54 From Classical Mechanics to Quantum Field Theory. A Tutorial
As usual, we can define a one-parameter family of unitary operators
{
F̂(t)
}
t∈R
such that:
exp
{
ıpQ̂t/
}
=F̂†(t)exp
{
ıpQ/̂
}
F̂(t)
exp
{
ıqP̂t/
}
=F̂†(t)exp
{
ıqP/̂
}
F̂(t)
(1.278)
which, in terms of the infinitesimal generatorĤ, can be written as:
F̂(t)=exp
{
−ıHt/̂
}
. (1.279)
Eqs. (1.277) imply the commutation relations:
[
P,̂Ĥ
]
=0,
[
Q,̂Ĥ
]
=
ı
m
P.̂ (1.280)
If we now look for a quantum operatorĤgiven by a quadratic function, as it
happens for the generators of linear and homogeneous canonical transformations,
i.e. by a Hamiltonian of the type:
Ĥ=aP̂^2 +bQ̂^2 +c
(
P̂Q̂+Q̂P̂
)
, (1.281)
it is easy to check that the solution of commutation relations (1.280) is given by:
Ĥ=
P̂^2
2 m
+λ̂I, (1.282)
wherêIis the identity operator andλany real constant. Thus, apart from this
constant term, the quantum operator associated with the time evolution is the
standard quantum Hamiltonian for a free particle of massm.
Example 1.3.8. Harmonic oscillator evolution.
From the classical equations of motion
q(t)=qcosωt+p
sinωt
mω
p(t)=pcosωt−qmωsinωt
, (1.283)
one finds thatF(t) is given, in this case, by:
F(t)=
∣∣
∣∣
∣∣
∣
cosωt sinωt
mω
−mωsinωtcosωt