with A. This compatibility implies that the set {bm} of possible values of B, evalu-ated from the initial vector |Ψ(t)⟩, is not affected by the prior measurement of the quantity A, which projects |Ψ(t)⟩ into the subspace associated with the eigenvalue ak of Â, thus generalizing the idea of projection (6):
where is the projector onto the subspace associated with the eigenvalue ak.
The existence of compatible physical quantities provides the possibility of com-pletely reconstructing the state vector |Ψ(t)⟩ from the measurements of quantities A and B, regardless of the order in which these measurements are performed. This property relies on the fact that the corresponding operators, Â and B̂, commute with each other. This means that their commutator is zero, i.e.:
[Â, B̂] = ÂB̂ − B̂Â = 0 soit ÂB̂ = B̂Â
In other words, the order of the measurement operations does not matter. Thus, if we measure the quantities A and B successively, the state of the system can be decomposed as follows:
where the vectors {|an, bm ⟩} are common eigenstates of the operators  and B̂. In these states, both quantities A and B are simultaneously well-defined, with values an and bm, respectively. These considerations generalize to any number of oper-ators, allowing for the consideration of situations where several quantities are simultaneously well-defined, or even necessary to fully specify the state of the system if the measurements of A and B have not done so.
One of the fundamental principles of quantum mechanics is that the position x and momentum p are incompatible quantities. Mathematically, this means that the corresponding operators do not commute with each other. This situation is expressed by the following formula:
.
In other words, it is impossible to find eigenvectors of the form |x, p⟩ that are common to both x̂ and p̂ operators, in which both quantities are simultaneously well-defined. This means that the measurement of one of these quantities neces-sarily disturbs the possible spectrum of values of the other. Indeed, it follows from the commutation relation that the spectra ∆x and ∆p of the possible values of x and p, respectively, are related by a fundamental relation:
(7)
This relation is known as the Heisenberg’s inequality22. It clearly shows that a meas-urement of x, which generally reduces the spectrum ∆x of possible values of this quantity23, extends the spectrum ∆p of possible values of the momentum p.
It is sometimes stated that the Heisenberg inequalities (7) prevent the simultaneous measurement of x and p, but this assertion is incorrect. In fact, it is always possible to measure any set of physical quantities of a quantum system simultaneously. The true limitation imposed by inequality (7) lies in the fact that the informa-tion obtained about one quantity, for example, the position x, necessarily affects the information that can be obtained about the other quantity, for example, the momentum p, which is then referred to as the conjugate quantity.