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Chapter 6: Q15E (page 224)

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

Short Answer

Expert verified

The reflection probability is 0.007

Step by step solution

01

Definition of Reflection Probability

Reflection probability or reflection coefficient is defined as the ratio of amplitude of reflected wave to that of incident wave.

R=k2-k1k2+k12

Where, localid="1657549250276" k2=2m(E-v0)2 andk1=2mE2

02

Given/known parameters

E=5eV andV0=2eV

03

Solution

Substituting the values in the formula:

R=5-(-2)-55-(-2)+52

R=0.007

04

Explanation and Conclusion

The reflection probability is 0.007,i.e., there is0.7% chance of particle being reflected back.

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Most popular questions from this chapter

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’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 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≪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 4×32KBT. 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[-e24πε0h4m3kBT]

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’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?

Particles of energy Eare incident from the left, where U(x)=0, and at the origin encounter an abrupt drop in potential energy, whose depth is -3E.

  1. Classically, what would the particles do, and what would happen to their kinetic energy?
  2. Apply quantum mechanics, assuming an incident wave of the formψinc=eikx, where the normalization constant has been given a simple value of 1, determine completely the wave function everywhere, including numeric values for multiplicative constants.
  3. What is the probability that incident particles will be reflected?

A particle moving in a region of zero force encounters a precipice---a sudden drop in the potential energy to an arbitrarily large negative value. What is the probability that it will “go over the edge”?

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.)
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