From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 55

Proceeding just as in the previous example, we obtain:

̂Wt(q,p)=exp

{

(ı/)

[

qP̂t+pQ̂t

]}

, (1.285)

with now:

Q̂t=Q̂cosωt+P̂sinωt

mω

P̂t=P̂cosωt−Qmω̂ sinωt

. (1.286)

Defining againF̂(t)=exp

{

−ıHt/̂ 

}

and working out the commutation rela-
tions, one gets:
[
Q,̂Ĥ

]

=

ı

m

P,̂

[

P,̂Ĥ

]

=−ımω^2 Q.̂ (1.287)

Thus a quadratic Hamiltonian must now have the form:

Ĥ=P̂

2

2 m

+

1

2

mω^2 Q̂^2 +λ̂I, (1.288)

which, again up to an additive multiple of the identity, is the standard quantum
Hamiltonian for the harmonic oscillator.

1.3.3.3 Quantum mechanics on phase space

For simplicity, in this Sect. we will work inS≈R^2 , since generalizations toS≈
R^2 nare easy to work out.
As a preliminary remark, let us observe that, for anyf∈L^2

(

R^2

)

,wehavethe
identity:
∫
dξdηdq′dp′
(2π)^2

f(q′,p′)e−ıω^0 ((q

′,p′),(ξ,η))/

eı(ξp+ηq)/=f(q,−p). (1.289)

Defining[ 19 ]the symplectic Fourier transformFs(f):

Fs(f)(η,ξ)=

∫

dqdp

2 π f(q,p)e

−ıω 0 ((q,p),(ξ,η)), (1.290)

where as usualω 0 ((q,p),(ξ,η)) =qη−pξ, we can rewrite the above expression as:
∫
dξdη
2 π

[

1

Fs(f)

(

η

,

ξ



)]

eı(ξp+ηq)/=f(q,−p). (1.291)

The Weyl map, which amounts to make, in Eq.(1.291), the replacement:

exp{ı(ξp+ηq)/}→exp

{

ı

(

ξP̂+ηQ̂

)

/

}

≡̂W(ξ,η), (1.292)