Quantum Mechanics for Mathematicians

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)