60 From Classical Mechanics to Quantum Field Theory. A Tutorial
as well as
∫
dqdp
2 π
f(q,p) = Tr [Ω (f]). (1.317)
This allows for a formal definition of atrace operationon phase space given by:
Tr [f]≡
∫
dqdp
2 π
f(q,p). (1.318)
Example 1.3.10.The 1D Harmonic oscillator.
We go back to the Hamiltonian
Ĥ=
P̂^2
2 m
+^1
2
mω^2 Q̂^2 , (1.319)
which has eigenvalues:En=ω(n+1/2) (n≥0) and eigenfunctions:
ψn(x)=
(mω
π
) 1 / (^41)
√
2 nn!
exp
(
−ζ^2 / 2
)
Hn(ζ), (1.320)
whereζ=x
√
mω/. We want to evaluate here the Wigner function associated
with the so called Boltzmann factorÔ=exp
(
−βĤ
)
.From:
〈
x
∣∣
∣e−βĤ
∣∣
∣x′
〉
=
∑∞
n=0
e−βEnψ∗n(x)ψn(x′), (1.321)
inserting the explicit form (1.320) of the eigenfunctions and manipulating the
expression (we refer to[ 14 ]for details), one finds that the matrix element (1.321)
can be expressed as:
〈
x
∣∣
∣e−β
Ĥ∣∣
∣x′
〉
=
√
mω
π
e−(ζ
(^2) +ζ′ (^2) )/ 2
√
z
1 −z^2
exp
[
2 zζζ′−z^2
(
ζ^2 +ζ′^2
)
1 −z^2
]
.
This yields the Wigner function:
Ω−^1
(
e−βĤ
)
(q,p)=
1
cosh (βω/2)
exp
{
−tanh (βω/2)
[
mω
q^2 +
p^2
mω
]}
.
(1.322)
Finally, using Eq. (1.322), we find with some long but elementary algebra:
Tr
[
Ω−^1
(
e−βĤ
)]
=
∫
dqdp
2 πΩ
− 1
(
e−βĤ
)
=
1
2sinh(βω/2), (1.323)
which is the expected result for the canonical partition function of the 1D harmonic
oscillator (see Example 1.3.1).