In the Dirac notation one has
ψ(qt,t) =〈qt|ψ(t)〉=〈qt|e−iHt|ψ(0)〉=〈qt|e−iHt
∫∞
−∞
|q 0 〉〈q 0 |ψ(0)〉dq 0
and the propagator can be written as
U(t,qt,q 0 ) =〈qt|e−iHt|q 0 〉
U(t,qt,q 0 ) can be computed for the free particle case by Fourier transform
of the momentum space multiplication operator:
ψ(qt,t) =
1
√
2 π
∫+∞
−∞
eikqtψ ̃(k,t)dk
=
1
√
2 π
∫+∞
−∞
eikqte−i
1
2 mk
(^2) t ̃
ψ(k,0)dk
=
1
2 π
∫+∞
−∞
eikqte−i
21 mk^2 t
(∫+∞
−∞
e−ikq^0 ψ(q 0 ,0)dq 0
)
dk
=
∫+∞
−∞
(
1
2 π
∫+∞
−∞
eik(qt−q^0 )e−i
21 mk^2 t
dk 0
)
ψ(q 0 ,0)dq 0
so
U(t,qt,q 0 ) =U(t,qt−q 0 ) =
1
2 π
∫+∞
−∞
eik(qt−q^0 )e−i
21 mk^2 t
dk (12.5)
Note that (as expected due to translation invariance of the Hamiltonian opera-
tor) this only depends on the differenceqt−q 0. Equation 12.5 can be rewritten
as an inverse Fourier transform with respect to this difference
U(t,qt−q 0 ) =
1
√
2 π
∫+∞
−∞
eik(qt−q^0 )U ̃(t,k)dk
where
U ̃(t,k) =√^1
2 π
e−i
21 mk^2 t
(12.6)
To make sense of the integral 12.5, the productitcan be replaced by a
complex variablez=τ+it. The integral becomes well-defined whenτ= Re(z)
(“imaginary time”) is positive, and then defines a holomorphic function inz.
Doing the integral by the same method as in equation 11.6, one finds
U(z=τ+it,qt−q 0 ) =
√
m
2 πz
e−
m
2 z(qt−q^0 )
2
(12.7)
Forz=τreal and positive, this is the kernel function for solutions to the
partial differential equation
∂
∂τ
ψ(q,τ) =
1
2 m
∂^2
∂q^2
ψ(q,τ)