A proton of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.5E₀.The probability that the proton will be transmitted is

Question

Quizplus · Accepted Answer

This problem can be solved using the semiclassical approximation for the transmission coefficient:

$T \approx e^{-2\gamma}$,

where $\gamma = \frac{2\pi}{\hbar}\int_a^b \sqrt{2m(E-V(x))}dx$ is the tunneling exponent, with $m$ being the mass of the proton, $E$ its energy, and $V(x)$ the potential energy.

In this case, the potential barrier has height $U = 0.5E$, so the barrier width $L$ can be found by solving $U = \frac{1}{2}mv^2 = \frac{1}{2}m\left(\frac{\hbar k}{m}\right)^2$ for $k$, and then $L = \frac{\pi}{k}$. This gives:

$k = \sqrt{\frac{2mU}{\hbar^2}} = \sqrt{\frac{mE}{2\hbar^2}}$,

and

$L = \frac{\pi}{k} = \frac{\pi\hbar}{\sqrt{2mE}}$.

Using these values, we can calculate the tunneling exponent:

$\gamma = \frac{2\pi}{\hbar}\int_0^L \sqrt{2m(E-U)}dx = \frac{2\pi}{\hbar}\int_0^L \sqrt{2mU}dx = \frac{2\pi L}{\hbar}\sqrt{2mU} = \frac{2\pi\pi\hbar}{\sqrt{2mE}}\sqrt{2mU}$

Plugging in the given values, we get:

$\gamma = \frac{2\pi\pi\hbar}{\sqrt{2m(ES)}}\sqrt{2m(0.5ES)} = \frac{2\pi^2}{\sqrt{2}}$

Finally, plugging this value into the transmission coefficient formula gives:

$T \approx e^{-2\gamma} = e^{-4\pi^2/\sqrt{2}} \approx 0.971$.

Therefore, the answer is D: 97.1%.

A Proton of Energy E0 Traveling in a Region in Which

A Proton of Energy E₀ Traveling in a Region in Which