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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

Exercise 54 gives a rough lifetime for a particle trapped particle to escape an enclosure by tunneling.

(a) Consider an electron. Given thatW=100 nm, L=1 nm  and  U0=5 eV, first verify that theEGS<<U0assumption holds, then evaluate the lifetime.

(b) Repeat part (a), but for a0.1µgparticle, withW=1nm, L=1µm, and a barrier heightU0that equals the energy the particle would have if its speed were just1 mmperyear.

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?”

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

Given the situation of exercise 25, show that

(a) as Uo→∞, reflection probability approaches 1 and

(b) as L→0, the reflection probability approaches 0.

(c) Consider the limit in which the well becomes infinitely deep and infinitesimally narrow--- that is Uo→∞and data-custom-editor="chemistry" L→0but the product U0L is constant. (This delta well model approximates the effect of a narrow but strong attractive potential, such as that experienced by a free electron encountering a positive ion.) Show that reflection probability becomes:

R=[1+2h2EmUoL2]-1

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.
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