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Chapter 4: Q17E (page 134)
Determine the Compton wavelength of the electron, defined to be the wavelength it would have if its momentum were.
Compton Wavelength of the electron
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Question: An electron beam strikes a barrier with a single narrow slit, and the electron flux number of electrons per unit time per unit area detected at the very center of the resulting intensity pattern is . Next, two more identical slits are opened, equidistant on either side of the first and equally “illuminated” by the beam. What will be the flux at the very center now? Does your answer imply that more than three times as many electrons pass through three slits than through one? Why or why not?
A crack between two walls is wide. What is the angular width of the central diffraction maximum when
(a) an electron moving at passes through?
(b) A baseball of mass and speed passes through?
(c) In each case, an uncertainty in momentum is introduced by the “experiment” (i.e., passing through the slit). Specifically, what aspect of the momentum becomes uncertain, and how does this uncertainty compare with the initial momentum of each?
What is the wavelength of a neutron of kinetic energy ? Of kinetic energy? (The difference is important for the fission of uranium, as a neutron will be more easily absorbed when its wavelength is very large compared with the dimensions of the wavelength behaves more like a diffuse wave than like a localized particle)
Equation (4-21) expresses a function as a sum of plane waves, each with a coefficient . Equation (4-22) finds the coefficients from the given function . The equations aren't independent statements; in fact, one is the inverse of the other. Equation (4-22) giveswhen is known, and (4-21) does the reverse. Example 4.7calculates from a specific. Show that when this is inserted into (4-21) , the original is returned. Use the Euler formula and the symmetry properties of odd and even functions to simplify your work.
Calculate the ratio of (a) energy to momentum for a photon, (b) kinetic energy to momentum for a relativistic massive object of speed u, and (e) total energy to momentum for a relativistic massive object. (d) There is a qualitative difference between the ratio in part (a) and the other two. What is it? (e) What are the ratios of kinetic and total energy to momentum for an extremelyrelativistic massive object, for which What about the qualitative difference now?
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