Download •Course: Introduction to Green functions in Physics •Lecturer: Mauro Ferreira •Recommended Bibliography:

•Course: Introduction to Green functions in Physics
•Lecturer: Mauro Ferreira
•Recommended Bibliography:
- “Green’s Functions in Quantum Physics” by
E. N. Economou
- “Introduction to wave scattering, Localization,
and Mesoscopic Phenomena”, by Ping Sheng
Syllabus:
1. Introduction
a. Solving general linear equations
b. Basic mathematical tools
c. A few examples in classical Physics:
i.
Harmonic oscillator
ii. Poisson equation
iii. Wave equation
iv. Diffusion equation
Syllabus:
2. Green function of quantum particles
a. Schroedinger equation
b. Continuous Green function
c. Lattice Green function
d. Perturbation theory
e. Scattering theory
f. Transport ( linear response theory )
Course prerequisites:
It is assumed that you have a working knowledge
of the following topics:
• Differential equations;
• Complex variables;
• Basic Quantum Mechanics;
• Introductory Solid State theory.
• Evaluation of GF
• Use of GF in relevant cases
• Emphasis on analytically solvable cases
• Numerical techniques for evaluating GF will
also be presented
• Mathematical formalism presented in as
much details as possible
Assessment method: Final year exam
Outline
• Mathematical background
• Physical Applications
(Classical Physics)
Consider the following mathematical problem
known “external” function
linear
operator
unknown
We introduce a function G(x,x’) defined as
Multiplying the eq. above by f(x’) and integrating over x’
By comparison with the original equation
Example: Consider the
following differential equation
?
?
Contour integrals on the
complex plane are ideal to
treat these type of problems
Cauchy’s theorem: Let C be a simple closed curve
on the complex plane z. If f(z) is analytic on and
inside C, then
d
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i
m
e
R
r
e
C
Cauchy’s theorem: Let C be a simple closed curve
on the complex plane z. If f(z) is analytic on and
inside C, then
C
If C contains a singular point (or pole) at z0 the contour
integral not necessarily vanishes. For example, if C is a
circle of radius ρ about z0, the integral
C
Now consider a function f(z) that can be expanded
around a singular point z0 in a series of the form
(Laurent series) :
In this case
The contour integral of a function with several
singularities (poles) inside a contour C can be
obtained by using the residue theorem.
Residue theorem
C
The advantage of the
residue theorem is that a
contour integral can be
found by only evaluating
the residues (at the poles)
Methods for finding the residue of a function
(A) Laurent series
(B) Simple pole
(C) Multiple poles
Methods for finding the residue of a function
(A) Laurent series
(B) Simple pole
(C) Multiple poles
Given
find the residue of f(z) at
z=1.
Methods for finding the residue of a function
(A) Laurent series
(B) Simple pole
(C) Multiple poles
If f(z) has a simple pole at
z=z0, we find the residue by
multiplying f(z) by (z-z0) and
evaluating the result at z=z0.
Methods for finding the residue of a function
(A) Laurent series
(B) Simple pole
(C) Multiple poles
If f(z) has a pole of order n at z=z0,
to find the residue we multiply f(z)
by (z-z0)n, differentiate the result
(n-1) times, divide it by (n-1)! and
evaluate the result at z=z0.
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a
…
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p
B
Is G(x-x’) a solution to
?

• Mathematical problem (not the simplest solution)
• What’s the relevance in Physics ?
Solving physical problems
• Harmonic oscillator
• Poisson equation
• Wave equation
• Diffusion equation