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For ultrarelativistic particles such as photons or
Chapter , Problem 13P(choose chapter or problem)
Problem 13P
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.
Questions & Answers
QUESTION:
Problem 13P
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.
ANSWER:
Solution
Step 1 of 4
(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.