16 From Classical Mechanics to Quantum Field Theory. A Tutorial
any monomial in the variablesx, pthat classically reads asxmpnhas to be re-
placed with the symmetric version obtained by writing the products ofmtimes
x-factors andntimesp-factors in all possible orders. Applying this rule to our
problem, we exactly obtain Eq. (1.50).
Thus, in the coordinate representation, we get:
Hˆ=ω
2(
−
d^2
dx^2+x^2 +1)
. (1.52)
Making use of (1.36), the latter can be rewritten as:
Hˆ=ω
2(
a†a+^1
2)
=ω
2(
N+^1
2
)
. (1.53)
Thus we see that the Hamiltonian of the 1D harmonic oscillator is (up to a con-
stant) the number operator of bosonic type, whose spectral problem has been
solved in the previous subsection.
1.2.1.4 Measurement and probability
Since the very beginning, the notion of measurement has played an important rˆole
in the discussion about the interpretation of QM[ 6 ], and is central to analyze pos-
sible connections with applications, such as in optimization and control problems
[ 32 ]. However, such a study goes well beyond the scope of these lectures^10 .For
completeness, we will give here only the definition of apositive operator valued
measure(POVM), that we will specialize in one example.
The definition rests on the hypothesis that the outcome of a quantum mea-
sure is a random variableX and therefore its possible values form a measure
space (X,μ). We will denote with Mthe set of all measurable subsets ofX
and withB+sathe set of positive self-adjoint bounded operators on the Hilbert
spaceH.
Postulate 4.A POVM is specified by a functionF:M→Bsa+such thatF(X)=
IHand with the property that the probability that a measurement on a (pure or
mixed) stateρyields a resultX∈A, for anyA∈M, is given by:
Pr{X∈A}=Tr[ρF(A)]. (1.54)(^10) For a thorough discussion of this topic see sect. 2 of the second part of this volume.