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Chapter 6: Q8CQ (page 223)

A ball is thrown straight up at 25ms-1. Someone asks “Ignoring air resistance. What is the probability of the ball tunneling to a height of1000m?” Explain why this is not an example of tunneling as discussed in this chapter, even if the ball were replaced with a small fundamental particle. (The fact that the potential energy varies with position is not the whole answer-passing through nonrectangular barriers is still tunnelirl8.)

Short Answer

Expert verified

This is not an example of tunneling.

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

In tunneling, we consider that the particle’s potential energy drops down to its original value after it has crossed the barrier and reached to the other side.

In the case given above, the potential energy of the ball increases by virtue of its height (PE=mgh). And when it reaches the highest point, i.e., the point where the kinetic energy is zero and potential energy is maximum, everything beyond this point is classically forbidden. But the ball cannot go beyond and hence this case cannot be considered under quantum tunneling.

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Most popular questions from this chapter

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).

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?”

For a general wave pulse neither E nor p (i.eneither ωnor k)areweIldefined. But they have approximate values E0andp0. Although it comprisemany plane waves, the general pulse has an overall phase velocity corresponding to these values.

vphase=ω0k0=E0/hp0/h=E0p0

If the pulse describes a large massive particle. The uncertaintiesarereasonably small, and the particle may be said to have energy E0and momentum p0.Usingtherelativisticallycorrect expressions for energy and momentum. Show that the overall phase velocity is greater thancand given by

vphase=c2uparticle

Note that the phase velocity is greatest for particles whose speed is least.

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

The plot below shows the variation of ω with k for electrons in a simple crystal. Where, if anywhere, does the group velocity exceed the phase velocity? (Sketching straight lines from the origin may help.) The trend indicated by a dashed curve is parabolic, but it is interrupted by a curious discontinuity, known as a band gap (see Chapter 10), where there are no allowed frequencies/energies. It turns out that the second derivative ofω with respect to k is inversely proportional to the effective mass of the electron. Argue that in this crystal, the effective mass is the same for most values of k, but that it is different for some values and in one region in a very strange way.
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