rd year undergraduate in a paper that is testing all of the physics studied during the degree (so in theory the question should be doable with almost just background-ish type knowledge of the subject).
Thank you
Answer
I left a hint yesterday and since no one has followed up on it, I'll provide the answer. Knowing that the radius of the neutron is about $10^{-15} m$(1 fm), one can write (using the uncertainty principle)$$\Delta x \Delta p >\frac {\hbar} {2}$$ Now using the fact that for a bound particle the uncertainty of momentum is approximately the RMS momentum (see the answer in the link in the first comment above), this means that a quark in the neutron has an RMS momentum $$p_{RMS}=\frac {\hbar}{2} fm^{-1}$$ There are three quarks in a neutron and if they are highly relativistic one can assume they are near massless. Massless particles have an energy of pc, so the quark contribution to the mass of the neutron should be $$M_q=\frac{3}{2} \hbar c fm^{-1}$$ Now every good theoretical nuclear physicist remembers that $\hbar c =197 MeV fm$, so the quark contribution to the neutron mass should be $$M_q = 295.5 MeV $$ in this approximation. The actual mass of the neutron is about 939.5 MeV so that leaves about 644 MeV for the gluon contribution to the neutron mass.
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