Solution for a Single Particle in a Semi-Infinite Quantum Well
Claimed by Adam Barletta 4/21/22
Introduction
As you may have already learned, the Single Particle in a Box problem is a greatly intuitive way to begin to understand some of the major concepts connecting classical mechanics and quantum mechanics. It is recommended that you first begin by analyzing the "Infinite Well" solution prior to this "Semi-Infinite Well". Once you have a solid understanding of the concepts utilized in that example, you will find yourself better able to understand this specific example and explore the nuances that come with it. Like the "Infinite Well" example, we begin with a particle in a well where the potential energy where [math]\displaystyle{ x \lt 0 }[/math] is infinity. We will call the region of infinite potential "Region I". The region where [math]\displaystyle{ 0 \lt x \lt L }[/math] ([math]\displaystyle{ L }[/math] is any arbitrary distance from [math]\displaystyle{ x }[/math]) possesses a potential energy of [math]\displaystyle{ 0 }[/math] ("Region II"), and the area [math]\displaystyle{ x \gt L }[/math] has an unchanging potential equal to the positive constant [math]\displaystyle{ V_{0} }[/math] ("Region III").
Background
Of course, we will start with Schrödinger's Equation: [math]\displaystyle{ \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+V\Psi=E\Psi }[/math]. This specific form of Schrödinger's Equation is known as a "1-Dimensional Time-Independent Schrödinger Equation" due to its singular spacial dimension and its lack of dependency on time. The reason for the singular spacial dimension is for simplicity. The equation does not depend on time due to the the potential regions being only dependent on position, allowing us to ignore time for now. The variable [math]\displaystyle{ \hbar }[/math] represents Planck's constant [math]\displaystyle{ h }[/math] ([math]\displaystyle{ 6.62607015\times 10^{-34} \frac{\text{m}^{2}\text{kg}}{\text{s}} }[/math]) divided by [math]\displaystyle{ 2\pi }[/math], [math]\displaystyle{ m }[/math] represents the mass of the singular particle, [math]\displaystyle{ \Psi }[/math] represents the wave function we are trying to find, [math]\displaystyle{ V }[/math] represents the potential energy, and [math]\displaystyle{ E }[/math] represents the energy of the particle. Due to the piece-wise nature of our potential, we will solve this equation in each region separately at first.
Method to Solve
The procedure we will use to solve this problem will go as follows:
- The general solution to the Schrödinger Equation will be found in all regions
- Each region extending to positive or negative infinity will be made finite to comply with the requirements of a proper wave function
- The solution will be made continuous at all boundaries and made differentiable at non-infinite potential boundaries
- Assure the equation is properly normalized
Following each of these steps is crucial for finding an applicable solution. Although this is an impossible real world scenario, following these steps will assure that this example is intuitive and beneficial to our understanding of quantum wave functions.