Question

Particles of energy \(E\) are incident from the left. where \(U(x)=0\), and at the origin encounter an abrupt drop in potential energy, whuse depth is \(-3 E\). (a) Classically. what would the particles do, and what would happen to their kinetic energy? (b) Apply quantum mechanics. Assuming an incident wave of the form \(\psi_{\text {ine }}(x)=1\) \(12^{i 6}\), where the normalization constant has been given the simple value of 1, determine completely the wave function everywhere. including numeric value: for the multiplicative constants. (c) What is the probability that incident particles will be reflected?

Short answer

Classically, the particles would move forward and their kinetic energy would increase. In terms of quantum mechanics, the full wave function for x < 0 is \( e^{i6x}\) and for x > 0 is \(e^{i6x}\). The probability of the particles being reflected is 0.

Step-by-step solution

Classical Analysis

(a) Classically, the particles encountering a drop in potential energy equivalent to -3 times their energy would continue forward. Their kinetic energy would increase because the potential energy decreases and energy is conserved.

Quantum Analysis

(b) To determine the wave function, we start with the given incident wave function, \(\psi_{\text {inc}}(x)=1 e^{i6x}\). There are three regions to consider: x < 0, x = 0, and x > 0. Firstly, for x < 0, introduce a reflected wave: \(\psi_{\text {r}}(x) = Re^{-i6x}\). Then, the total wave function for x < 0 is \(\psi_{<}(x)= \psi_{\text {inc}}(x) + \psi_{\text {r}}(x)= 1e^{i6x} + Re^{-i6x}\). When x > 0, the wave function is \(\psi_{>}(x) = Te^{i6x}\), where \(T\) is the transmitted wave amplitude. Now we need to match the wave functions at x = 0 to find \(R\) and \(T\). To evaluate \(R\) and \(T\), we must impose continuity of \(\psi_(x)\) and its derivative at x = 0. That results in two equations: \(1 + R = T \) and \(6i(1 - R) = 6i T\). Solving these gives \(R = 0\) and \(T = 1\). So, the full wave function: For x < 0, \(\psi_{<}(x) = e^{i6x}\). For x > 0, \(\psi_{>}(x)= e^{i6x}\).

Reflection Probability

(c) The reflection probability is given by \(|R|^2\), which equates to 0 in this case. Thus, there is no chance that particles will be reflected.

Key concepts

Potential Energy in Quantum Mechanics

In the realm of quantum mechanics, potential energy is a core concept that plays a crucial role when examining particle behavior. It influences the shape and behavior of wave functions, which are mathematical descriptions of particles. Potential energy is often denoted by the symbol \(U(x)\), signifying its dependence on position, \(x\). Unlike in classical mechanics, where a decrease in potential energy directly translates to an increase in kinetic energy, quantum systems often display more nuanced behavior.

For instance, in the textbook exercise, an abrupt drop in potential energy to \(-3E\) at the origin might lead to the expectation that particles would have increased kinetic energy classically. Quantum mechanically, the situation prompts an analysis of wave function behavior and transmission-reflection probabilities. The potential serves as a boundary condition that influences how wave functions match on either side of a boundary, affecting the probability of transmission and reflection, which is a very subtle but essential difference from classical interpretations.

Wave Function Analysis

Wave function analysis in quantum mechanics is used to predict the behavior of particles. The wave function, represented as \(\psi(x)\), encapsulates all information about a particle’s state. The square modulus of the wave function, \(|\psi(x)|^2\), gives the probability density of finding the particle at position \(x\).

When faced with a problem of determining the wave function across different regions, as in the exercise provided, the continuity of the wave function and its derivative at the boundary is crucial. This continuity represents the physical requirement that particles do not abruptly disappear or appear as they move from one region to another. The specification of the wave function in each region—\(\psi_{<}(x)\) for \(x < 0\) and \(\psi_{>}(x)\) for \(x > 0\)—and their subsequent matching grants insight into the possible reflection and transmission of particles at the boundary. To fully integrate wave function analysis into learning, understanding these matching conditions - including both the wave function and its derivative - is imperative for predicting the motion of quantum particles.

Quantum Reflection Probability

In quantum mechanics, reflection probability is a measure of the likelihood that a particle encountering a potential barrier will be reflected back rather than transmitted through the barrier. This probability is denoted by \(|R|^2\), where \(R\) is the reflection amplitude determined by boundary conditions at the potential discontinuity or barrier.

In the context of the given exercise, the reflection probability calculation was based on the continuity equations at the point of potential change. The resulting values of \(R\) and \(T\) indicate, respectively, reflection and transmission amplitudes. In this specific scenario, the calculation indicated that \(R = 0\), which implies that there is no reflection. This result is surprising from a classical standpoint but aligns with quantum mechanics principles where the potential barrier is not a simple obstacle but rather a modifier of the wave function's behavior. Such counterintuitive results underscore the significance of solutions that are faithful to the peculiarities of quantum theory and highlight the importance of understanding reflection in quantum mechanics.