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Chapter 6: Q7CQ (page 223)
Given the same particle energy and barrier height and width, which would tunnel more readily: a proton or an electron? Is this consistent with the usual rule of thumb governing whether classical or non-classical behavior should prevail?
The answer is electron.
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What fraction of a beam of electrons would get through a wide electrostatic barrier?
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?
For the E>U0 potential barrier, the reflection, and transmission probabilities are the ratios:
Where A, B, and F are multiplicative coefficients of the incident, reflected, and transmitted waves. From the four smoothness conditions, solve for B and F in terms of A, insert them in R and T ratios, and thus derive equations (6-12).
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.
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).
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