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Chapter 6: Q2CQ (page 223)

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?

Short Answer

Expert verified

The answer depends on the nature of particles: Quantum or Classical.

Step by step solution

01

Definition of tunneling

Classically, it is not allowed that a particle with energy less than the energy of a barrier would pass through it but quantum mechanically, it is possible. This phenomenon is called tunneling.

02

Explanation and conclusion

The solution to any energy greater than U1 would be sinusoidal. So, in the middle of the barrier to the right side, the transmission would be allowed.

But this would be possible only for a quantum particle. A classical particle would need energy greater than U2 to get past the barrier.

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Most popular questions from this chapter

Calculate the reflection probability 5evfor an electron encountering a step in which the potential drop by 2ev

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."

The equations for Rand T in the E>U0barrier 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,32KBT. and the Coulomb potential energy they would have when 2fm apart, roughly the separation at which they stick, show that a temperature of about 1019K would be needed.

b) The Sun's core is only about 10k. If nuclei can鈥檛 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 energyE. As rdecreases, the Coulomb potential energy increases, until the separation ris roughly the nuclear radius rnuc. Whereupon the potential energy is Umaxand then quickly drops down into a very deep "hole" as the strong-force attraction takes over. Given then E鈮�Umax, the point b, where tunnelling must begin. will be very large compared with rnuc, so we approximate the barrier's width Las simply b. Its height, U0, we approximate by the Coulomb potential evaluated at b2. Finally. for the energy Ewhich fixes b, let us use 4脳32KBT. 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

exp[-e24蟺蔚0h4m3kBT]

c)Using the proton mass for , evaluate this factor for a temperature of107K. Then evaluate it at3000K. about that of an incandescent filament or hot flame. and rather high by Earth standards. Discuss the consequences.

Example 6.3 gives the refractive index for high-frequency electromagnetic radiation passing through Earth鈥檚 ionosphere. The constant b, related to the so-called plasma frequency, varies with atmospheric conditions, but a typical value is8脳1015鈥�rad2/s2 . Given a GPS pulse of frequency1.5鈥�GHz traveling through 8kmof ionosphere, by how much, in meters, would the wave group and a particular wave crest be ahead of or behind (as the case may be) a pulse of light passing through the same distance of vacuum?
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