From Classical Mechanics to Quantum Field Theory

76 From Classical Mechanics to Quantum Field Theory. A Tutorial

That number is extremely small if compared with macroscopic scales. This is the
ultimate reason why the incompatibility ofLxandLzis negligible for macroscopic
systems.
Direct inspection proves thatσ(Sa)={±/ 2 }. Similarlyσ(La)={n|n∈Z}.
Therefore, differently from CM, the values of angular momentum components form
a discrete set of reals in QM. Again notice that the difference of two closest values
is extremely small if compared with typical values of the angular momentum of
macroscopic systems. This is the practical reason why this discreteness disappears
at macroscopic level.

Just a few words about the time evolution and composite systems already discussed
in the first part are however necessary now, a wider discussion on the time evolution
will take place later in this paper.

2.1.3 Time evolution

Among the class of observables of a quantum system described in a given iner-
tial reference frame, an observableHcalled the (quantum)Hamiltonianplays a
fundamental rˆole. We are assuming here that the system interacts with a station-
ary environment. The one-parameter group of unitary operators associated toH
(exploiting (2.7) to explain the notation)

Ut:=e−itH:=

∑

h∈σ(H)

e−ithPh(H),t∈R (2.13)

describes thetime evolution of quantum statesas follows. If the state at time
t= 0 is represented by the unit vectorψ∈H, the state at the generic timetis
represented by the vector

ψt=Utψ.

Remark 2.1.7.Notice thatψthas norm 1 as necessary to describe states, since
Utis norm preserving it being unitary.

Taking (2.13) into account, this identity is equivalent to

i

dψt

dt =Hψt. (2.14)

Equation (2.14) is nothing but a form of the celebratedSchr ̈odinger equation.If
the environment is not stationary, a more complicated description can be given
whereHis replaced by a class of Hamiltonian (selfadjoint) operators parametrized
in time,H(t), witht∈R. This time dependence accounts for the time evolution
of the external system interacting with our quantum system. In that case, it is