From Classical Mechanics to Quantum Field Theory

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

∣∣

∣∣

∣∣

∣

. (1.284)