From Classical Mechanics to Quantum Field Theory

144 From Classical Mechanics to Quantum Field Theory. A Tutorial

whereNis an eigenvector basis forρ,pφ∈[0,1]for anyφ∈N,and
∑

φ∈N

pφ=1.

The stated proposition allows us to introduce some notions and terminology rel-
evant in physics. First of all, extreme elements inS(H) are usually calledpure
statesby physicists. We shall denote their set is denotedSp(H). Non-extreme
states are instead called mixed states,mixturesornon-pure states.If

ψ=

∑

i∈I

aiφi,

withIfinite or countable (and the series converges in the topology ofHin the
second case), where the vectorsφi∈Hare all non-null and 0=ai∈C, physicists
say that the state 〈ψ|〉ψ is called ancoherent superpositionof the states
〈φi, 〉φi/||φi||^2.
The possibility of creating pure states by non-trivial combinations of vectors
associated to other pure states is called, in the jargon of QM, superposition
principle of (pure) states.
There is however another type of superposition of states. Ifρ∈S(H)satisfies:

ρ=

∑

i∈I

piρi

withIfinite,ρi∈S(H), 0=pi∈[0,1] for anyi∈I,and

∑

ipi=1,thestateρ
is calledincoherent superpositionof statesρi(possibly pure).
Ifψ,φ∈Hsatisfy||ψ||=||φ||= 1 the following terminology is very popular:
The complex number〈ψ,φ〉is thetransition amplitudeorprobability am-
plitudeof the state〈φ, 〉φon the state〈ψ, 〉ψ, moreover the non-negative real
number|〈ψ,φ〉|^2 is thetransition probability of the state〈φ, 〉φon the state
〈ψ, 〉ψ. We make some comments about these notions. Consider the pure state
ρψ∈Sp(H), writtenρψ=〈ψ, 〉ψfor someψ∈Hwith||ψ||=1. Whatwewant
to emphasise is that this pure state is also an orthogonal projectorPψ:=〈ψ,〉ψ,
so it must correspond to an elementary observable of the system (anatomusing
the terminology of Theorem 2.3.13). The na ̈ıve and natural interpretation of that
observable is this: “the system’s state is the pure state given by the vectorψ”.
We can therefore interpret the square modulus of the transition amplitude〈φ, ψ〉
as follows. If||φ||=||ψ||= 1, as the definition of transition amplitude imposes,
tr(ρψPφ)=|〈φ, ψ〉|^2 ,whereρψ:=〈ψ,〉ψandPφ=〈φ,〉φ. Using (4) we conclude:
|〈φ, ψ〉|^2 is the probability that the state, given (at timet) by the vectorψ,following
a measurement (at timet) on the system becomes determined byφ.