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

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

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

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

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

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

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

Propagators and Green’s Functions

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’) = 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 Defining relation Solution in terms of propagator 1 q(x) 1 q(-x)

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

Contour Integrals in the Complex Plane y -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

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

Contour Integrals in the Complex Plane Kramers-Kronig Relations