A Short Course on Quantum Mechanics and Methods of Quantization 59
(2) Setting:Ô=Q̂, we find at once:
Ω−^1
(
Q̂
)
(q,p)=q. (1.312)
(3) Consider nowÔ=|φ〉〈ψ|, which is the simplest example of a finite-rank
operator. Then it is immediate to see that:
Ω−^1 (|φ〉〈ψ|)(q,p)=
∫∞
−∞
dξeıpξ/φ(q+ξ/2)ψ∗(q−ξ/2). (1.313)
It is also easy to check formula (1.310):
∫
dp
2 π
e{−ıp(x−x
′)/}
Ω−^1
(
Ô
)(x+x′
2
,p
)
=φ(x)ψ∗(x′)=〈x|φ〉〈ψ|x′〉.
(4) We can now proceed to consider a self-adjoint operator with discrete spec-
trum: Ô|φn〉=λn|φn〉, with〈φn|φm〉=δnm,
∑
n|φn〉〈φn|=I.Thenwe
can write:
Ω−^1
(
Ô
)
(q,p)=
∑
n
λn
∫
dξeıpξ/φn(q+ξ/2)φ∗n(q+ξ/2). (1.314)
The most interesting consequence of what was seen above is the fact that
the Weyl and Wigner mapsestablish a bijection[ 19 ]between Hilbert-Schmidt
operators and square-integrable functions on phase space, which is also strongly
bicontinuous.
Indeed the following theorem holds^27 :
f will be square-integrable if and only ifΩ(f)is Hilbert-Schmidt. Similarly,
Ω−^1
(
Â
)
will be square-integrable if and only ifÂis Hilbert-Schmidt.
We notice that, since [Fs(η,ξ)]∗=Fs(−η,−ξ)and̂W†(ξ,η)=̂W(−ξ,−η),
the Weyl and Wigner maps preserve conjugation:
Ω(f∗)=Ω(f)†, Ω−^1
(
Ô†
)
=Ω−^1
(
Ô
)∗
, (1.315)
so guaranteeing thatfis real iff Ω (f) is at least a symmetric operator.
Before looking at some examples, we also observe that Eq. (1.310) implies
Trx
[
Ô
]
≡
∫
dx〈x|O|x〉=
∫
dqdp
2 π
Ω−^1
(
Ô
)
(q,p), (1.316)
(^27) See[ 14 ]for a proof.