Mathematical Foundations of Quantum Mechanics 69- nowadays much more used – is thereduced Planck constant, pronounced
“h-bar”,
:h
2 π=1. 054571726 × 10 −^34 J·s.The physical dimensions ofh(or)arethoseofanaction, i.e.energy×time.A
rough check on the appropriateness of a quantum physical description for a given
physical system is obtained by comparying the value of some characteristic action
of the system with. For a macroscopic pendulum (say, length∼ 1 m,mass∼ 1 kg
maximal speed∼ 1 ms−^1 ), multiplying the period of oscillations and the maximal
kinetic energy, we obtain a typical action of∼ 2 Js >>h. In this case quantum
physics is expected to be largely inappropriate, exactly as we actually know from
our experience of every days. Conversely, referring to a hydrogen electron orbiting
around its proton, the firstionization energymultiplied with the orbital period
of the electron (computed using the classical formula with a value of the radius
of the order of 1 ̊A) produces a typical action of the order ofh. Here quantum
mechanics is necessary.
2.1.1.2 General properties of quantum systems
Quantum Mechanics (QM) enjoys a triple of features which seem to be very far
from properties of Classical Mechanics (CM). These remarkable general properties
concern the physical quantities of physical systems. In QM physical quantities are
calledobservables.
(1) Randomness.When we perform a measurement of an observable of a quan-
tum system, the outcomes turn out to bestochastic: Performing measurements
of the same observableAon completely identical systems prepared in thesame
physical state, one generally obtains different resultsa, a′,a′′....
Referring to the standard interpretation of the formalism of QM (see[ 2 ]for a
nice up-to-date account on the various interpretations), the randomness of mea-
surement outcomes should not be considered as due to an incomplete knowledge
of the state of the system as it happens, for instance, in Classical Statistical Me-
chanics. Randomness is notepistemic, but it isontological. It is a fundamental
property of quantum systems.
On the other hand,QM permits one to compute theprobability distributionof
all the outcomes of a given observable, once the state of the system is known.
Moreover, it is always possible to prepare a stateψawhere a certain observable
Aisdefinedand takes its valuea. That is, repeated measurements ofAgive rise
to the same valueawith probability 1. (Notice that we can perform simultaneous