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

• Quanta: Observations or measurements will record many discrete quantities. For example, a detector will detect an electron or detect no electron but it will not detect 0.5 electrons. The quantum nature of particles allows for a probability of detection that can span continuously the values from 0-> 1 but independent of the probability an actual measurement finds that an entire electron either present or not present.
• Amplitudes: To account for the wave nature of QM amplitudes are used to describe “how much”. For example, for a two slit experiment, one needs to know the amplitude for the particle to pass through slit 1 and arrive at location y on the screen. Amplitudes are used to describe:
• the contribution of various sequences of events or paths to a final result,
• the content of a state.

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

• Vector spaces: In order to incorporate the nature of QM correctly the states are viewed as constituting a vector space. One might notice that classically light exhibits wave behavior. A very interesting example of how light behaves is the impact of polarizing filters. An interesting result is the transmission of light through two perpendicular filters (zero transmission) when a third filter is inserted in between the two perpendicular ones. In this case the transmission need not be zero. Although the result might be surprising it is understandable because of the vector nature of the electric field. Quantum states are vectors in Hilbert space. Some states are fundamentally different from other states (orthogonal) a particle at location x1 is not at all the same as a particle at location x2. However a particle at location x1 does and a particle of definite momentum do share or overlap properties. Both of these states may result in a particle being detected at x1. The behavior of QS is encapsulated in the mathematics of vector spaces.
• Vector addition: The sum of two QS with weight factors (amplitudes) is a valid QS. You can any state A to any state B and the result is a valid QS.

• Basis: There exists sets of QS that are linearly independent and from which any conceivable QS can be built using the above addition rule.
• Inner product: An inner product can be introduced by adding a dual space and a product rule.

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.

• Orthonormal: using the inner product one can impose the condition of normality and orthogonality for a basis.

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.

• Two slit experiment: Feynman uses this example to introduce QM. There are several great websites that discuss the ideas.
• Filters: Again Feynman devotes considerable time to the topic and several interesting web sites can be found under Stern Gerlach Spin filters e. g.

http://www.freeinfosociety.com/pdfs/science/stern-gerlachexperiment.pdf

• Polarizing light filters are an example of these types of filters.

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