1. Propagators and Green’s Functions
Diffusion equation (B 175)
Fick’s law combined with continuity equation
Fick’s Law
Continuity Equation
j flux of solute, heat, etc.
y solute, heat, etc concentration
r solute, heat, etc source density
D diffusion constant

2. 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’) = t > t’
Ko(x,t,x’,t’) = δ(x-x’) t = t’ (equal times)
Ko(x,t,x’,t’) →0 as |x| → ∞ open BC

3. Propagators and Green’s Functions
If propagator satisfies defining relations, solution is generated

4. Propagators and Green’s Functions
Propagator for 1-D diffusion equation with open BC
Representations of Dirac delta function in 1-D

5. Propagators and Green’s Functions
Check that propagator satisfies the defining relation

6. Propagators and Green’s Functions
Ko(x,t> )
Consider limit of Ko as t tends to zero t
Ko(x,t>0.001) t
Ko(x,t>0.1) t

7. Propagators and Green’s Functions
Solution of diffusion equation by separation of variables
Expansion of propagator in eigenfunctions of
Sin(kx)e-k2Dt k=10
Sin(kx)e-k2Dt k=15

8. Propagators and Green’s Functions

9. 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 (r ≠ 0)
Go satisfies
Go(x,t,x’,t’) = t < t’
Go(x,t,x’,t’) = δ(x-x’) t = t’ (equal times)
Go(x,t,x’,t’) →0 as |x| → ∞ open BC

10. Propagators and Green’s Functions
Translational invariance of space and time
Defining relation
Solution in terms of propagator 1
q(x) 1
q(-x)

11. Propagators and Green’s Functions
Check that defining relation is satisfied
Exercise: Show that the solution at time t is

12. Green’s Function for Schrödinger Equation
Time-dependent single-particle Schrödinger Equation
Solution by separation of variables

13. 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= d

14. Green’s Function for Schrödinger Equation
Single-particle Green’s function
time
Add particle
Remove particle
t > t’ t’
t’ > t t

15. Green’s Function for Schrödinger Equation
Eigenfunction expansion of Go for an added particle (M 40)

16. Green’s Function for Schrödinger Equation
Eigenfunction expansion of Go for an added particle

17. Green’s Function for Schrödinger Equation
Eigenfunction expansion of Go for an added hole

18. Green’s Function for Schrödinger Equation
Poles of Go in the complex energy plane
Im(e)
Re(e)
x x xx xxx x x xxx
xxx xx xx xxx x x xxx eF
Advanced (holes)
Retarded (particles)

19. Contour Integrals in the Complex Plane
Exercise: Fourier back-transform the retarded Green’s function

20. Green’s Function for Schrödinger Equation
Spatial Fourier transform of Go for translationally invariant system

21. Functions of a Complex Variable
Cauchy-Riemann Conditions for differentiability (A 399) x y
Complex plane
f(z) = u(x,y)+iv(x,y)
z = x + iy = reif

22. Functions of a Complex Variable
Non-analytic behaviour
A pole in a function renders the function non-analytic at that point

23. Functions of a Complex Variable
Cauchy Integral Theorem (A 404) y C x

24. Functions of a Complex Variable
Cauchy Integral Formula (A 411) x y zo C

25. Functions of a Complex Variable
Cauchy Integral Formula (A 411) x y zo C C2

26. 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 x y zo C

27. 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 x y zo C

28. Contour Integrals in the Complex Plane
Cauchy Residue Theorem (A 444)

29. Contour Integrals in the Complex Plane
Cauchy Residue Theorem (A 444)

30. 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
enclosed pole x y -R +R

31. Contour Integrals in the Complex Plane
Integration along real axis in complex plane
Theta function (M40) x y C
t >0

32. Contour Integrals in the Complex Planey -R +R C1 C2
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

33. Contour Integrals in the Complex Planezo žo x y
-R→-
Kramers-Kronig Relations (A 469)

34. Contour Integrals in the Complex Plane
Kramers-Kronig Relations