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Chapter 6: Q21E (page 224)
What fraction of a beam of electrons would get through a wide electrostatic barrier?
The required answer is
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The matter wave dispersion relation given in equation (6-23) is correct only at low speed and when mass/internal energy is ignored.
(a) Using the relativistically correct relationship among energy, momentum and mass, show that the correct dispersion relation is
(b) Show that in the limit of low speed (small p and k) and ignoring mass/internal energy, this expression aggress with that of equation (6-23).
The plot below shows the variation of 蝇 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 of蝇 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.
A beam of particles of energy incident upon a potential step of,is described by wave function:
The amplitude of the wave (related to the number of the incident per unit distance) is arbitrarily chosen as 1.
A beam of particles of energy E incident upon a potential step ofE is described by wave function:
The potential energy barrier in field emission is not rectangular, but resembles a ramp, as shown in Figure 6.16. Here we compare tunnelling probability calculated by the crudest approximation to that calculated by a better one. In method 1, calculate T by treating the barrier as an actual ramp in which U - E is initially, but falls off with a slop of M. Use the formula given in Exercise 37. In method 2, the cruder one, assume a barrier whose height exceeds E by a constant (the same as the average excess for the ramp) and whose width is the same as the distance the particle tunnels through the ramp. (a) Show that the ratio T1/T2 is . (b) Do the methods differ more when tunnelling probability is relatively high or relatively low?
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