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Chapter 9: 42E (page 405)
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For a roomtall, by roughly what percent does the probability of an air molecule being found at the ceiling differ from that of an equal speed molecule being found at the floor? Ignore any variation in temperature from floor to ceiling.
Equation (9-27) gives the density of states for a system of oscillators but ignores spin. The result, simply one state per energy change ofbetween levels, is incorrect if particles are allowed different spin states at each level, but modification to include spin is easy. From Chapter 8, we know that a particle of spinis allowedspin orientations, so the number of states at each level is simply multiplied by this factor. Thus,
.
(a) Using this density of states, the definition, and
calculate the parameterin the Boltzmann distribution (9-31) and show that the distribution can thus be rewritten as
(b) Argue that if,the occupation number is much less than 1 for all E.
Figure 9.8 cannot do justice to values at the very highspeed end of the plot. This exercise investigates how small it really gets. However, although integrating the Maxwell speed distribution over the full range of speeds from 0 to infinity can be carried out (the so-called Gaussian integrals of Appendix K), over any restricted range, it is one of those integrals that. unfortunately. cannot be done in closed form. Using a computational aid of your choice. show that the fraction of molecules moving faster thanis; faster thanis; and faster thanis. where" from Exercise 41, is. (Exercise 48 uses these values in an interesting application.)
The Stirling approximation., is very handy when dealing with numbers larger than about . Consider the following ratio: the number of ways Nparticles can be evenly divided between two halves of a room to the number of ways they can be divided withon the right andon the left.
(a) Show, using the Stirling approximation, that the ratio is approximatelyfor large.
(b) Explain how this fits with the claim that average behaviours become more predictable in large systems.
The fact that a laser's resonant cavity so effectively sharpens the wavelength can lead to the output of several closely spaced laser wavelengths, called longitudinal modes. Here we see how. Suppose the spontaneous emission serving as the seed for stimulated emission is of wavelength , but somewhat fuzzy, with a line width of roughly either side of the central value. The resonant cavity is exactly long. (a) How many wavelengths fit the standing-wave condition'? (b) If only u single wavelength were desired, would change the length of the cavity help? Explain.
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