Propagators and Green’s Functions
• Diffusion equation (B 175)
• Fick’s law combined with continuity equation
j(r, t )  D (r, t )
 (r, t )
 .j(r, t )   (r, t )
t
 (r, t )
 D. (r, t )   (r, t )
t
L  
L

 D 2
t
j


D
Fick’s Law
Continuity Equation
flux of solute, heat, etc.
solute, heat, etc concentration
solute, heat, etc source density
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  (x, t)   dx' K o (x, t, x' , t' )a(x' )

Governs how any initial conditions (IC) will evolve
Solutions to homogeneous problem  (r, t )  0 for particular IC, a(x)
Subject to specific boundary conditions (BC)

L  0  ( x, t  0)  a( x ) L   D 2
t
•
•
•
•
Ko satisfies
LKo(x,t,x’,t’) = 0
Ko(x,t,x’,t’) = δ(x-x’)
Ko(x,t,x’,t’) →0 as |x| → ∞
t > t’
t = t’
(equal times)
open BC
Propagators and Green’s Functions
• If propagator satisfies defining relations, solution is generated

using  (x, t)   dx' K o (x, t, x' , t' )a(x' )

Check L  0 and  (x, t  -)  a(x)

L   dx' LK o (x, t, x' , t' )a(x' )  0
t  t' 





 (x, t  t o )   dx' K o (x, t o , x' , t o )a(x' )   dx'  (x - x' )a(x' )  a(x)
t  t'
Propagators and Green’s Functions
• Propagator for 1-D diffusion equation with open BC
1/ 2
 1 
K o ( , )  

4

D



  x  x '   t - t'
2

e 4 D
• Representations of Dirac delta function in 1-D
Gaussian
 (x) 
lim

0
1

e x / 
 /
Lorentzian  ( x ) 
  0 x2   2
lim
2
2
Fourier
c
1
 (x) 
2
a f(x)  (x  b)dx  f(b)

ikx
dk
e

-
iff a  b  c
Propagators and Green’s Functions
• Check that propagator satisfies the defining relation
1/ 2
 1 
K o ( , )  

 4D 
  x  x '   t - t'
2

e 4 D


LK o ( , )    D 2 K o ( , )
 t

2

 4 D
e

1 
1

  3 / 2 
K o ( , ) 

4 D 5 / 2 
4D  2
2
D
2

 4 D
e

1 
1


K
(

,

)



o
3/ 2
 2
4 D 5 / 2 
4D  2
2
 
2 

 D 2 K o ( , )  LK o ( , )  0
 
 
2
Propagators and Green’s Functions
Ko(x,>0.00001)
• Consider limit of Ko as  tends to zero
1/ 2
 1 
K o ( , )  

4

D



  x  x '   t - t'
2

e 4 D

Ko(x,>0.001)
  2 D  2  4D
K o ( ,  0) 
lim
2


1
 0

2
e     
K o ( , )   x  x ' at equal times
D0

Ko(x,>0.1)
1;
Plot3D Sqrt 1
PlotRange
4 Pi D0 t
0, 1
Exp
xx
4 D0 t
, x, 1, 1 , t, 0.1, 1 ,

Propagators and Green’s Functions
• Solution of diffusion equation by separation of variables
2
• Expansion of propagator in eigenfunctions of 
Sin(kx)e-k2Dt k=10


2 
2 
  D 2 ψ(x, t)    D 2 X(x)T(t)  0
x 
x 
 t
 t
X' ' (x)   k 2 X T' (t)   k 2 DT
X(x)  e
ikx
T(t)  e
ψ(x, t)  X(x)T(t)  e
k 2Dt
2
ikx  k Dt
e
K o (x, t; x' , t' )   φ p (x)φ (x' )e
*
p
Sin(kx)e-k2Dt k=15
k 2p D t -t'
p
φ p (x) is an eigenfunct ion of  2
k p is wavenumbe r allowed by the BC
Propagators and Green’s Functions
For open BC there is no restrictio n on k p

K o (x, t; x' , t' )   dk e
ik(x -x' ) k 2p D t -t'
e

For equal times t  t'

K o (x, t; x' , t'  t)   dk eik(x -x' )   (x  x' )

Satisfacti on of defining relation
  2  
ik(x -x' ) k 2p D t -t'
e
LK o (x, t; x' , t' )    2   dk e
 t x  
  k 2 D  D(ik) 2 K o (x, t; x' , t' )  0


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)
L    (r, t  0)  a(r )

L   D 2
t
• Go satisfies


LG o     2 G o (r, t, r ' , t' )  δ(r  r ' ) δ(t  t' )
 t

• 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
Propagators and Green’s Functions
• Translational invariance of space and time
G o  G o R,  R  r - r'   t - t'
• Defining relation

2 
  D R G o R,    R    
 t

G o R,   0   0 causality G o  0  R  
• Solution in terms of propagator
(x)
1
G o R,     K o R, 
    1   0 Heaviside step function
    0   0
d  
   
d
0
(-x)
1
0
Propagators and Green’s Functions
• Check that defining relation is satisfied
  



2 
K o R,       D 2R K o R, 
  D R   K o R,  

 t

 t

    K o R,     LK o R, 
    K o R,  0   0
     R 
• Exercise: Show that the solution at time t is
t
ψ(r, t)   dt'  dr 'G o r, t, r ' , t' ρr ' , t'    dr 'G o r, t, r ' , t o ψr ' , t o 
to
Green’s Function for Schrödinger Equation
• Time-dependent single-particle Schrödinger Equation
 

i

Ĥ
(
r
)

Ψ(r, t)  0
 t

2
Ĥ(r )   V̂(r )
2
• Solution by separation of variables
Ψ(r, t)   c n n (r )e
n
Ĥ(r ) n (r )   n n (r )
 n  n   1
- i n t
Green’s Function for Schrödinger Equation
• Defining relation for Green’s function
 

 i  Ĥ(r ) G o (r, r ' , t, t' )   r - r ' t - t' 
 t

• Eigenfunction expansion of Go
iG o r, r' , t  t'    n (r) n* (r' )e
- i n t - t' 
 t - t' 
n
• Exercise: Verify that Go satisfies the defining relation LGo= 
Green’s Function for Schrödinger Equation
• Single-particle Green’s function
G o r, r' , t  t'   θt  t' K o r, r' , t  t' 
Add particle
Remove particle
t’
Remove particle
t > t’
time
Add particle
t
t’ > t
time
Green’s Function for Schrödinger Equation
• Eigenfunction expansion of Go for an added particle (M 40)
G o r, r ' , t  t'   i
*

(
r
)

 n n (r' )e
- i n t - t' 
 t - t' 
nunocc

G o r, r ' ,     d t - t' G o r, r ' , t  t' e
 i  t - t' 

 i


 n (r ) n* (r ' )  d e
nunocc
- i  n   
  
-
-1
- i  n   
*
 i   n (r ) n (r ' )
e
i  n   
nunocc
 e- i  n   
1

*
  n (r ) n (r ' )




n 

nunocc
n






0
Green’s Function for Schrödinger Equation
• Eigenfunction expansion of Go for an added particle
G o r, r ' ,    i


 n (r ) n* (r ' )  d e
nunocc
- i  n  i   
  
-
i.e.  n   n  i  positive infinitesi mal
 t - t'   0 for t - t' finite  t - t'    for t - t'  
 i

 n (r ) n* (r ' )
nunocc
-1
- i  n  i   
e
i  n  i   
 n (r ) n* (r ' )
G r, r ' ,    
Retarded Green' s Function
nunocc    n  i

o
poles at    n  i i.e. below real axis in complex plane

0
Green’s Function for Schrödinger Equation
• Eigenfunction expansion of Go for an added hole
- i t - t' 
G - r, r ' , t  t'   i  (r ) * (r ' )e n
 t'-t
o

nocc
n
n

G -o r, r ' ,    i   n (r ) n* (r ' )  d e
nocc
- i  n  i   
-
0
 i   n (r ) n* (r ' )  d e
nocc
   
- i  n  i   
-


-1
 i   n (r ) n* (r ' )
 0 
nocc
 i  n  i    
*

(
r
)

n (r ' )
G -o r, r ' ,     n
Advanced Green' s Function
nocc    n  i
Poles at    n  i i.e. above real axis in complex plane
Green’s Function for Schrödinger Equation
• Poles of Go in the complex energy plane
Im()
F
Advanced (holes)
x x xx xxx x x xxx
xxx xx xx xxx x x xxx
Retarded (particles)
Re()
Contour Integrals in the Complex Plane
• Exercise: Fourier back-transform the retarded Green’s function

d  i 
G (r, r ' , )  
e
G o (r, r ' ,  )
2


o
d e  i
G (r, r ' , )   ψ n (r ) ψ(r ' ) 
2  -  n  i
nunocc



o
*
n
Fill in the missing steps to obtain
 i  n  i 
G o (r, r ' , )  i  ψ n (r )ψ*n (r ' )e
 ( )
nunocc
Green’s Function for Schrödinger Equation
• Spatial Fourier transform of Go for translationally invariant system
1 ik.r
ψ k (r ) 
e
' box normalisat ion' on V
V
 2 


 L 
ik .( r -r ')
1
e
 (k - k F )
G o (r, r ' ,  )  
V k
   k  i

D

k
  d Dk L3  V
1
d 3k


V k
2 3

ik .( r -r ')
3
d
k
e
 (k - k F ) -ik '.(r-r ')
G o (k ' ,  )  
d(
r

r
'
)
e
3 
   k  i
- 2   


d(r  r ' ) i (k-k ').( r-r ')  (k - k F )
e
3
   k  i
  2 
  d 3k 
-

 (k - k F )
  d 3k (k  k ' )
   k  i
-
 (k  k F )
 (k F - k )
G o (k ,  ) 
G o (k ,  ) 
   k  i
   k  i

d(r  r ' ) i (k-k ').( r -r ')
  (k  k ' )
 2 3 e
Functions of a Complex Variable
• Cauchy-Riemann Conditions for differentiability (A 399)
z  x  iy
f(z)  u(x, y)  iv(x, y)
u  iv u iv u iv
lim f(z  δz)  f(z)
f ' (z)  δz0



δz
δx  iy δx iy iy δx
limit must be independen t of direction in which δz  δx  iy
approaches zero
u v
v
u

and

x y
x
y
y
x
Complex plane
f(z) = u(x,y)+iv(x,y)
z = x + iy = reif
are the Cauchy - Riemann conditions
If f is differenti able in a small region about z o , f(z) is analytic at zo
If f is differenti able in the whole complex plane, f(z) is an entire function
Functions of a Complex Variable
• Non-analytic behaviour
20
10

I1 
dx f(x)
 x - x o
1.0
0.5
0.5
1.0
1.5
2.0

I2 
dx f(x)
 x - x o  i
10
20
Plot Sin 4 x
x 1 , x, 1, 2 , PlotRange
A pole in a function renders the function non-analytic at that point
20, 20
Functions of a Complex Variable
• Cauchy Integral Theorem (A 404)
f(z) analytic (and  single - valued) in some simply connected region, R
For every closed path C in R
 f(z)dz  0
y
C
C x
Proved using Stokes' theorem
Functions of a Complex Variable
• Cauchy Integral Formula (A 411)
C is a closed contour in the complex plane
f(z) analytic on the contour
f(z)dz
C z - zo is well defined since z is on C and zo is not
f(z)
is not analytic at z  z o unless f(z o )  0
z - zo
y
1 f(z)dz
 f(z o )

2i C z - z o
Direction of integratio n is anti - clockwise
Changing direction of integratio n changes sign of integral
zo
C x
Functions of a Complex Variable
• Cauchy Integral Formula (A 411)
From Cauchy integral theorem
f(z)dz
f(z)dz

C z - z o C z - z o  0
2
y
zo C2
z - z o  r ei dz  ir ei d
lim f(z  r ei )ir ei d
f(z)dz
r0
r0 o
 2if(z o )
i

z - zo
re
C2
C2
lim
f(z)dz
C z - z o  2if(z o )
1 f(z)dz
 f(z o )

2i C z - z o
If z o lies outside C then
f(z)
f(z)dz
 g(z) is analytic and 
0
z - zo
z - zo
C
C x
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

f(z)   a n (z - z o )
n 0
n
f n (z o )
1
f(z' )dz'
an 

n!
2i C (z'-zo ) n 1
y
zo
C x
f(z)  a o  a1 (z - z o )  a 2 (z - z o ) 2  ...
• Expansion applies for |z-zo| < |z-z1| where z1 is nearest non-analytic point
• See exercises for proof of expansion coefficients
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
y
zo
f(z) 

 a n (z - z o ) n
n  
an 
1
f(z' )dz'
2i C (z'-zo ) n 1
f(z)  a o  a1 (z - z o )  a 2 (z - z o ) 2  ... 
C x
a -1
a -2

 ...
2
z - z o z - z o 
• 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
Contour Integrals in the Complex Plane
• Cauchy Residue Theorem (A 444)

 f(z)dz   a  (z - z
n  
C
n
m th order pole
n
o
) dz
C
n 1 z1
(z - z o )
a n  (z - z o ) dz  a n
n 1
C
n
 0 n  1
z1
Let z - z o  r ei
2
i
ir e d
 2i a -1 n  1
i
re
0
a -1  (z - z o ) -1 dz  a -1 
C
a -1 is the residue of f(z) at z o
 f(z)dz  2i  residues
C

1
d m-1
m


a 1 
z

z
f(z)
o
m -1
(m  1)! dz
simple pole (m  1)
a 1  z  z o f(z) z  z o
See tutorial sheet for proof

z zo
Contour Integrals in the Complex Plane
• Cauchy Residue Theorem (A 444)
Example : f(z) has two simple poles
f(z) 
1
z  a z  b 
A
B


z  a  z  b 
by partial fractions
lim
z a
z  a A  z  a B
z  a f(z)  lim
z a
z  a  z  b 
1

z  b 
1

 A  a 1
a  b 
lim
z a
Contour Integrals in the Complex Plane
• Integration along real axis in complex plane
 f(z)dz  lim
R 
R
 f(x)dx  lim
-R
R 

i
i
f(Re
)
i
Re
d

y
0
 2i  residues in UHP (upper half plane)
enclosed pole
-R
• Provided:
• f(z) is analytic in the UHP
• f(z) vanishes faster than 1/z

 f(x)dx  2i  residues in UHP (upper half
plane)
-
• 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
x
+R
Contour Integrals in the Complex Plane
• Integration along real axis in complex plane
• Theta function (M40)

d e  i
 ( )  
 1 (  0)
2

i


i


y
 0 (  0)
 i  i 
e
e  
pole at   i residue 

2i
2i
close contour in lower 1/2 plane for   0
t >0
x
  i
C

d e  i
d e  i
d e  i
C 2i   i   2i   i  LHP 2i   i
d e  i
e  
 
 0  2i
 e   (  0)
2i   i
2i


close contour in upper 1/2 plane for   0
0
(  0)
Contour Integrals in the Complex Plane
• 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?
x o δ

 f(z)dz   f(x)dx   f(z)dz  
-
C1
x o δ
y
C2
C1
-R
f(x)dx   f(z)dz  2i  enclosed residues
C2
Let z - x o  re i dz  irei d
dz
ire i d
C z - x o   rei  i clockwise, similarly  i anticlockw ise
1
0

• If small semi-circle C1 in/excludes pole contribution appears twice/once
lim
 0
x o δ


-
x o δ
-
 f(x)dx   f(x)dx  P  f(x)dx
x
+R
Contour Integrals in the Complex Plane
y
• Kramers-Kronig Relations (A 469)
1 f(z)dz
 f(z o )

2i C z - z o
1
2i
1
2i
zo
Cauchy integral formula
-R→-

f(x)dx
- x - z o  f(z o )


-
f(x)dx
_
0
pole at z o  x o  iy o in side contour
_
pole at z o  x o  iy o outside contour
x - zo

 1
1
1 

f(z o ) 
f(x)dx

_ 


2i -
 x - zo x - zo 
1
f(z o ) 
i
1

x - zo
1
_
x - zo

2x - x o 
x - x o 2  y o2
f(x) x - x o dx
1
f(x)dx
(y

0)

P
(y o  0)
- x - x o 2  yo2 o

i - x - x o


žo
x
+R →+
Contour Integrals in the Complex Plane
• Kramers-Kronig Relations
1 x-x o f(x)dx
f(z o )  
f(x, y)  u(x, y)  iv(x, y)
2
2
i - x-x o   y o

1
f(z o )  u(x o , y o )  iv(x o , y o ) 
i
x-x o v(x,0)dx
 - x-x o 2  y o2

1 x-x o u(x,0)dx
v(x o , y o )   
 - x-x o 2  y o2
u(x o , y o ) 
1
1

v(x o ,0)  

Equate real parts (y o  0)
Equate imaginary parts (y o  0)

v(x,0)dx
u(x o ,0) 
P
 - x-x o 
1
x-x o (u(x,0)  iv(x,0))dx
-
x-x o 2  y o2

Equate real parts (y o  0)

u(x,0)dx
x-x o 
-
P
Equate imaginary parts (y o  0)