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Chapter 6: Q1CQ (page 223)
Could the situation depicted in the following diagram represent a particle in a bound state? Explain.
The answer is no.
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As we learn in physical optics, thin-film interference can cause some wavelengths of light to be strongly reflected while others not reflected at all. Neglecting absorption all light has to go one way or the other, so wavelengths not reflected are strongly transmitted. (a) For a film, of thickness t surrounded by air, what wavelengths λ (while they are within the film) will be strongly transmitted? (b) What wavelengths (while they are “over” the barrier) of matter waves satisfies condition (6-14)? (c) Comment on the relationship between (a) and (b).
A beam of particles of energy E incident upon a potential step ofE is described by wave function:
Show that if you attempt to detect a particle while tunneling, your experiment must render its kinetic energy so uncertain that it might well be "over the top."
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.
Verify that the reflection and transmission probabilities given in equation (6-7) add to 1.
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