While mechanics in its classical form describes the world with determinism, even if predictability is undermined, when turning to statistical methods, this is drastically changed in a quantum mechanical setting. Already well established and tangible concepts such as particles are recast in the setting of quantum mechanics using wave functions. This makes eﬀects describable, which are completely unavailable to the classical setting, without changing the Newtonian structure of the universe with its absolute time.

Quantum mechanics is often introduced in a detached manner with little to no connection to the classical settings supporting utterly alien eﬀects which even border on the bizarre. It is therefore interesting to note that the governing structures so painstakingly uncovered by scientists over the course of time for the classical case are still present in the quantum setting. This comes as no surprise in light that the macroscopic world of every day life follows classical rules, wherefore the descriptions of quantum systems should always yield classical behaviour, when taken to the appropriate limit.

The setting of the quantum world is chosen to be a complex Hilbert space from which functions are chosen to represent the state of a system. The structure of this complex Hilbert space together with a link to the macroscopic world is suﬃcient to extract the governing equation of the system.

The wave functions , which are drawn from the complex Hilbert space are acted upon mappings, which are often called operators in this context, to describe transitions of states. While the are capable of encoding the state of a quantum system, they elude observations completely. At ﬁrst this seems to contradict the requirement that scientiﬁc theories must be falsiﬁable [93] and thus accessible to some form of experiments and measurement. However, observables can be constructed from in a systematic fashion, which are again accessible to measurements and thus to testing of the theory. Without touching upon the, for many researchers sensitive, topic of the many diﬀerent interpretations of quantum mechanics [94][95][96][97] the wave functions need to meet the following property

The prescription of constructing observables yields expectation values (Deﬁnition 101) of previously classically determined entities such as location or momentum. It is thus a reasonable demand that this mapping shall produce entities with their expected structures, e.g., appropriate tensors (Deﬁnition 54).

As in the classical case, a diﬀerential form (Deﬁnition 59) is derivable with a function on the Hilbert space . At a point for a tangent vector (Deﬁnition 48) using an arbitrary parameter this gives

In case is self adjoint with respect to the scalar product, this can be reshuﬄed to

Thus the structures governing the evolution of the classical case, in the form of Hamiltonian vector ﬁelds, are also recognizable in a quantum system. While all of the Hilbert space is endowed with the symplectic structure, the normalization requirement conﬁnes the selection of states to the unit hyper-sphere. However, this restriction is not suﬃcient to remove all ambiguity, since it still remains in the form of uni-modular factors

In case of the Hamiltonian operator, the prescription 5.60 of assigning vector ﬁelds to operators yields exactly Schr¨odinger’s equation as can be seen by

which is the main governing equation of the quantum system. Thus the Hamiltonian vector ﬁeld deﬁned in this fashion is indeed responsible for the evolution of the quantum system in the Schr¨odinger picture of quantum mechanics, where operators remain constant and the states evolve.

While a function captures a state completely, descriptions usually employ a representation using either position or momentum representations, which may be obtained by (utilizing Bra-Ket notation introduced in Section 4.6.2)

where is the spatial dimension of the problem under investigation.

These descriptions account for half the degrees of freedom implicitly, while exposing only the other half. It is, however, possible to construct a representation, which explicitly recovers all degrees of freedom. The density operator and the density matrix are such representations. The density operator is given by

Utilizing such a representation, which provides information about the correlation of all of the points in the problem domain, it is possible to also construct a representation which uses the position and the momentum , or wave vector associated by

Since the transform linking the representations in terms of and is linear, the evolution of the system may be expressed in either of them equivalently. The structures inherent to the space of quantum mechanics also allow to formulate the evolution of a physical quantity, such as for the as