The topic of this chapter is the wave function – what it is, how it is to be interpreted and how information can be extracted from it. To this end, the notion of operators in quantum physics is introduced. And the statistical interpretation called the Born interpretation is discussed. This discussion also involves terms such as expectation values and standard deviations. The first part, however, is dedicated to a brief outline of how quantum theory came about – who were the key people involved, and how the theory grew out of a need for understanding certain natural phenomena. Parallels are drawn to the historical development of our understanding of light. At a time when it was generally understood that light is to be explained in terms of travelling waves, an additional understanding of light consisting of small quanta turned out to be required. It was in this context that Louis de Broglie introduced the idea that matter, which finally was known to consist of particles – atoms – must be perceived as waves as well. Finally, formal aspects such as Dirac notation and inner products are briefly addressed. And units are introduced which allow for convenient implementations in the following chapters.

The purpose of this chapter is to clearly define the mathematical objects that describe particles of various kinds: bosons (spin-0 and spin-1) or spin-1/2 fermions. Starting from the Schrödinger equation, the Klein–Gordon equation, the Dirac equation and the Maxwell equations are detailed, leading to the description of the associated quantised field – a well-adapted framework to treat states composed of many particles that can be created or annihilated when they interact. The notion of 4-current is introduced, and the quantisation of the various fields is presented. With the Dirac equation, the spinor’s properties are described extensively. The interpretation of the solutions of the Dirac equation in terms of antiparticles and spin or helicity degrees of freedom is then detailed. Helicity and chirality are also treated carefully. Finally, the Maxwell field and the Proca field are described, highlighting their specificities in terms of polarisation degrees of freedom.