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Chapter 6: Q13E (page 224)
Show that is equivalent to , provided that .
Hence, the proof for the equation is obtained.
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Your friend has just finished classical physics and can鈥檛 wait to know what lies ahead. Keeping extraneous ideas and postulates to a minimum, Explain the process of Quantum-mechanical tunneling.
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?
Consider a particle of mass m inside the well as shown in the figure. If bound, its lowest energy state would of course be the ground state, but would it be bound? Assume that for a while, it at least occupies the ground state, which is much lower than, and the barriers qualify as wide. Show that a rough average time it would remain bound is given by: where.
The plot below shows the variation of 蠅 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蠅 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.
What fraction of a beam of electrons would get through a wide electrostatic barrier?
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