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Chapter 6: Q6CQ (page 223)

How should you answer someone who asks, “In tunneling through a simple barrier, which way are particles moving, in the three regions--before, inside, and after the barrier?”

Short Answer

Expert verified

Particles aren’t moving until detected.

Step by step solution

01

Definition of quantum tunneling

The quantum phenomenon, in which a particle can penetrate a barrier and pass through it even though it is forbidden classically, is known as quantum tunneling.

02

Explanation and conclusion

If we talk about before the barrier, the particles can be detected to go towards the barrier as an incident wave and away from it as a reflected wave. After the barrier, the particles can be detected to move away from the barrier as a transmitted wave.

Inside the barrier, the particles tunnel like a wave and thus it can be said that they do not possess real momentum inside the barrier. So, their direction is not specific.

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Most popular questions from this chapter

Suppose the tunneling probability is10-12for a wide barrier when E is 1100U0

(a) About how much smaller would it be if ’E’ were instead 11000U0?

(b) If this case does not support the general rule that transmission probability is a sensitive function of E, what makes it exceptional?

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?

Example 6.3 gives the refractive index for high-frequency electromagnetic radiation passing through Earth’s ionosphere. The constant b, related to the so-called plasma frequency, varies with atmospheric conditions, but a typical value is8×1015 rad2/s2 . Given a GPS pulse of frequency1.5 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?

A method for finding tunneling probability for a barrier that is "wide" but whose height varies in an arbitrary way is the so-called WKB approximation.

T=exp[−2ℏ∫122m(U(x)−E)dx]

Here U(x) is the height of the arbitrary potential energy barrier.Whicha particle first penetrates at x=0 and finally exits at x=L. Although not entirely rigorous, show that this can be obtained by treating the barrier as a series of rectangular slices, each of width dx (though each is still a "wide" barrier), and by assuming that the probability of tunneling through the total is the product of the probabilities for each slice.

Reflection and Transmission probabilities can be obtained from equations (6-12). The first step is substituting -iαfork'. (a) Why? (b) Make the substitutions and then use definitions of k and α to obtain equation (6-16).
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