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Chapter 6: Q3CQ (page 223)
Why is the topic of normalization practically absent from Sections 6.1 and 6.2?
We don’t normalize with multiple particles as with a single particle.
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Show that is equivalent to , provided that .
The diagram below plots ω(k) versus wave number for a particular phenomenon. How do the phase and group velocities compare, and do the answer depend on the central value of k under consideration? Explain.
The equations for and T in the barrier essentially the same as light through a transparent film. It is possible to fabricate a thin film that reflects no light. Is it possible to fabricate one that transmits no light? Why? Why not?
Fusion in the Sun: Without tunnelling. our Sun would fail us. The source of its energy is nuclear fusion. and a crucial step is the fusion of a light-hydrogen nucleus, which is just a proton, and a heavy-hydrogen nucleus. which is of the same charge but twice the mass. When these nuclei get close enough. their short-range attraction via the strong force overcomes their Coulomb repulsion. This allows them to stick together, resulting in a reduced total mass/internal energy and a consequent release of kinetic energy. However, the Sun's temperature is simply too low to ensure that nuclei move fast enough to overcome their repulsion.
a) By equating the average thermal kinetic energy that the nuclei would have when distant,. and the Coulomb potential energy they would have when fm apart, roughly the separation at which they stick, show that a temperature of about K would be needed.
b) The Sun's core is only about . If nuclei can’t make it "over the top." they must tunnel. Consider the following model, illustrated in the figure: One nucleus is fixed at the origin, while the other approaches from far away with energy. As decreases, the Coulomb potential energy increases, until the separation is roughly the nuclear radius . Whereupon the potential energy is and then quickly drops down into a very deep "hole" as the strong-force attraction takes over. Given then , the point , where tunnelling must begin. will be very large compared with , so we approximate the barrier's width as simply b. Its height, , we approximate by the Coulomb potential evaluated at . Finally. for the energy which fixes , let us use . which is a reasonable limit, given the natural range of speeds in a thermodynamic system.Combining these approximations, show that the exponential factor in the wide-barrier tunnelling probability is
c)Using the proton mass for , evaluate this factor for a temperature of. Then evaluate it at. about that of an incandescent filament or hot flame. and rather high by Earth standards. Discuss the consequences.
For particles incident from the left on the potential energy shown below, what incident energies E would imply a possibility of later being found infinitely far to the right? Does your answer depend on whether the particles behave classically or quantum-mechanically?
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