Question

For a finite square well, you have seen solutions for particle energies greater than and less than the well depth. Show that these solutions are equal outside the potential well if the particle energy is equal to the well depth. Explain your answer and the possible difficulty with it.

Short answer

Show that the solutions for particle energies greater than and less than the well depth are equal outside the potential well when the particle energy is equal to the well depth. To show this, we analyzed the time-independent Schrödinger equation for a finite square well potential and compared the wavefunctions and their continuity conditions. We found that the wavefunction solutions inside the well have the form: \[\psi_{inside}(x) = A\sin(k_1x) + B\cos(k_1x)\] And outside the well: \[\psi_{outside}(x) = Ce^{ik_1x} + De^{-ik_1x}\] When the particle energy is equal to the well depth, their wave vector coefficients are equal, \(k_1=k_2\), and the wavefunction must be decaying in the outside regions to ensure normalizability. The continuity conditions of the wavefunction and its first derivative at the boundaries ensure that the outside solutions are equal for particle energies equal to the well depth. However, it may cause difficulty in solving for the energy eigenvalues and their corresponding wavefunctions when energy is equal to the well depth, and numerical methods or approximation techniques might be necessary in practice.

Step-by-step solution

1. Setup the Problem

Consider a particle in a finite square well with potential \(V(x)\) given by: \[ V(x) = \begin{cases} -V_0 & -a \le x \le a \\ 0 & \text{otherwise} \end{cases} \] Where \(V_0\) is the well depth and \(2a\) is the width of the well. Note that the particle energy E is equal to the well depth, i.e., \(E = V_0\).

2. Schrödinger Equation inside and outside the well

The time-independent Schrödinger equation for this problem is given by: \[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\] Inside the well (\(-a\le x\le a\)), the potential is \(-V_0\), and the equation becomes: \[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} - V_0\psi(x) = V_0\psi(x)\] Outside the well (\(x<-a\) or \(x>a\)), the potential is \(0\), and the equation becomes: \[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} = V_0\psi(x)\]

3. Analyzing the solutions

The general form of the solutions inside and outside the well are given by: Inside the well: \[\psi_{inside}(x) = Ae^{ik_1x} + Be^{-ik_1x}\] Where \(k_1 = \frac{\sqrt{2mV_0}}{\hbar}\). We can write this as: \[\psi_{inside}(x) = A\sin(k_1x) + B\cos(k_1x)\] Outside the well: \[\psi_{outside}(x) = Ce^{ik_2x} + De^{-ik_2x}\] Where \(k_2 = \frac{\sqrt{2mE}}{\hbar}\). Since the energy is equal to the well depth, we can write \(k_2=k_1\). Also, note that the wavefunction must be decaying in the outside regions (i.e., \(C=0\) for \(x<-a\) and \(D=0\) for \(x>a\) to ensure normalizability). Hence the solutions outside the well simply become: \[\psi_{outside}(x) = Ce^{ik_1x} + De^{-ik_1x}\]

4. Continuity conditions and conclusion

At the boundaries \(x = \pm a\), the wavefunction and its first derivative must be continuous, which includes the outside and inside solutions. Thus, we have the following conditions: - At \(x = -a\): \[\psi_{inside}(-a) = \psi_{outside}(-a)\] \[\frac{d\psi_{inside}}{dx}\Big|_{x=-a} = \frac{d\psi_{outside}}{dx}\Big|_{x=-a}\] - At \(x = a\): \[\psi_{inside}(a) = \psi_{outside}(a)\] \[\frac{d\psi_{inside}}{dx}\Big|_{x=a} = \frac{d\psi_{outside}}{dx}\Big|_{x=a}\] Comparing the continuity conditions, we can see that the outside solutions are equal for particle energies equal to the well depth. However, this equality may cause difficulty in solving for the energy eigenvalues and their corresponding wavefunctions. In practice, one needs to use numerical methods or approximation techniques to find the allowed energies and wavefunctions for a finite square well when the energy is equal to the well depth.

Key concepts

Schrödinger Equation

The Schrödinger equation is a fundamental concept in quantum mechanics, governing the behavior of quantum systems. It is a differential equation that provides a way to calculate the wavefunction of a system, which contains all the information about the state of a particle.

For a non-relativistic particle, the time-independent Schrödinger equation can be written as:
\[\begin{equation}-\frac{\text{\textnormal{\(\backslash \)hbar}}^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\end{equation}\]Where \text{\textnormal{\(\backslash \)hbar}} is the reduced Planck's constant, \textnormal{\(m\)} is the particle's mass, \textnormal{\(V(x)\)} is the potential energy at position \textnormal{\(x\)}, \textnormal{\(\psi(x)\)} is the wavefunction, and \textnormal{\(E\)} is the energy of the particle. This equation tells us how the wavefunction changes with space under the influence of a potential.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of matter and light on the atomic and subatomic scale. It describes phenomena that cannot be explained by classical mechanics, such as the discrete energy levels of electrons in an atom and the dual wave-particle nature of particles.

One of the core principles of quantum mechanics is the probabilistic nature of physical systems. The wavefunction obtained from solving the Schrödinger equation gives us probabilities, not certainties, about where a particle might be found or how it might behave under measurement. Understanding quantum phenomena requires us to accept the inherent uncertainties involved with microscopic systems.

Particle in a Potential Well

A particle in a potential well is a classic problem in quantum mechanics, serving as a model for a variety of systems, from electrons in a quantum dot to neutrons in a potential trap. The well represents a region where the potential energy is lower than surrounding areas, effectively 'trapping' the particle to some degree.

In the finite square well, the potential energy inside the well is negative relative to the energy outside, where it is zero. The finite square well allows for both bound states (where the particle's energy is less than the potential at the well's boundaries) and scattering states (where the particle's energy is higher).

Wavefunction Continuity

Wavefunction continuity is a crucial concept in quantum mechanics and it arises from the requirement that the wavefunction must be single-valued and continuous everywhere. This includes not only the wavefunction itself but also its first spatial derivative.

When dealing with a potential well, the continuity condition must be applied at the edges of the well (\text{\textnormal{\(\backslash \)pm}} a). This implies that at these boundary points, both the value of the wavefunction and the slope of the wavefunction inside the well must match those outside the well, to form a physically plausible wavefunction that can be normalized (i.e., has a finite probability of finding the particle in space).

Energy Eigenvalues

The concept of energy eigenvalues pertains to the allowed energy levels that a quantum system, such as a particle in a potential well, can occupy. These energies are discrete (quantized) for bound states and are associated with specific wavefunctions called eigenfunctions.

For a finite square well, finding energy eigenvalues typically involves solving the Schrödinger equation under boundary conditions and matching continuity conditions. Such eigenvalues are essential for determining the stationary states of a system and predicting how it will behave over time. Due to the complexity of matching conditions particularly when the particle energy is equal to the well depth, numerical methods or approximations are often used to solve for these energy levels in practice.