But before entering into these considerations, we will briefly set out the funda-mental elements of the formalism of quantum mechanics, often called postulates. This will serve as a starting point for discussing the conceptual implications of the theory. We will present these elements as neutrally and objectively as possible, without immediately delving into more philosophical considerations.

The Postulates of Quantum Mechanics

Here, we state the postulates of quantum mechanics, without claiming exhaus-tiveness. The objective is to present those most pertinent for the forthcoming discussion.

State of a Physical System

At a given instant t, a quantum system is described by a state vector, denoted |Ψ(t)⟩, which belongs to a space called Hilbert space, denoted 𝓗. This space is equipped with an inner product that associates, with any two state vectors |Ψ1(t)⟩ and |Ψ2(t)�, a complex number denoted ⟨Ψ1(t)|Ψ2(t)⟩. This inner product repre-sents the “overlap” between two states and indicates the extent to which one state can be considered similar to another. Thus, the identity of states corresponds to ⟨Ψ1(t)|Ψ2(t)⟩ = 112, while total difference corresponds to orthogonality, that is ⟨Ψ1(t)|Ψ2(t)⟩ = 0. Physically, as we will see later, the inner product ⟨Ψ1(t)|Ψ2(t)⟩ is closely related to the transition between the corresponding states; for this reason, it is also called the transition amplitude between the states |Ψ1(t)⟩ and |Ψ2(t)⟩.

Physical Quantities

Any measurable physical quantity, often called an observable—such as position, energy, etc.—and generically denoted A, is represented in the Hilbert space 𝓗, by a Hermitian operator13, denoted Â, which acts on the state vectors. A funda-mental characteristic of quantum mechanics is that a physical quantity can only take certain precise numerical values, corresponding to the eigenvalues of the operator Â14. The Hermitian nature of these operators is essential: it ensures that the eigenvalues of  are real, in accordance with observed results. Furthermore, the eigenvectors associated with distinct eigenvalues are orthogonal, meaning their inner product is zero. Finally, depending on the physical context, these eigenvalues can be discrete or continuous. For simplicity, and unless otherwise indicated, we will assume that the eigenvalues are discrete and denote the set of all possible values of the quantity A by {an}15. In this case, we say that the values of A are quantized and that its spectrum is discrete. This is one of the essential aspects of quantum mechanics, breaking from classical mechanics where almost all physical quantities take continuous values16.

Spectral Decomposition

To each eigenvalue an of an operator  is associated one or more eigenvectors. In the simplest case, there exists a unique eigenvector |an⟩ associated with the eigen-value an, which is then said to be “non-degenerate”. The vectors {|an⟩} then form an orthogonal basis of the state space, allowing any state |Ψ⟩ of the system to be decomposed as follows:

(1)

Where represents a summation over the integer n. Expression (1) is of funda-mental importance. Indeed, due to the structure of the vector space 𝓗, it is pos-sible to decompose the state |Ψ⟩ into a superposition of states {|an ⟩} where the coef-ficients {cn(t)} are complex numbers that indicate the relative weight of each state in the spectral decomposition of |Ψ(t)⟩. This ability to combine or superimpose multiple states, called the principle of superposition, is one of the most remarkable