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Chapter 5: Bound States: Simple Cases

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Modern Physics
Pages: 141 - 194
Modern Physics

Modern Physics

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

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Illustration

91 Questions for Chapter 5: Bound States: Simple Cases

  1. When is the temporal part of the wave function 0 ? Why is this important?

    Found on Page 186

  2. Simple models are very useful. Consider the twin finite wells shown in the figure, at First with a tiny separation. Then with increasingly distant separations, In all case, the four lowest allowed wave functions are planned on axes proportional to their energies. We see that they pass through the classically forbidden region between the wells, and we also see a trend. When the wells are very close, the four functions and energies are what we might expect of a single finite well, but as they move apart, pairs of functions converge to intermediate energies.

    Found on Page 186

  3. In the harmonic oscillators eave functions of figure there is variation in wavelength from the middle of the extremes of the classically allowed region, most noticeable in the higher-n functions. Why does it vary as it does?

    Found on Page 186

  4. Summarize the similarities are differences between the three simple bound cases considered in this chapter.

    Found on Page 186

  5. Consider a particle bound in a infinite well, where the potential inside is not constant but a linearly varying function. Suppose the particle is in a fairly high energy state, so that its wave function stretches across the entire well; that is isn’t caught in the “low spot”. Decide how ,if at all, its wavelength should vary. Then sketch a plausible wave function.

    Found on Page 186

  6. The quantized energy levels in the infinite well get further apart as n increases, but in the harmonic oscillator they are equally spaced.

    Found on Page 186

  7. In several bound systems, the quantum-mechanically allowed energies depend on a single quantum number we found in section 5.5 that the energy levels in an infinite well are given by, En=a1n2wheren=1,2,3.....andis a constant. (Actually, we known whata1is but it would only distract us here.) section 5.7 showed that for a harmonic oscillator, they areEn=a2(n−12), wheren=1,2,3.....(using ann−12with n strictly positive is equivalent towith n non negative.) finally, for a hydrogen atom, a bound system that we study in chapter 7,En=−a3n2, wheren=1,2,3.....consider particles making downwards transition between the quantized energy levels, each transition producing a photon, for each of these three systems, is there a minimum photon wavelength? A maximum ? it might be helpful to make sketches of the relative heights of the energy levels in

    Found on Page 187

  8. Quantum-mechanical stationary states are of the general form Ψ(x,t)=ψ(x)e-iωt. For the basic plane wave (Chapter 4), this is Ψ(x,t)=Aeikxe-iωt=Aei(kx-ωt), and for a particle in a box it is Asinkxe-iωt. Although both are sinusoidal, we claim that the plane wave alone is the prototype function whose momentum is pure-a well-defined value in one direction. Reinforcing the claim is the fact that the plane wave alone lacks features that we expect to see only when, effectively, waves are moving in both directions. What features are these, and, considering the probability densities, are they indeed present for a particle in a box and absent for a plane wave?

    Found on Page 187

  9. Under what circumstance does the integral ∫x0∞xbdxdiverge? Use this to argue that a physically acceptable wave function must fall to 0 faster than|x|−1/2 does as xgets large.

    Found on Page 187

  10. Quantization is an important characteristic of systems in which a particle is bound in a small region. Why "small," and why "bound"?

    Found on Page 185

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