From Classical Mechanics to Quantum Field Theory

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).