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Chapter 6: Unbound States: Steps Tunneling, and Particle-Wave Propagation

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Modern Physics
Pages: 195 - 230
Modern Physics

Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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Illustration

47 Questions for Chapter 6: Unbound States: Steps Tunneling, and Particle-Wave Propagation

  1. Particles of energy Eare incident from the left, where U(x)=0, and at the origin encounter an abrupt drop in potential energy, whose depth is -3E.

    Found on Page 224

  2. In the E>Uopotential barrier, there should be no reflection when the incident wave is at one of the transmission resonances. Prove this by assuming that a beam of particles is incident at the first transmission resonance, E=Uo+(π2h2/2mL2), and combining continuity equations to show thatB=0. (Note: k’ is particularly simple in this special case, which should streamline your work.)

    Found on Page 224

  3. As we learn in physical optics, thin-film interference can cause some wavelengths of light to be strongly reflected while others not reflected at all. Neglecting absorption all light has to go one way or the other, so wavelengths not reflected are strongly transmitted. (a) For a film, of thickness t surrounded by air, what wavelengths λ (while they are within the film) will be strongly transmitted? (b) What wavelengths (while they are “over” the barrier) of matter waves satisfies condition (6-14)? (c) Comment on the relationship between (a) and (b).

    Found on Page 225

  4. Given the situation of exercise 25, show that

    Found on Page 225

  5. For the E>U0 potential barrier, the reflection, and transmission probabilities are the ratios:

    Found on Page 225

  6. Reflection and Transmission probabilities can be obtained from equations (6-12). The first step is substituting -iαfork'. (a) Why? (b) Make the substitutions and then use definitions of k and α to obtain equation (6-16).

    Found on Page 225

  7. Question: An electron bound in an atom can be modeled as residing in a finite well. Despite the walls. When many regularly spaced atoms are relatively close together as they are in a solid-all electrons occupy alltheatoms. Make a sketch of a plausible multi-atom potential energy and electron wave function.

    Found on Page 226

  8. The diagram below plots ω(k) versus wave number for a particular phenomenon. How do the phase and group velocities compare, and do the answer depend on the central value of k under consideration? Explain.

    Found on Page 223

  9. The plot below shows the variation of ω with k for electrons in a simple crystal. Where, if anywhere, does the group velocity exceed the phase velocity? (Sketching straight lines from the origin may help.) The trend indicated by a dashed curve is parabolic, but it is interrupted by a curious discontinuity, known as a band gap (see Chapter 10), where there are no allowed frequencies/energies. It turns out that the second derivative ofω with respect to k is inversely proportional to the effective mass of the electron. Argue that in this crystal, the effective mass is the same for most values of k, but that it is different for some values and in one region in a very strange way.

    Found on Page 223

  10. Show that ψ(x)=A'eikx+B'e-ikxis equivalent to ψ(x)=Asinkx+Bcoskx, provided that A'=12(B-iA)B'=12(B+iA).

    Found on Page 224

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