Classical Mechanics can be formulated in configuration space (Lagrangian formulation) or in phase space (Hamiltonian formulation). Via the Legendre transform both formulations are in duality to each other. Similarly, there is a variety of different formulations available for quantum mechanics.
This chapter gives an overview of the formulations used in this thesis. We first discuss different formulations of Schrödinger's quantum mechanics. Then we make the transition to the Wigner formulation which retains many of the concepts and notions of classical mechanics.