Propagators and Green’s Functions

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 • PropagatorKo(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

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>0.00001) • 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 functionGo(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’) = 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

10. q(x) 1 q(-x) 1 0 0 Propagators and Green’s Functions • Translational invariance of space and time • Defining relation • Solution in terms of propagator

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

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

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

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

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

26. y zo 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 • Expansion applies for |z-zo|<|z-z1| where z1 is nearest non-analytic point • See exercises for proof of expansion coefficients

27. y zo C x 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

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. y enclosed pole x -R +R 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

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

32. y C2 C1 x -R +R 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? • If small semi-circle C1in/excludes pole contribution appears twice/once

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

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