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Chapter 4: Q22E (page 135)
Roughly speaking for what range of wavelengths would we need to treat an electron relativistically, and what would be the corresponding range of accelerating potentials? Explain your assumptions.
Rane of Wavelength and Range of Accelerating potentials:
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In the hydrogen atom, the electron鈥檚 orbit, not necessarily circular, extends to a distance of a about an angstrom from the proton. If it is to move about as a compact classical particle in the region where it is confined, the electron鈥檚 wavelength had better always be much smaller than an angstrom. Here we investigate how large might be the electron鈥檚 wavelength. If orbiting as a particle, its speed at could be no faster than that for circular orbit at that radius. Why? Find the corresponding wavelength and compare it to . Can the atom be treated classically?
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?
Classically and nonrelativistically, we say that the energyof a massive free particle is just its kinetic energy. (a) With this assumption, show that the classical particle velocityis. (b) Show that this velocity and that of the matter wave differ by a factor of 2. (c) In reality, a massive object also has internal energy, no matter how slowly it moves, and its total energyis, where.Show thatisand thatisIs there anything wrong with it? (The issue is discussed further in Chapter 6.)
Question: Atoms in a crystal form atomic planes at many different angles with respect to the surface. The accompanying figure shows the behaviors of representative incident and scattered waves in the Davisson-Germer experiment. A beam of electrons accelerated through 54 V is directed normally at a nickel surface, and strong reflection is detected only at an angle of 500.Using the Bragg law, show that this implies a spacing D of nickel atoms on the surface in agreement with the known value of 0.22 nm.
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 鈥渋lluminated鈥� 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?
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