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 con lined inside a box of width 10−15m. It was once thought that atomic nuclei might cont ain 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 Lobe in the lowest-energy state), and divide by c2 to obtain an estimate of the nucleon mass.

Step 1 of 4</p>

(a) Find a formula for the allowed energies of an ultrarelativistic particle confined to a one-dimensional box of length L .

The given formula for energy of an ultrarelativistic particle is,

Where E is energy which is quantized, P is momentum and c is speed of light.

The value of momentum is also quantized due to the property of wavelength quantization.That is,

Using the relation, where h is planck’s constant and is wavelength.

Step 2 of 4</p>

The wavelength in terms of width L is given by

Therefore, above equation becomes

………...1

Hence, the above formula gives the allowed energies of an ultrarelativistic particle confined to a one-dimensional box of length L