Question

Suppose the tunneling probability is${\mathbf{10}}^{\mathbf{-}\mathbf{12}}$for a wide barrier when E is $\frac{\mathbf{1}}{\mathbf{100}}{\mathit{U}}_{\mathbf{0}}$

(a) About how much smaller would it be if ’E’ were instead $\frac{\mathbf{1}}{\mathbf{1000}}{\mathit{U}}_{\mathbf{0}}$?

(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

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 :

$T=16\frac{E}{U_0}\left(1-\frac{E}{U_0}\right)e^{-\frac{2L}{\nparallel }\left(\sqrt{2m\left(U_{0}E\right)}\right)}$

Given Parameters

$T_i={10}^{-12}\mathrm{for}E=\frac{1}{100}{U}_0$

Step  3: Solution

We are asked to find out the Tunneling Probability(T) when $E=\frac{1}{1000}{U}_0$

Initially,

$$\begin{gathered} {10}^{12}=16.\frac{1}{100}.\left(1-\frac{1}{100}\right).e^{-\frac{2L}{\nparallel }\sqrt{\frac{2m.99}{100}}{U}_0} \\ \iff \frac{2L\sqrt{2m{U}_0}}{\nparallel }=25.918 \end{gathered}$$

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

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

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.