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Chapter 9: Q7CQ (page 402)
When would a density of states be needed: in a sum over states? in a sum over energies? in an integral over energies? in an integral over states?
Thus, among the choices, density of states is most useful when we consider integral over energies.
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The Fermi velocityis defined by, whereis the Fermi energy. The Fermi energy for conduction electron in sodium is 3 I. (a) Calculate the Fermis velocity (b) What would be the wavelength of an electron with this velocity? (c) If each sodium atom contributes one conduction electron to the electron gas and sodium atom are spaced roughlyapart. If it is necessary, by the criteria of equation (9-43), to treat the conduction electron gas as quantum gas?
This problem investigates what fraction of the available charge must be transferred from one conductor to another to produce a typical contact potential. (a) As a rough approximation treat the conductors as 10 cm square plates apart-a parallel-plate capacitors so that , where . How much charge must be transferred from one plate to the other to produce a potential difference of ?(b) Approximately what fraction would this be of the total number of conduction electrons in a piece of copper. which has one conduction electron per atom?
The exact probabilities of equation (9-9) rest on the claim that the number of ways of addingdistinct non-negative integer to give a total of is . One way to prove it involves the following trick. It represents two ways that distinct integers can add to and, respectively. In this special case.
The X's represent the total of the integers, -each row has . The represent "dividers" between the distinct integers of which there will of course be each row has . The first row says that is (three before the divider between it and ), is (no between its left divider with and its right divider with ), ) is . through are , is , and and are . The second row says that is . is , is , and all other are . Further rows could account for all possible ways that the integers can add to . Argue that properly applied, the binomial coefficient (discussed in Appendix ) can be invoked to give the correct total number of ways for any and .
Exercise 52 gives the Boltzmann distribution for the special case of simple harmonic oscillators, expressed in terms of the constant . Exercise 53 gives the Bose-Einstein and Fermi-Dirac distributions in that case. Consider a temperature low enough that we might expect multiple particles to crowd into lower energy states:. How many oscillators would be expected in a state of the lowest energy,? Consider all three-classically distinguishable. boson, and fermion oscillators - and comment on the differences.
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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