For ultrarelativistic particles such as photons or high-energy electrons, the relation between energy and momentum is not E = p2/2m but rather E = pc. (This formula is valid for massless particles, and also for massive particles in the limit E » mc2.)
(a) Find a formula for the allowed energies of an ultrarelativistic particle confined to a one-dimensional box of length L.
(b) Estimate the minimum energy of an electron confined inside a box of width 10-15 m. It was once thought that atomic nuclei might contain electrons; explain why this would be very unlikely.
(c) A nucleon (proton or neutron) can be thought of as a bound state of three quarks that are approximately massless, held together by a very strong force that effectively confines them inside a box of width 10-15 m. Estimate the minimum energy of three such particles (assuming all three of them to be in the lowest-energy state), and divide by c2 to obtain an estimate of the nucleon mass.