Solution for a Single Particle in an Infinite Quantum Well - Darin

Introduction

The particle in a box problem is a classic way of understanding yet another difference between the classical and quantum worlds. Imagine a box whose sides represent infinite potential, so no particle has enough energy to climb them and escape. Now say we put in an ordinary ball with some initial velocity and it bounces between the walls. The ball is no more likely to be in one place than another. But by solving the Schrödinger equation we'll see that a quantum particle isn't so egalitarian.

Mathematical Setup

[math]\displaystyle{ \frac{-\hbar \lt sup\gt 2\lt /sup\gt }{2m} \frac{\partial \lt sup\gt 2\lt /sup\gt \Psi}{\partial x \lt sup\gt 2\lt /sup\gt } + V \Psi = E \Psi }[/math]
To describe the state of the quantum particle in the potential well we will find a function [math]\displaystyle{ \Psi }[/math] that satisfies the barrier conditions of the well and the Schrödinger equation (above), a differential equation.

Barrier Conditions

For the region [math]\displaystyle{ x ≤ 0 }[/math] the wave function must be zero because the particle cannot overcome an infinite potential barrier. Similarly for [math]\displaystyle{ x ≥ L }[/math] . Now, we just need to construct the solution/wave-function [math]\displaystyle{ \Psi }[/math] for the region [math]\displaystyle{ 0 ≤ x ≥ 0 }[/math].