/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q31E Suppose the tunneling probabilit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Q31E (page 225)

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

(a) About how much smaller would it be if 鈥橢鈥� 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?

Short Answer

Expert verified

Probability of transmission/tunneling reduces to a factor of 100, in the new case

Write answer for both subaprts

Step by step solution

01

Definition

Transmission probability can be defined as the probability with which the particle (whose Total energy is less than the threshold potential energy of the barrier) can tunnel/transmit through the barrier.

Its formula is given by :

T=16EU01-EU0e-2L鈭�2mU0E

02

Given Parameters

Ti=10-12forE=1100U0

03

Step  3: Solution

We are asked to find out the Tunneling Probability(T) when E=11000U0

Initially,

1012=16.1100.1-1100.e-2L鈭�2m.99100U0鈬�2L2mU0鈭�=25.918

Now, plugging in this constants value to get our new T, we do:

T=16X11000X9991000Xe-25.918X9991000

Finally, comparing both:

TiT=100

04

Explanation and Conclusion:

From the calculations, we can see that when the energy of the particle reduces, so does its probability of getting through the barrier, i.e., T is sensitive on E.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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?

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

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.

Why is the topic of normalization practically absent from Sections 6.1 and 6.2?

The plot below shows the variation of 蠅 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 of蠅 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.
See all solutions

Recommended explanations on Physics Textbooks

Turning Points in Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Radiation

Read Explanation

Rotational Dynamics

Read Explanation

Quantum Physics

Read Explanation

Electromagnetism

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.