We start by reviewing the differences between states in classical mechanics, in stochastic mechanics, and states in quantum mechanics. On the theoretical side, this brings us to the study of the Hilbert space, and on the experimental side we look at the Stern-Gerlach experiment, and discuss the spin of a particle in the context of quantum states and measurement.
Next, we study how states walk around in Hilbert space, and how transformations in Hilbert space take place. We describe this in term of the Stone theorem. Time evolution of quantum states is a prime example for this, and we will investigate both the Schrodinger and the Heisenberg formalisms. Another example of a transformation is a rotation. We investigate how spin states change under rotation, how combining several spins into one object is done, and how the combined object acts under rotation.
We then add statistical uncertainty back to the pure (quantum) states, leading to a description in terms of density matrices. These are shown to obey the Von Neumann equation and to allow calculation of quantum statistical expectation values.
Finally, Feynman's path integral theory is introduced as an alternative way to describe quantum mechanics, and this different angle of incidence is linked back to the Schrodinger formalism.