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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 ’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?

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∦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∦2m.99100U0⇔2L2mU0∦=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.

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Most popular questions from this chapter

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.

A beam of particles of energy E incident upon a potential step ofU0=(5/4)E is described by wave function:ψinc(x)=eikx

  1. Determine the reflected wave and wave inside the step by enforcing the required continuity conditions to obtain their (possibly complex) amplitudes.
  2. Verify the explicit calculation the ratio of reflected probability density to the incident probability density is 1.

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.)

Show that ψ(x)=A'eikx+B'e-ikxis equivalent to ψ(x)=Asinkx+Bcoskx, provided that A'=12(B-iA)B'=12(B+iA).

Exercise 39 gives a condition for resonant tunneling through two barriers separated by a space width of2s, expressed I terms of factorβgiven in exercise 30. Show that in the limit in which barrier widthL→∞, this condition becomes exactly energy quantization condition (5.22) for finite well. Thus, resonant tunneling occurs at the quantized energies of intervening well.
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