A free particle of energy E approaches a square,...

A free particle of energy $E$ approaches a square, one-dimensional potential well of depth $V_{0}$ and width $2 a$. Show that the probability of being reflected by the well vanishes when $K a=n \pi / 2$, where $n$ is an integer and $K=\left(2 m\left(E+V_{0}\right) / \hbar^{2}\right)^{1 / 2}$. Explain this phenomenon in physical terms.

Key Concepts

Resonant Transmission

Resonant transmission is the phenomenon where, under specific conditions, a particle can transmit through a potential without any reflection. In the given problem, the condition Ka = n?/2 ensures that the phase acquired by the wave traversing the potential well leads to constructive interference in the transmitted wave and destructive interference in the reflected wave, effectively cancelling the reflection probability.

Wave Interference

Wave interference is central to understanding the vanishing reflection in this scenario. The particle's wave function accumulates phase as it propagates through the well, and when the phase difference between multiple internal reflections results in destructive interference for the reflected components, the net amplitude for reflection becomes zero. This effect is a direct result of the coherent nature of quantum waves and the precise phase relationships enforced by the boundary conditions.

Quantum Mechanical Scattering

Quantum mechanical scattering deals with the interaction between a particle and a potential, where the particle is described by a wave function that can split into reflected and transmitted components. When a particle encounters a potential well, its behavior (such as reflection and transmission probabilities) is determined by solving the Schrödinger equation and matching the wave function and its derivative at the boundaries of the potential region.

Potential Well and Energy States

A potential well is a region in space where the potential energy is lower than the surrounding area. In this context, the well alters the kinetic energy of the particle inside it (as seen by the modified wavenumber K). The energy of the particle relative to the well depth changes the propagation characteristics within the well compared to the free region outside, leading to different phase accumulations over the well's width.

Transcript

Please give Ace some feedback