A Short Course on Quantum Mechanics and Methods of Quantization 17This means that the outcomeXof a measure onρis described by a probability
density distribution given by
p(X)=Tr[ρF(X)] =∑nj=1pj〈ψj|F(X)|ψj〉. (1.55)where to write the last expression we have taken into account Eq. (1.6).
Example 1.2.8. Projection valued measures.
This is a particular case of the above definition whenFgives projection operators.
This emerges, for example, when considering an observableOˆ: the possible out-
comes of a measure are its eigenvalues, so that the spaceXis given by its spectrum
σ(Oˆ)⊂R, whileFis the map that associates to any measurable subsetA⊂σ(Oˆ)
the projection operatorPA on the corresponding eigenspace. More specifically,
assuming for simplicity thatOˆhas only a discrete non-degenerate spectrum so
that it can be decomposed by means of thespectral decomposition (1.8), we have
thatFis the map that associates to any pointλα∈σ(Oˆ) the projectorPαon the
corresponding one-dimensional eigenspace. Thus Eq. (1.55) reads:
pα=p(λα)=Tr[ρPα]=〈ψα|ρ|ψα〉. (1.56)If considering a pure state ρφ = |φ〉〈φ|, Eq. (1.56) simply becomes: pα =
|〈φ|ψα〉|^2.
Notice that probability densities are defined by using the notion of the trace,
which depends on the Hermitean product defined in H, and gives the pairing
between states and observables mentioned at the beginning of this Sect.
1.2.2 Geometric quantum mechanics
In this Sect. we will describe the geometrical structures that appear in QM. To do
so, we will stick to the case of a finiten-level system, so thatH=Cn. The starting
observation is that, being a vector space,His a manifold and there is a natural
identification of the tangent space at any pointψ∈HwithHitself:TψH≈H,so
that we have the identification:TH≈H×H, withTHthe tangent bundle ofH.
Thus, in the following, vectors of the Hilbert space play a double rˆole, as points of
the space and as tangent vectors at a given point^11. Also, to discuss the geometry
of the space of states in QM in more detail, we consider the realificationHRofH,
whose tangent bundle isTHR≈HR×HR,sotolookatHas to a real manifold.
(^11) Here we will write vectors asψ, φ,···, instead of using the Dirac notation|ψ〉,|φ〉,···.