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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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To obtain a rough estimate of the mean time required for uranium-238 to alpha-decay, let us approximate the combined electrostatic and strong nuclear potential energies by rectangular potential barrier half as high as the actual 35 Mev maximum potential energy. Alpha particles (mass 4 u) of 4.3 Mev kinetic energy are incident. Let us also assume that the barrier extends from the radius of nucleus, 7.4 fm to the point where the electrostatic potential drops to 4.3 Mev (i.e., the classically forbidden region). Because , this point is 35/4.3 times the radius of the nucleus, the point at which U(r) is 35 Mev. (a) Use these crude approximations, the method suggested in Section 6.3, and the wide-barrier approximation to obtain a value for the time it takes to decay. (b) To gain some appreciation of the difficulties in a theoretical prediction, work the exercise “backward” Rather than assuming a value for U0, use the known value of the mean time to decay for uranium-238 and infer the corresponding value of U0, Retain all other assumptions. (c) Comment on the sensitivity of the decay time to the height of the potential 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?
As we learned in example 4.2, in a Gaussian function of the formis the standard deviation or uncertainty in position.The probability density for gaussian wave function would be proportional tosquared:. Comparing with the timedependentGaussian probability of equation (6-35), we see that the uncertainty in position of the time-evolving Gaussian wave function of a free particle is given by
. That is, it starts atand increases with time. Suppose the wave function of an electron is initially determined to be a Gaussian ofuncertainty. How long will it take for the uncertainty in the electron's position to reach, the length of a typical automobile?
Consider a potential barrier of height . (a) Find a width aroundfor which there will be no reflection of electrons incident upon the barrier. (b) What would be the reflection probability for electrons incident upon the same barrier? (Note: This corresponds to a difference in speed of less than
From equations (6-23) and (6-29) obtain the dispersion coefficient for matter waves (in vacuum), then show that probability density (6-35) follows from (6-28)
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