A Short Course on Quantum Mechanics and Methods of Quantization 49By Stone’s theorem[ 34 ], there exists an essentially self-adjoint generatorĜ(z):
̂W(αz)=exp{
ıαĜ(z)/}
, (1.239)
withĜ(αz)=αĜ(z). Thus Eq. (1.233) reads:
eıαĜ(z)/eıβĜ(z′)/
=eıαβω(z,z′)/
eıαĜ(z)/eıβĜ(z′)/
, (1.240)
giving, at the infinitesimal order, the following commutation relation:
[
Ĝ(z),Ĝ(z′)
]
=−ıω(z,z′). (1.241)Example 1.3.5. The free particle.
The simplest case we can consider is given byS=R^2 with coordinatesz=(q,p)
and the standard symplectic formω=dq∧dp,sothatω((q,p),(q′,p′)) =qp′−q′p.
In this case a Weyl system satisfies:
̂W((q,p)+(q′,p′)) =Ŵ(q,p)̂W(q′,p′)exp{
−
ı
2 (qp′−q′p)}
. (1.242)
Now ifS 1 ={z 1 ≡(q,0)}andS 2 ={z 2 ≡(0,p)}, one has:
̂W(q,p)=̂W((q,0) + (0,p)) =̂W(q,0)̂W(0,p)exp{−ıqp/ 2 }. (1.243)Defining the infinitesimal generators as:
Û(q)=̂W(q,0)≡exp{
ıqP/̂}
V̂(p)=̂W(0,p)≡exp{
ıpQ/̂}, (1.244)
one immediately finds that Eq. (1.241) gives the standard CCR:
[
Q,̂P̂
]
=ıI. (1.245)Thus, using the already-mentioned Baker-Campbell-Hausdorff formula, it is
straightforward to see that the Weyl system is given by the map:
Ŵ(q,p)=exp{
ı(
qP̂+pQ̂)
/
}
. (1.246)
To construct a concrete realization of this Weyl system, we can consider wave
functionsψ∈L^2 (R,dx) and define the families of operators (1.244) via:
(
Û(q)ψ
)
(x)=ψ(x+q)
(
V̂(p)ψ)
(x)=exp{ıpx/}ψ(x), (1.247)
which are one-parameter strongly continuous groups of unitary transformations
satisfying:
(
Û(q)V̂(p)ψ
)
(x)=exp{ıqp/}(
V̂(p)Û(q)ψ)
(x). (1.248)