Question

Consider an attractive square-well potential, \(U(x)=0\) for \(x<-\alpha, U(x)=-U_{0}\) for \(-\alpha \leq x \leq \alpha\) where \(U_{0}\) is a positive constant, and \(U(x)=0\) for \(x>\alpha .\) For \(E>0,\) the solution of the Schrödinger equation in the 3 regions will be the following: For \(x<-\alpha, \psi(x)=e^{i \kappa x}+R e^{-i \kappa x}\) where \(\kappa^{2}=2 m E / \hbar^{2}\) and \(R\) is the amplitude of a reflected wave. For \(-\alpha \leq x \leq \alpha, \psi(x)=A e^{i \kappa^{\prime} x}+B e^{-i \kappa^{\prime} x}\) and \(\left(\kappa^{\prime}\right)^{2}=2 m\left(E+U_{0}\right) / \hbar^{2}\). For \(x>\alpha, \psi(x)=T e^{i \kappa x}\) where \(T\) is the amplitude of the transmitted wave. Match \(\psi(x)\) and \(d \psi(x) / d x\) at \(-\alpha\) and \(\alpha\) and find an expression for \(R\). What is the condition for which \(R=0\) (that is, there is no reflected wave)?

Short answer

Answer: There is no reflection when the wave number inside the potential (\(\kappa\)) is equal to the wave number outside the potential (\(\kappa'\)), i.e., \(\kappa = \kappa'\).

Step-by-step solution

Write down the boundary conditions for \(\psi(x)\)

At \(x=-\alpha\) and \(x=\alpha\), the wave functions and their first-derivatives must be continuous. Write down these boundary conditions: 1. \(\psi(x=-\alpha)\): \(e^{i \kappa (-\alpha)}+R e^{-i \kappa (-\alpha)} = A e^{i \kappa' (-\alpha)}+B e^{-i \kappa' (-\alpha)}\) 2. \(\psi(x=\alpha)\): \(T e^{i \kappa \alpha} = A e^{i \kappa' \alpha}+B e^{-i \kappa' \alpha}\) 3. \(\frac{d\psi}{dx}(x=-\alpha)\): \(i \kappa (e^{i \kappa (-\alpha)}-R e^{-i \kappa (-\alpha)}) = i \kappa' (A e^{i \kappa' (-\alpha)}-B e^{-i \kappa' (-\alpha)})\) 4. \(\frac{d\psi}{dx}(x=\alpha)\): \(i \kappa T e^{i \kappa \alpha} = i \kappa' (A e^{i \kappa' \alpha}-B e^{-i \kappa' \alpha})\)

Solve the system of equations

We have four equations with four unknowns (R, T, A, and B). Solve this system of equations in order to find the amplitude of the reflected wave R. Divide equation 3 by equation 1 and equation 4 by equation 2, to eliminate A and B. 5. \(\frac{i \kappa (e^{i \kappa (-\alpha)}-R e^{-i \kappa (-\alpha)})}{e^{i \kappa (-\alpha)}+R e^{-i \kappa (-\alpha)}} = \frac{i \kappa' (A e^{i \kappa' (-\alpha)}-B e^{-i \kappa' (-\alpha)})}{A e^{i \kappa' (-\alpha)}+B e^{-i \kappa' (-\alpha)}}\) 6. \(\frac{i \kappa T e^{i \kappa \alpha}}{A e^{i \kappa' \alpha}+B e^{-i \kappa' \alpha}} = \frac{i \kappa' (A e^{i \kappa' \alpha}-B e^{-i \kappa' \alpha})}{T e^{i \kappa \alpha}}\) Now, we can find R by solving equations 5 and 6.

Find the expression for R

Divide equation 6 by equation 5 in order to eliminate all the terms without R. Then, solve for R: \(R = \frac{\kappa - \kappa'}{\kappa + \kappa'}\)

Find the condition for R = 0

In order to find the condition for which there is no reflected wave (\(R=0\)), we will set R equal to zero and solve for the relationship between \(\kappa\) and \(\kappa'\): \(\frac{\kappa - \kappa'}{\kappa + \kappa'}=0\) This condition is satisfied when \(\kappa=\kappa'\), which means the wave has no reflection, and all of the wave energy is transmitted through the potential.

Key concepts

Schrödinger Equation

The Schrödinger Equation is a fundamental component of quantum mechanics, providing a way to calculate the quantum state of a system over time. In essence, it is akin to Newton’s laws of motion in classical mechanics but applicable at the quantum level. This equation characterizes how wave functions, which describe the probabilities of a particle's position and momentum, evolve. For a time-independent scenario, the equation can be written as follows:
\[ -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + U(x) \psi = E \psi \]
Here, \(\psi\) denotes the wave function, \(U(x)\) represents the potential energy, and \(E\) is the total energy of the system. The equation indicates that the behavior and properties of particles are determined by the wave function \(\psi\). This wave function is crucial for understanding phenomena like tunneling and quantization in quantum systems.

Wave Functions

Wave functions are mathematical descriptions used in quantum mechanics to provide vital information about particles. These functions are solutions to the Schrödinger Equation, where the square of the modulus of a wave function, \(|\psi(x)|^2\), represents the probability density for finding a particle at a particular position \(x\) in space.
Since particle behavior in quantum mechanics is probabilistic, these wave functions help predict the likelihood of different outcomes.

• The wave function must be continuous and smooth.
• It contains information about both the amplitude and phase of the wave.
• The normalization condition ensures that the total probability is one.

In summary, wave functions provide comprehensive insights into the quantum state of particles. When working with wave functions, it is paramount to adhere to conditions such as continuity and normalization.

Square-Well Potential

In quantum mechanics, a square-well potential is a potential energy function characterized by constant values in different regions of space, like a box in one dimension.
This scenario is often described in three regions as follows:

• For \(x < -\alpha\), the potential \(U(x) = 0\).
• For \(-\alpha \leq x \leq \alpha\), the potential \(U(x) = -U_0\), where \(U_0\) is a positive constant.
• For \(x > \alpha\), the potential \(U(x) = 0\).

This model simulates how a particle behaves under such potential barriers and wells, mimicking idealized cases like particles trapped in nuclear or material lattices. The square-well potential helps elucidate fundamental quantum mechanics principles by highlighting how particles can exhibit reflection and transmission even in the presence of potential energy changes.

Boundary Conditions

Boundary conditions are critical in solving quantum mechanics problems, specifically when analyzing wave functions within a given potential, like the square-well potential scenario. For wave functions to be physically meaningful, they must satisfy specific boundary conditions at the borders of regions with different potentials.
Key conditions include:

• Continuity of wave function \(\psi(x)\).
• Continuity of its first derivative \(\frac{d\psi}{dx}\).

These conditions ensure that solutions – wave functions – are valid across different potential regions. By meticulously applying these boundary conditions at specified points (such as \(-\alpha\) and \(\alpha\) in the square-well potential model), we can determine critical attributes of wave functions, including amplitude ratios of reflected and transmitted waves, such as the condition for no reflection.