determined by the action of the Hamiltonian operator on that same state vector. The Schrödinger equation, along with the initial condition |Ψ(t = 0)⟩ = |Ψ0⟩, allows for a complete determination of the system’s state at any given time t.

Born’s Rule

While, in the formalism of quantum mechanics, a system is represented by an abstract entity, the state vector |Ψ(t)⟩, which belongs to a Hilbert space 𝓗 and evolves over time according to the Schrödinger equation (4), the rule—or postu-late—of Born establishes the correspondence between the formal content and the operational content of the theory:

The probability of finding, during a measurement of the physical quantity A, per-formed on the normalized state19 |Ψ(t)⟩, the value ak , is given by:

(5)

where denotes the square of the modulus of the complex number . As we can see, is nothing other than the square of the tran-sition amplitude between the states |Ψ(t)⟩ and |ak⟩. If, for example, we consider the continuous quantity position 𝑥̂, then the probability of finding the system in a position within an interval of size dx centered at x, at time t, is given by20:

c k ( t )
d P ( x ) = | ⟨ x | Ψ ( t ) ⟩ | ² d x = | Ψ ( x , t ) | ² d x .

It is immediately apparent, from the expression (5), that it is at least formally pos-sible to reconstruct the state |Ψ(t)⟩, i.e., to determine all the coefficients {cn} of the decomposition (1)21. This requires, first of all, preparing a very large number N of systems in the same state |Ψ(t)⟩, and then measuring the quantity A. These meas-urements provide a set of results {an} associated with the occurrence numbers {Nn} and, consequently, with respective frequencies of occurrence {fn} = {Nn/N}. These statistical frequencies are then equated with the probabilities {𝓟(an)}, and thus, via (5), with the squares of the moduli of the coefficients {cn}. It is thus observed that in quantum mechanics, since predictions are probabilistic in nature, this implies, in practice, considering a statistical ensemble of systems that are said to be “prepared” identically.

State Vector Reduction

The postulate of state vector reduction specifies the form of the state vector imme-diately after measurement:

Immediately after the measurement that results in the outcome ak for the quan-tity A, the state vector is projected onto the eigenvector |ak⟩ associated with the eigenvalue ak :

(6)

Compatible Physical Quantities

Among the set of measurable physical quantities, some play a privileged role: these are the ones that are said to be compatible with each other. Physically, this means that the measurement of one does not affect the result of the measurement of the other (see below). This notion of compatibility is crucial, especially in situa-tions where the measurement of a quantity A does not lead to the determination of a unique vector, but rather to the determination of a subspace of the Hilbert space. This case corresponds to the situation of a “degenerate” eigenvalue ak .

In this context, in order to completely specify the state vector after the measure-ment of A, it is necessary to consider another physical quantity, B, that is compatible

| c k ( t ) | 2 = c k ( t ) c k ¯ ( t )
| Ψ ( t ) ⟩ = ∑ n c n ( t ) | a n ⟩ ⟶ | a k ⟩ .
P ( a k )
P ( a k ) = | ⟨ a k | Ψ ( t ) ⟩ | ² = | c k ( t ) | ²

92