Chapter III - Waves and Particles: Matter Waves

3.2) A beam of low energy electrons is directed normally at a metal surface, and strong reflection is detected only at an angle of 35 degrees. What is l/D, the ratio of the electron wavelength l to the interatomic spacing D at the surface?

3.11) A beam of electrons, kinetic energy = 54 eV, is directed normally at a nickel surface, and strong reflection is detected only at an angle of 50 degrees. Determine the spacing of nickel atoms on the surface.

3.13) In the hydrogen atom, the electron's orbit, not necessarily circular, extends to a distance of about an Angstrom from the proton. If it is to move about as a compact classicla particle in the region where it is confined, the electron's wavelength had better always be much smaller than an Angstrom. Is it? How large might be the electron's wavelength - how small might be its speed? If orbiting as a particle, its speed at 1 Angstrom could be no faster than that for circular orbit at that radius. (Why?) Find the corresponding wavelength and compare it to 1 Angstrom. Can the atom be treated classically?

3.26) To how small a region must an electron be confined for borderline relativistic speeds - say, 0.05 c - to become reasonably likely? On the basis of this, would you expect relativistic effects to be prominent for hydrogen's electron, of orbit radius near 10^-10 m? For a lead atom "inner-shell" electron, of orbit radius 10^-12 m?

3.32) A particle is connected to a spring and undergoes one-dimensional motion. (a) Write an expression for the total (kinetic plus potential energy) of the particle in terms of its position x, its mass m, its momentum p, and the force constant k of the spring. (b) Now treat the particle as a wave. Assume that the product of the uncertainties in position and momentum is governed by an uncertainty relation DpDx = h/4p. Also assume that since x is on average zero, the uncertainty Dx is roughly equal to a typical value of |x|. Similarly assume that Dp = |p|. Eliminate p in favor of x in the energy expression. (c) Find the minimum possible energy for the wave.

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