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").