Question

A beam of particles of energy \(E\) and incident upon a potential step of \(U_{0}=\\{E\) is described by the wave function $$ \psi _{inc} (x) = 1e^{ill} $$ (a) Determine completely the reflected wave and the wave inside the step by enforcing the required continuity conditions to obtain their (possibly complex) amplitudes, (b) Verify by explicit calculation that the ratio of reflected probability density to the included probability density is 1.

Short answer

The reflected wave will be the same as the incident wave, thereby carrying an amplitude of 1. The wave inside the potential step will have an amplitude of \(e^{2ikx}\). Both the incident and reflected waves have a probability density of 1, confirming that the ratio of reflected to incident probability densities is indeed 1.

Step-by-step solution

Wave on the left of step

We start by considering the wave on the left of the step. The wave function for a free particle of energy \(E\) is \(\psi = A e^{ikx} + B e^{-ikx}\), where \(A\) and \(B\) are complex coefficients representing the amplitudes of the wave moving in the +x and -x directions respectively, and \(k = \sqrt{2mE}/\hbar\). The given incident wave function \(\psi_{inc}(x) = 1e^{ikx}\) suggests that we only have a wave moving in the +x direction, so we can equate this our wave function and set \(A = 1\) and \(B = 0\). Hence, \(\psi(x) = e^{ikx}\) for \(x < 0\).

Wave on the right of step

Next, we examine the wave on the right of the step. Again, the wave function for a free particle of energy \(E\) is \(\psi = C e^{ik'x} + D e^{-ik'x}\), where now \(k' = \sqrt{2m(E-U_0)}/\hbar\) because the particle now has a potential energy of \(U_0\). As \(U_0 = E\), the wave only has a wave moving in the -x direction, so \(C = 0\) and \(D\) is some complex number. Hence, \(\psi(x) = D e^{-ik'x}\) for \(x \geq 0\).

Identify the wave inside the step

The wave inside the step is \(\psi(x) = D e^{-ik'x}\) for \(x \geq 0\) where \(D\) is some complex number that we have yet to determine.

Enforcing the continuity conditions

At the point of the potential step, the wave function and its first derivative must be continuous. This means that the wave function just on the left and just on the right of the step must be equal, and their first derivatives must also be equal. Hence, equating the wave functions gives \(e^{ikx} = D e^{-ik'x}\) which simplifies to \(D = e^{2ikx}\). Likewise equating the first derivative gives \(ike^{ikx} = -ik'D e^{-ik'x}\), which simplifies to \(k' = k\). Therefore the wave function for \(x \geq 0\) can be written as \(\psi(x) = e^{2ikx}\), and the wave inside the step has a complex amplitude of \(e^{2ikx}\).

Calculating the reflected and incident probability densities

The probability density of a wave function \(\psi\) is given by \(|\psi|^2\). Hence, the reflected probability density is \(|\psi_{reflected}|^2 = |1|^2 = 1\). Similarly, the incident probability density is \(|\psi_{inc}|^2 = |e^{ikx}|^2 = 1\).

Verifying the ratio

Finally, we verify whether the ratio of reflected probability density to incident probability density is 1. As both densities are 1, their ratio is indeed 1.

Key concepts

Potential Step

A potential step in quantum mechanics is like an energy threshold that particles encounter. Imagine a hill where particles with less energy cannot climb over, while those with enough might pass. In our scenario of potential step, a beam of particles hits this energy boundary, where the potential energy is denoted as \( U_0 \).
For a potential step, the threshold energy \( U_0 \) is crucial since it dictates the particle's behavior at the boundary. If the particle's energy \( E \) is greater than \( U_0 \), the particle has the energy needed to cross. However, if \( E = U_0 \) — the case in our exercise — interesting quantum behavior occurs, which we'll explore through wave functions and continuity conditions.
In essence, potential steps allow us to study quantum behaviors like reflection and transmission of particles at known energy boundaries.

Wave Functions

In quantum mechanics, wave functions represent the state of a particle. They are fundamental because they contain all the information about a system. The wave function, typically denoted as \( \psi(x) \), indicates how a particle's amplitude behaves through space.
The incident wave function on the left of the potential step for our particle is \( \psi_{inc}(x) = 1e^{ikx} \). This expression tells us that:

• There's a wave moving to the right (positive x-direction) with a specific momentum.
• The wave number \( k \) relates to the particle's energy through the equation \( k = \sqrt{\frac{2mE}{\hbar}} \).

On the right side of the step, the wave’s dynamics depend on whether the particle has penetrated the barrier, described by a different wave function.
By analyzing these wave functions, we can predict how much of the wave gets reflected versus transmitted, leading us to understand the behavior of particles at boundaries.

Probability Density

Probability density is an important concept, measuring where a particle associated with a wave function is likely to be found. Mathematically, if a wave function is \( \psi(x) \), the probability density is given by \( |\psi(x)|^2 \).
For instance, if we consider our incident wave \( \psi_{inc}(x) = 1e^{ikx} \), the probability density for the incoming wave is \( |1e^{ikx}|^2 = 1 \). This indicates a uniform probability of finding the particle along its path before hitting the potential barrier.

• The reflected wave probability density is the same — \( |\psi_{reflected}|^2 = 1 \).
• This shows all particles encountering the potential get reflected back with equal probability.

Understanding and comparing probability densities help us determine whether particles predominantly pass through or are reflected by the potential step.

Continuity Conditions

Continuity conditions play a crucial role in linking the behavior of wave functions at boundaries. When a wave function encounters a boundary, like a potential step, both the wave function and its derivative must remain continuous.
Why is this important? It ensures particles' conservation laws are maintained — meaning no sudden "jumps" in particle probability.
Let's break it down:

• The wave function \( \psi(x) \) from the left must equal the wave function from the right at the boundary.
• Similarly, their derivatives must be equal, ensuring smooth transitions between different regions.

In the exercise, applying continuity leads us to the realization that \( D = e^{2ikx} \), refining our solution by ensuring both sides of the step seamlessly connect. Such conditions allow predictions of how much wave gets reflected or transmitted at interfaces.