Chapter 6: Q31E (page 225)

Suppose the tunneling probability is $10^{- 12}$ for a wide barrier when E is $\frac{1}{100} U_{0}$
(a) About how much smaller would it be if ’E’ were instead $\frac{1}{1000} U_{0}$ ?
(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

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 :

$T = 16 \frac{E}{U_{0}} (1 - \frac{E}{U_{0}}) e^{- \frac{2 L}{∦} (\sqrt{2 m (U_{0} E)})}$

Given Parameters

$T_{i} = 10^{- 12} for E = \frac{1}{100} U_{0}$

Step 3: Solution

We are asked to find out the Tunneling Probability(T) when $E = \frac{1}{1000} U_{0}$

Initially,

$10^{12} = 16 . \frac{1}{100} . (1 - \frac{1}{100}) . e^{- \frac{2 L}{∦} \sqrt{\frac{2 m . 99}{100}} U_{0}} \Leftrightarrow \frac{2 L \sqrt{2 m U_{0}}}{∦} = 25.918$

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

$T = 16 X (\frac{1}{1000}) X (\frac{999}{1000}) X e^{- 25.918 X \sqrt{\frac{999}{1000}}}$

Finally, comparing both:

$\frac{T_{i}}{T} = 100$

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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Short Answer

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Definition

Given Parameters

Step 3: Solution

Explanation and Conclusion:

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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

Step by step solution

Definition

Given Parameters

Step 3: Solution

Explanation and Conclusion:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Radiation

Circular Motion and Gravitation

Electric Charge Field and Potential

Conservation of Energy and Momentum

Magnetism

Study anywhere. Anytime. Across all devices.