Our goal is to obtain an overview of the essential ingredients in QM. Here are some salient aspects:




To determine the probability of an event one squares the amplitude.





Familiar vector space of 3-d does not typically use the notion of a dual space when introducing the inner or dot product. However 4-d spacetime can be conveniently cast into the above form and the minus sign for the time component, when calculating length, can be introduced by allowing the space and its dual to have opposite signs for the t-component.


These are somewhat formal mathematical rules but they are essential.  If one can learn to formulate QM as vector space with amplitudes and bases, then the behavior of QS can be extracted and a certain intuition can be built.


There are two types of idealized problems that are usually introduced to provide examples of quantum behavior. 





Typically these two idealized systems give the students a good introduction to the properties of quantum systems.