From Classical Mechanics to Quantum Field Theory

14 From Classical Mechanics to Quantum Field Theory. A Tutorial

maximally entangled state. We can measure the entanglement of a state by using
physical quantities, called entanglement witnesses. For instance, for a pure state
described by a density matrixρAB, one can define the von Neumann entropy:

S(ρAB)≡TrA[ρAlogρA], (1.39)

whereρA=TrB[ρAB] is the partial trace over the systemB(of course one can
interchange the role ofAandBto get the same result). We will not further discuss
this topic and refer the interested reader for example to[ 7 ], where a discussion of
the geometric aspects of entanglement is also presented.

1.2.1.3 Dynamical evolution

Postulate 3.The dynamical evolution of a quantum state|ψ(t)〉∈His specified
by a suitable self-adjoint operatorHˆ, called Hamiltonian, and governed by the
Schr ̈odinger equation:

ı

d

dt

|ψ(t)〉=Hˆ|ψ(t)〉. (1.40)

In the following, we will always assume thatHˆ is time-independent. In this
case,Hˆbeing self-adjoint, the equation can be solved by introducing the (strongly-
continuous) one-parameter group of unitary operators (the evolution operator):

U(t)=e−

ıtHˆ

, (1.41)

since then:

|ψ(t)〉=U(t)|ψ(0)〉. (1.42)

Notice that, sinceU(t) is unitary, scalar products, hence probabilities, are con-
served.
The (possibly generalized) eigenvectors ofHˆare the only stationary states :
Hˆ|ψλ〉=λ|ψλ〉⇔|ψλ(t)〉=e−ıEλt|ψλ〉. (1.43)

Using the fact that the eigenstates form an orthonormal basis inH,itiseasyto
see that the evolution of any state can be written in the form:

|ψ(t)〉=

∑

λ

cλe−

ıEλ

 t|ψλ〉 if |ψ(0)〉=

∑

λ

cλ|ψλ〉. (1.44)

Example 1.2.6. Free particle onRn.
This system is easily described in the coordinate representation whenH={ψ(x)},
by the Hamiltonian operator:

Hˆ 0 =

pˆ 

2

2 m

=−

^2

2 m

∇^ x^2 , (1.45)

wheremis the mass of the particle.