A Short Course on Quantum Mechanics and Methods of Quantization 43
Recalling that (i) the classical action is given by:
S=
∫t
0
dt′
m
2
(
dx
dt
) 2
(1.208)
and (ii) the solution of the classical equation of motiond^2 x(t)/dt^2 = 0, satisfying
the boundary conditionsx(0) =x′,x(t)=x, yields the classical trajectory:
xcl(t′)=
t′(x−x′)
t
+x′, x ̇cl(t′)=
x−x′
t
, (1.209)
one can easily verify that:
Sclas(x, x′)≡S|xcl=
∫t
0
dt′
m
2
(
x−x′
t′
) 2
. (1.210)
Comparing with (1.207), we see that the quantum kernel is given by the exponen-
tial of the classical action:
〈x|e−ıtH|x′〉=F(t)exp
[ı
Sclas(x, x′)
]
,F(t)=
√
m
2 πit
. (1.211)
up to a pre-factorF(t) which depends only on time. It is an interesting fact that
it can be calculated exactly by also makinguse of a semi-classical stationary phase
approximation[ 35 ].
Example 1.3.3. The 1D harmonic oscillator.
Similar calculations can be done for the Hamiltonian of the 1D harmonic oscillator,
to get:
K(x, x′;t)=Fho(t)exp
[ı
Sclas(x, x′)
]
, (1.212)
with
Scl(x, x′)=
ωm
2 sin(ωt)
[
(x+x′)^2 cos(ωt)− 2 xx′
]
, (1.213)
Fho=
√
mω
2 πısin(ωt). (1.214)
As in the previous example,Fhocan be calculated exactly by means of the sta-
tionary phase approximation[ 35 ].
1.3.2.2 Feynman integral in imaginary time and partition function
For a quantum system, the (canonical) partition function can be calculated as:
Z≡Tr
[
e−βH
]
=
∫
dx〈x|e−βH|x〉. (1.215)