Mathematical Foundations of Quantum Mechanics 71destructive.) If the measured state isψ, immediately after the measurement of an
observableAobtaining the valueaamong a plethora of possible valuesa, a′,a′′,...,
the state changes toψ′generally different formψ. In the new stateψ′, the distri-
bution of probabilities of the outcomes ofAchanges to 1 for the outcomeaand 0
for all other possible outcomes.Ais thereforedefinedinψ′.
When we perform repeated and alternated measurements of a pair of incom-
patible observables,A,B, the outcomes disturb each other: If the first outcome
ofAisa, after a measurement ofB, a subsequent measurement ofAproduces
a′=ain general. Conversely, ifAandBare compatible, the outcomes of their
subsequent measurements do not disturb each other.
In CM there are measurements that, in practice, disturb and change the state
of the system. It is however possible to decrease the disturbance arbitrarily, and
nullify it in ideal measurements. In QM it is not always possible as for instance
witnessed by (2.1).
In QM, there are two types of time evolution of the state of a system. One is
the usual one due to the dynamics and encoded in the famousSchr ̈odinger equation
we shall see shortly. It is nothing but a quantum version of classicalHamiltonian
evolutionas presented in the first part. The other is the sudden change of the state
due to measurement procedure of an observable, outlined in (3): Thecollapse of
the state(orwavefunction) of the system.
The nature of the second type of evolution is still a source of an animated debate
in the scientific community of physicists and philosophers of Science. There are
many attempts to reduce the collapse of the state to the standard time evolution
referring to the quantum evolution of the whole physical system, also including
the measurement apparatus and the environment (de-coherence processes)[2; 12].
None of these approaches seem to be completely satisfactory up to now.
Remark 2.1.1.Unless explicitly stated, we henceforth adopt a physical unit sys-
tem such that=1.
2.1.2 Elementary formalism for the finite dimensional case
To go on with this introduction, let us add some further technical details to the
presented picture to show how practically (1)-(3) have to be mathematically in-
terpreted (reversing the order of (2) and (3) for our convenience). The rest of
the paper is devoted to make technically precise, justify and widely develop these
ideas from a mathematically more advanced viewpoint than the one of the first
part.
To mathematically simplify this introductory discussion, throughout this sec-
tion, except for Sect 2.1.5, we assume thatHdenotes afinite dimensionalcomplex