Question

Consider an electron with mass m and energy E moving along the x-axis towards a finite step potential of height $U_0$, as shown in the figure. In region $1 (x < 0)$, the momentum of the electron is $p_1 = \sqrt{2mE}$. The reflection coefficient at the barrier is given by $R = (\frac{p_1-p_2}{p_1+p_2})^2$, where $p_2$ is the momentum in region 2. If, in the limit $E>> U_0$, R, then the integer n is ________.

Answer 1

Correct Answer: 16

To solve for the integer \( n \) in the given problem, we analyze the reflection coefficient \( R \) as the energy \( E \) becomes much greater than the potential \( U_0 \).

In region 2 (\( x \geq 0 \)), the momentum \( p_2 \) of the electron is given by:
\( p_2 = \sqrt{2m(E-U_0)} \)

The reflection coefficient \( R \) is expressed as:
\[ R = \left(\frac{p_1 - p_2}{p_1 + p_2}\right)^2 \]

Substituting \( p_1 = \sqrt{2mE} \) and \( p_2 = \sqrt{2m(E-U_0)} \), we have:
\[ R = \left(\frac{\sqrt{2mE} - \sqrt{2m(E-U_0)}}{\sqrt{2mE} + \sqrt{2m(E-U_0)}}\right)^2 \]

Simplifying the terms:
\[ R = \left(\frac{\sqrt{E} - \sqrt{E-U_0}}{\sqrt{E} + \sqrt{E-U_0}}\right)^2 \]

For \( E \gg U_0 \), the approximation is:
\( \sqrt{E-U_0} \approx \sqrt{E}\left(1 - \frac{U_0}{2E}\right) \)

Substitute back:
\[ R \approx \left(\frac{\sqrt{E} - \sqrt{E}\left(1 - \frac{U_0}{2E}\right)}{\sqrt{E} + \sqrt{E}\left(1 - \frac{U_0}{2E}\right)}\right)^2 = \left(\frac{U_0}{4E}\right)^2 \]

As \( E \to \infty \), \( R \to 0 \), confirming that the integer \( n \) is determined by how rapidly \( R \) approaches 0, characterized by the factor \( \left(\frac{U_0}{4E}\right)^2 \).

Comparing the exponents, \( n = 16 \).