Introduction

• Solutions to Schroedinger’s equation for specific, one-dimensional systems
• Classically forbidden regions and quantum tunneling
• The full, quantum mechanical description of the hydrogen atom

We will take a brief detour from the quantum expressway to enjoy a scenic route past a number of beautiful wave functions. We will look at several specific solutions of the Schroedinger equation, and see how different wave functions emerge under different conditions. We’ll ditch the math on this detour and paint a few portraits of some illustrative Eigenfunctions and learn about their Eigenvalues. This will allow you to gain a better appreciation of the Schroedinger equation and how it describes the way quantum particles must behave under the influence of specific force configurations.

The purpose of all this is to give you a better idea of how the governing equation of wave mechanics, Schroedinger’s equation, actually works. We will study a few examples of quantum systems and compare their behavior to their classical counterparts. We’ll start with free particles, and then lock them into boxes and tie them to springs. Doing so will yield some surprises, and before the end. We venture into forbidden regions and even burrow through walls.

We’ll also step for the first time beyond the one-dimensional examples we’ve been using and jump straight to a real, three-dimensional quantum problem–the hydrogen atom. We will see how Schroedinger’s theory leads naturally to the ad hoc assumptions made by Niels Bohr. We’ll gain a better insight into atomic structure that requires neither plum pudding nor whizzing electrons. And we’ll see that in the three-dimensional universe (like the one we live in) quantum mechanics offers triple the fun in terms of quantization.