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Chapter 7: Q49E (page 282)
Question: Verify the correctness of the normalization constant of the radial wave function given in Table 7.4 as
Answer
It has been proved that the normalization for the case 2pstateis correct.
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Spectral lines are fuzzy due to two effects: Doppler broadening and the uncertainty principle. The relative variation in wavelength due to the first effect (see Exercise 2.57) is given by
Where T is the temperature of the sample and m is the mass of the particles emitting the light. The variation due to the second effect (see Exercise 4.72) is given by
Where, is the typical transition time
(a) Suppose the hydrogen in a star has a temperature of . Compare the broadening of these two effects for the first line in the Balmer series (i.e., ). Assume a transition time of 10-8s. Which effect is more important?
(b) Under what condition(s) might the other effect predominate?
Classically, it was expected that an orbiting electron would emit radiation of the same frequency as its orbit frequency. We have often noted that classical behaviour is observed in the limit of large quantum numbers. Does it work in this case? (a) Show that the photon energy for the smallest possible energy jump at the “low-n-end” of the hydrogen energies is , while that for the smallest jump at the “high-n-end” is , where is hydrogen’s ground-state energy. (b) Use F=ma to show that the angular velocity of a classical point charge held in orbit about a fixed-point charge by the coulomb force is given by . (c) Given that , is this angular frequency equal to the minimum jump photon frequency at either end of hydrogen’s allowed energies?
Exercise 81 obtained formulas for hydrogen like atoms in which the nucleus is not assumed infinite, as in the chapter, but is of mass, while is the mass of the orbiting negative charge. (a) What percentage error is introduced in the hydrogen ground-state energy by assuming that the proton is of infinite mass? (b) Deuterium is a form of hydrogen in which a neutron joins the proton in the nucleus, making the nucleus twice as massive. Taking nuclear mass into account, by what percent do the ground-state energies of hydrogen and deuterium differ?
A hydrogen atom in an n = 2 state absorbs a photon,
A mathematical solution of the azimuthal equation (7-22) is , which applies when is negative, (a) Show that this simply cannot meet itself smoothly when it finishes a round trip about the z-axis. The simplest approach is to consider and . (b) If were , equation (7-22) would say simply that the second derivative of is . Argue than this too leads to physically unacceptable solution, except in the special case of being constant, which is covered by the , case of solutions (7-24).
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