Propagators and Green’s Functions
• Diffusion equation (B 175)• Fick’s law combined with continuity equation
Fick’s Law
Continuity Equation
2Dt
L
L
),(),(.Dt
),(
),(),(.t
),(
),(D),(
ttt
ttt
tt
rrr
rrjr
rrj
j flux of solute, heat, etc. solute, heat, etc concentration solute, heat, etc source densityD diffusion constant
Propagators and Green’s Functions
• Propagator Ko(x,t,x’,t’) for linear pde in 1-D
• Evolves solution forward in time from t’ to t• Governs how any initial conditions (IC) will evolve• Solutions to homogeneous problem for particular IC, a(x)
• Subject to specific boundary conditions (BC)
• Ko satisfies
• LKo(x,t,x’,t’) = 0 t > t’
• Ko(x,t,x’,t’) = δ(x-x’) t = t’ (equal times)
• Ko(x,t,x’,t’) →0 as |x| → ∞ open BC
2DL )x()0tx,( 0L
t
a
0),( tr
))a(x't',x't,(x,Kdx't)(x, o
Propagators and Green’s Functions
t't a(x)))a(x'x'-(xdx'))a(x't,x',t(x,Kdx')tt(x,
t't 0))a(x't',x't,(x,LKdx'L
a(x) )-t(x, 0L
))a(x't',x't,(x,Kdx't)(x,
oooo
o
o
andCheck
using
• If propagator satisfies defining relations, solution is generated
• Propagator for 1-D diffusion equation with open BC
• Representations of Dirac delta function in 1-D
Propagators and Green’s Functions
t'-t 'xx
D4
1),(K 4
22/1
o
De
cba f(b)b)dx(xf(x) x
/
0
lim)x(
edk 2
1)x( x1
0
lim)x(
22
-
ikx 2/2
iffLorentzian
FourierGaussian
c
a
e
Propagators and Green’s Functions
0),(LK),(K
42
1
4
1 ),(K
42
1
4
1 ),(K
),(K),(LK
t'-t 'xx
D4
1),(K
oo2
2
2/5
2
2/3o2
2
2/5
2
2/3o
o2
o
2/1
o
4
2
4
2
4
2
D
eDD
D
eDD
Dt
e
D
D
D
• Check that propagator satisfies the defining relation
Propagators and Green’s Functions
times equal at 'xx),(K
1
0
lim)0,(K
4D D2
t'-t 'xx
D4
1),(K
o
o
2
2/1
o
2
2
4
2
e
e D
• Consider limit of Ko as tends to zero
D0 1;
Plot3DSqrt1 4 Pi D0 t Exp x x 4 D0 t, x, 1, 1, t, 0.1, 1,PlotRange 0, 1
Ko(,>0.00001)
Ko(,>0.001)
Ko(,>0.1)
Propagators and Green’s Functions
BC theby allowed r wavenumbeis k
of ioneigenfunct an is
p
2p
p
*ppo
22
2
2
2
2
(x)φ
)e(x'(x)φφ)t',x't;(x,K
ee X(x)T(t)t)ψ(x,
eT(t) eX(x)
DTk(t)T' Xk(x)'X'
0X(x)T(t)x
Dt
t)ψ(x,x
Dt
t'-tD2k
Dt 2kkx
Dt 2kkx
p
i
i
• Solution of diffusion equation by separation of variables • Expansion of propagator in eigenfunctions of
2 Sin(kx)e-k2Dt k=10
Sin(kx)e-k2Dt k=15
Propagators and Green’s Functions
0)t',x't;(x,Kk)D(Dk
eedk xt
)t',x't;(x,LK
)x'(xedk t)t',x't;(x,K
t' t
eedk )t',x't;(x,K
o22
2
2
o
o
o
t'-tD2k)x'-ik(x
)x'-ik(x
t'-tD2k)x'-ik(x
p
p
i
relation defining of onSatisfacti
times equal For
k on nrestrictio no is there BC open For p
Propagators and Green’s Functions
• Green’s function Go(r,t,r’,t’) for linear pde in 3-D (B 188)
• Evolves solution forward in time from t’ to t in presence of sources
• Solutions to inhomogeneous problem for particular IC a(r)
• Subject to specific boundary conditions (BC)
• Heat is added or removed after initial time ( ≠ 0)
• Go satisfies
• Go(x,t,x’,t’) = 0 t < t’
• Go(x,t,x’,t’) = δ(x-x’) t = t’ (equal times)
• Go(x,t,x’,t’) →0 as |x| → ∞ open BC
2DL )()0t,( L
t
a rr
)t'δ(t )'δ()t','t,,(Gt
LG o2
o
rrrr
Propagators and Green’s Functions
• Translational invariance of space and time
• Defining relation
• Solution in terms of propagator
t'- t ' - ,GG oo rrRR
0G 0 0 ,G
,GDt
oo
o2
RR
RRR
causality
d
d
0 0
0 1
,K ,G oo
function step Heaviside
RR
1
0
(x)
0
1
(-x)
Propagators and Green’s Functions
R
R
RR
RRR RR
0 0,K
,LK ,K
,KDt
,K ,KDt
o
oo
o2
oo2
• Check that defining relation is satisfied
• Exercise: Show that the solution at time t is
oooo
o
t,'ψt,'t,,G'dt','ρt','t,,G'dt
t
dt't),ψ( rrrrrrrrr
Green’s Function for Schrödinger Equation
• Time-dependent single-particle Schrödinger Equation
• Solution by separation of variables
)(V2
-)(H
0t),Ψ()(Ht
2
rr
rr
i
1
)()()(H
ti-)e(ct),Ψ(
nn
nnn
n
nnn
rrr
rr
Green’s Function for Schrödinger Equation
• Defining relation for Green’s function
• Eigenfunction expansion of Go
• Exercise: Verify that Go satisfies the defining relation LGo=
t'-t ' - )t't,,',(G)(Ht o rrrrr
i
t'-tt'-ti-
)e'()(t't,',Gn
n*nno rrrri
Green’s Function for Schrödinger Equation
• Single-particle Green’s function
t't,',Kt'tθt't,',G oo rrrr
time
Add particle Remove particle
t > t’t’
time
Remove particle Add particle
t’ > tt
Green’s Function for Schrödinger Equation
• Eigenfunction expansion of Go for an added particle (M 40)
nn
n
unoccn
*nn
unoccn 0
n
n
*nn
unoccn -
n*nn
oo
unoccn
n*nno
1-
e)'()(
-e
1-)'()(
-e d)'()(
t'-tet't,',Gt'-td,',G
t'-tt'-ti-
)e'()(t't,',G
i
ii
i
ii
i
i
rr
rr
rr
rrrr
rrrr
Green’s Function for Schrödinger Equation
• Eigenfunction expansion of Go for an added particle
planecomplex in axis realbelow i.e.at poles
Function sGreen' Retarded
forfinitefor
malinfinitesi positive
)'()(
,',G
-e
1-)'()(
t'- t t'-t t'- t 0t'-t
..
-e d)'()(,',G
n
unoccn n
*nn
o
unoccn 0
n
n
*nn
nn
unoccn -
n*nno
i
i
iiii
i
iei
iii
rrrr
rr
rrrr
Green’s Function for Schrödinger Equation
• Eigenfunction expansion of Go for an added hole
planecomplex in axis real above i.e. at Poles
Function sGreen' Advanced
i
i
iii
iii
iii
i
n
occn n
*nn
o
occn n
*nn
occn
0
-
n*nn
occn -
n*nno
occn
n*nno
)'()(
,',-G
01-
)'()(
-e d)'()(
-e d)'()(,',-G
t'-tt'-ti-
)e'()(t't,',-G
rrrr
rr
rr
rrrr
rrrr
Green’s Function for Schrödinger Equation
• Poles of Go in the complex energy plane
Im()
Re()x x xx xxx x x xxxxxx xx xx xxx x x xxx
F
Advanced (holes)
Retarded (particles)
Contour Integrals in the Complex Plane
• Exercise: Fourier back-transform the retarded Green’s function
)(
ie)'()ψ(ψ),',(G
i-
e
2
d)'ψ()(ψ),',(G
),',(Ge2
d),',(G
n
unoccn
*nno
nunoccn
*nno
oo
ii
i
i
rrrr
rrrr
rrrr
obtain to steps missing the in Fill
Green’s Function for Schrödinger Equation
• Spatial Fourier transform of Go for translationally invariant system
ii
i
i
i
i
-i
-i
i-i
i
i
kk
k
k
k
kk k
kk
k-kk
kkk
kkrrkk
kkk
kkrrk
kkrr
kk
kkkrr
kr
rrkk
rrkk
rrkrrk
rrk
k.r
)( ),(G
)( ),(G
)'( e2
)'d(
)-()'(d
)-(
e2
)'d(d
e)-(e
)'d(2
d),'(G
2
d
V
1
)-(e
V
1),',(G
VL dL
2 e
V
1)(ψ
Fo
Fo
3F
-
3
F3
-
3
F
-3
3
o
3
3F
o
3DD
)'-).('(
)'-).('(
)'-'.()'-.(
)'-.(
Von ion'normalisatbox '
Functions of a Complex Variable
• Cauchy-Riemann Conditions for differentiability (A 399)
x
y
Complex planef(z) = u(x,y)+iv(x,y)z = x + iy = rei
functionentire
z atanalytic o
an is f(z) plane,complex wholethe in abledifferenti is f If
is f(z) ,z about region small a in abledifferenti is f If
conditions Riemann-Cauchy the areand
zero approaches
whichin direction of tindependen be must limit
o
y
u
x
v
y
v
x
u
yi δx δz
δx
v
y
u
y
v
δx
u
yδx
vu
δz
f(z)δz)f(z (z)' f
y)v(x,y)u(x, f(z)
y x z
lim0δz
i
ii
i
i
i
i
i
Functions of a Complex Variable
• Non-analytic behaviour
io2
o1
x-x
f(x)dx I
x-x
f(x)dx I
1 .0 0 .5 0 .5 1 .0 1 .5 2 .0
20
10
10
20
PlotSin4 x x 1, x, 1, 2, PlotRange 20, 20A pole in a function renders the function non-analytic at that point
Functions of a Complex Variable
• Cauchy Integral Theorem (A 404)
R in path closedevery For
R region, connectedsimply some in valued)-single (andanalytic f(z)
C
x
y
C
0f(z)dz C
theorem Stokes' using Proved
0 )f(z unless z z atanalytic not is z-z
f(z)
not is z and C on is z since defined wellis z-z
f(z)dz
contour the onanalytic f(z)
planecomplex the in contour closed a is
ooo
o
C o
C
Functions of a Complex Variable
• Cauchy Integral Formula (A 411)
integral of sign changes nintegratio of direction Changing
clockwise-anti is nintegratio of Direction
x
y
zo
C
)f(zz-z
f(z)dz
2
1o
C o
i
Functions of a Complex Variable
• Cauchy Integral Formula (A 411)
0z-z
f(z)dz g(z)
z-z
f(z)
)f(zz-z
f(z)dz
2
1
)f(z2z-z
f(z)dz
)f(z2er
der )er f(zlim
z-z
f(z)dzlim
der zer z-z
0z-z
f(z)dz
z-z
f(z)dz
C oo
o
C o
o
C o
o
C
o
C o
o
C oC o
22
2
0r0r
d
andanalytic isthen outside lies z If
theorem integralCauchy From
o C
i
i
ii
i
i
ii
ii
x
y
zoC
C2
Functions of a Complex Variable
• Taylor Series (A 416)• When a function is analytic on and within C containing a point zo
it may be expanded about zo in a Taylor series of the form
• Expansion applies for |z-zo| < |z-z1| where z1 is nearest non-analytic point• See exercises for proof of expansion coefficients
Cn i 1n
o
on
n0
non )z'-z(
)dz'f(z'
2
1
!n
)(zfa )z-z(af(z)
x
y
zo
C
...)z-(za)z-(zaaf(z) 2o2o1o
Functions of a Complex Variable
• Laurent Series (A 416)
• When a function is analytic in an annular region about a point zo
it may be expanded in a Laurent series of the form
• If an = 0 for n < -m < 0 and a-m = 0, f(z) has a pole of order m at zo
• If m = 1 then it is a simple pole• Analytic functions whose only singularities are separate poles are termed
meromorphic functions
Cn i 1n
on
non )z'-z(
)dz'f(z'
2
1a )z-z(af(z)
x
y
zo
C
...
z-z
a
z-z
a...)z-(za)z-(zaaf(z) 2
o
2-
o
1-2o2o1o
Contour Integrals in the Complex Plane
• Cauchy Residue Theorem (A 444)
proof for sheet tutorial See
1) (m pole simple
pole order mth
o
o
zzo1
zz
mo1-m
1-m
1
f(z)zza
f(z)zzdz
d
)!1m(
1a
residues
atof the is
Let
2f(z)dz
z f(z) a
1n a 2e
e adz)z-(za
e z-z
1n 01n
)z-(zadz)z-(za
dz)z-(zaf(z)dz
o1-
1-
2
0i
i
1-1-
o1-
io
1no
nn
on
non
1
1
i
ir
dir
r
C
C
z
zC
n CC
residue
Contour Integrals in the Complex Plane
• Cauchy Residue Theorem (A 444)
1
limaz
limaz
limaz
aAba
1
bz
1
bz
Baz
az
Aazf(z)az
bz
B
az
A
bzaz
1f(z)
fractions partialby
poles simple two has f(z) :Example
Contour Integrals in the Complex Plane
• Integration along real axis in complex plane
• Provided:
• f(z) is analytic in the UHP
• f(z) vanishes faster than 1/z
• Can use LHP (lower half plane) if f(z) vanishes faster than 1/z and f(z) is analytic there
• Usually can do one or the other, same result if possible either way
plane) half(upper in UHP residues 2
dRe)f(Relimf(x)dxlimf(z)dz0
R
R-RR
i
i ii
enclosed polex
y
-R +R
plane) half(upper in UHP residues 2 f(x)dx-
i
Contour Integrals in the Complex Plane
• Integration along real axis in complex plane• Theta function (M40)
)0( 0
0
)0( e 2
e20
e
2
d
e
2
d
e
2
de
2
d
0 2
e
2
e
)0( 0
)0( 1 e
2
d)(
for plane 1/2 upper in contour close
for plane 1/2 lower in contour close
residueat pole
ii
i
i
i
i
i
ii
i
ii
i
i
ii
iii
i
i
i
LHPC
x
y
C
i
t >0
• Integration along real axis in complex plane• Principal value integrals – first order pole on real axis• What if the pole lies on the integration contour?
• If small semi-circle C1 in/excludes pole contribution appears twice/once
Contour Integrals in the Complex Plane
iseanticlockw similarly clockwise, re
dre
x-z
dz
dre dzre x- zLet
residues enclosed 2f(z)dzf(x)dxf(z)dzf(x)dxf(z)dz
0
o
o
δx
δx
-
1
o 21
o
iii
i
i
i
i
ii
C
CC
x
y
-R +R
C1
C2
-δx
δx
-
f(x)dxf(x)dxf(x)dxlimo
o
0P
Contour Integrals in the Complex Plane
• Kramers-Kronig Relations (A 469)
0) (y x-x
f(x)dx1 0)(y
yx-x
dxx-xf(x)1)f(z
yx-x
x-x2
z-x
1
z-x
1
z-x
1
z-x
1f(x)dx
2
1)f(z
iyxz 0z-x
f(x)dx
2
1
iyxz )f(zz-x
f(x)dx
2
1
)f(zz-z
f(z)dz
2
1
o
- oo
-2o
2o
oo
2o
2o
o
o
_
o- o
_
o
o
ooo
- o
_
oooo
- o
o
C o
Pii
i
i
i
i
contour side at pole
contour side at pole
formula integralCauchy
_
out
in
+R →+
zo
žo
x
y
-R→-
Contour Integrals in the Complex Plane
• Kramers-Kronig Relations
0) (y xx
u(x,0)dx1,0)v(x
0) (y xx
v(x,0)dx1 ,0)u(x
0) (y yxx
u(x,0)dxxx1)y,v(x
0) (y yxx
v(x,0)dxxx1)y,u(x
yxx
v(x,0))dx(u(x,0)xx1)y,v(x)y,u(x)f(z
y)v(x,y)u(x, y)f(x, yxx
f(x)dxxx1)f(z
o
- oo
o
- oo
o
-2o
2o
ooo
o
-2o
2o
ooo
-2o
2o
oooooo
-2o
2o
oo
partsimaginary Equate
parts real Equate
partsimaginary Equate
parts real Equate
-P
-P
-
-
-
-
-
i-
ii
i-
-
i