Propagators and Green’s Functions

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


• 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


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


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


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


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)


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


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


• 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


• 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)


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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


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


• 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→-


• 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

Documents

Propagators and Green’s Functions