Question #43713

A beam of particles, each of mass $m$ and kinetic energy $E$, is incident from the left ($x < 0$) on a one-dimensional potential step defined by $V(x) = 0$ for $x < 0$ and $V(x) = 2E$ for $x \ge 0$. The incident wavefunction is $\psi_{\text{inc}}(x) = A e^{ikx}$ and the reflected wavefunction is $\psi_{\text{ref}}(x) = B e^{-ikx}$. The reflection amplitude ratio is $r = B/A = e^{i\delta}$, where $\delta \in (-\pi, \pi]$ is the phase shift upon reflection. In the classically forbidden region ($x \ge 0$), the probability density decays exponentially. Let $\Delta x$ be the penetration depth, defined as the distance over which the probability density decreases by a factor of $1/e^2$. Using the minimum uncertainty relation $\Delta x \Delta p = \hbar/2$, determine the phase shift $\delta$ and the minimum momentum uncertainty $\Delta p$ in the region $x \ge 0$.