Question

An electron has a $0.0100$ probability (a $1.00\%$ chance) of tunneling through a potential barrier. If the width of the barrier is doubled, will the tunneling probability be$0.0050,0.0025or0.0001$? Explain.

Short answer

The width of the barrier is doubled , will the tunneling probability be$0.0050,0.0025or0.0001$is$P_{tunnel,2}=0.0001$.

Step-by-step solution

Given Information

We have to given that an electron has a $0.0100$, the width of the barrier is doubled, will the tunneling probability be $0.0050,0.0025or0.0001$.

Simplify

$P_{tunnel,2}=$The probability that the electron is the tunnel through the barrier is given as the equation:

$P_{tunnel}=e^{\raisebox{1ex}{$-2\omega $}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}$

and we are given that the initial probability is $\left(0.01\right)$, so let the probability after the width of the barrier is doubled be $P_{tunnel,2}$since:

${\omega }_1$is the initial width after doubling. Notice that $e^{ab}={\left(e^a\right)}^b$. Then:

role="math" localid="1650132194389" $$\begin{gathered} P_{tunnel,2}=exp\left(-\frac{4{\omega }_1}{\eta }\right) \\ =exp{\left(-\frac{2{\omega }_1}{\eta }\right)}^2 \\ ={\left(P_{tunnel,1}\right)}^2 \\ {P}_{tunnel,2}=0.0001 \end{gathered}$$