It is important to study a number of simple systems as it provides our first exposure to quantum mathematics in action and because so many real systems build upon these simple systems. We can interpret many of the details of complex systems in terms of what we learn from the model systems.
An appreciation of the mathematical basis of quantum theory is assumed and you should understand the axioms of quantum mechanics. Nevertheless, it is hoped that you will come to understand both the mathematical machinery and the theoretical interpretations better by studying these models. PLEASE! Work through the questions and examples on paper, even if it means just copying down the answers; even the act of your hand going through the motions helps to drive the point home. And if you actually THINK about the problem... well, there's no telling how much you might actually understand!!
Study each of the following in turn and later refer back to them.