Examples of frequency spectra 549(Shankar, 1994), where the action of ˆaand ˆa†on an energy eigenstate of the oscillator
is
aˆ|n〉=
√
n|n− 1 〉, aˆ†|n〉=√
n+ 1|n+ 1〉. (14.4.2)These operators satisfy the commutation relation [ˆa,ˆa†] = 1. In terms of the creation
and annihilation operators, the Hamiltonian for a harmonic oscillator of frequencyω 0
can be written as
Hˆ 0 =(
aˆ†ˆa+1
2
)
̄hω 0. (14.4.3)In the interaction picture, the operators ˆaand ˆa†evolve according to the equations of
motion
dˆa
dt=
1
i ̄h[ˆa,Hˆ 0 ] =−iω 0 aˆdˆa†
dt=
1
i ̄h[ˆa†,Hˆ 0 ] =iω 0 ˆa†. (14.4.4)Eqns. (14.4.4) are readily solved to yield
aˆ(t) = ˆae−iω^0 t, ˆa†(t) = ˆa†eiω^0 t. (14.4.5)Using eqn. (10.4.19), the correlation function〈xˆ(0)ˆx(t)〉can be written as
〈ˆx(0)ˆx(t)〉=̄h
2 mω 01 −e−β ̄hω^0
e−β ̄hω^0 /^2×
∑∞
n=0e−(n+1/2)β ̄hω^0 〈n|(ˆa+ ˆa†)(ˆae−iω^0 t+ ˆa†eiω^0 t)|n〉. (14.4.6)After some algebra, we find that
〈xˆ(0)ˆx(t)〉=̄h
4 mω 0 sinh(β ̄hω 0 /2)[
eiω^0 teβ ̄hω^0 /^2 + e−iω^0 te−β ̄hω^0 /^2]
. (14.4.7)
Similarly, the correlation function〈xˆ(t)ˆx(0)〉can be shown to be
〈xˆ(t)ˆx(0)〉=̄h
4 mω 0 sinh(β ̄hω 0 /2)[
e−iω^0 teβ ̄hω^0 /^2 + eiω^0 te−β ̄hω^0 /^2]
. (14.4.8)
When we combine eqns. (14.4.7) and (14.4.8), the symmetric correlation function is
found to be
1
2
〈
[ˆx(0),ˆx(t)]+〉
=
̄h
2 mω 0tanh(β ̄hω 0 /2) cos(ω 0 t). (14.4.9)A comparison of eqn. (14.4.9) and eqn. (13.3.39) reveals that the quantum and classical
correlation functions are related by
1
2〈
[ˆx(0),ˆx(t)]+〉
=
β ̄hω 0
2tanh(β ̄hω 0 /2)〈x(0)x(t)〉cl. (14.4.10)The connection established in eqn. (14.4.10) between the quantum and classical posi-
tion autocorrelation functions of a harmonic oscillator can be exploited as a method