Relativistic Electron Theory
The Dirac Equation
Mathematical Physics Project
Karolos POTAMIANOS
Universit Libre de Bruxelles
e
Abstract
This document is about relativistic quantum mechanics and more
precisely about the relativistic electron theory. It pre

Chapter 2
Second Quantisation
In this section we introduce the method of second quantisation, the basic framework for
the formulation of many-body quantum systems. The rst part of the section focuses on
methodology and notation, while the remainder is dev

Chapter 1
From Particles to Fields
The aim of this section is to introduce the language and machinery of classical and quantum eld theory through its application to the problem of lattice vibrations in a solid. In
doing so, we will become acquainted with

18
CHAPTER 1. FROM PARTICLES TO FIELDS
1.3.2
Answers
1. Applying the Euler-Lagrange equation dt (n L) n L = 0 to the discrete La
grangian of the lattice model, one nds that the N equations of motion take the
form of a three-term dierence equation,14
mn =

CHAPTER 10
Molecules and Solids
10.1 Molecular Bonding and Spectra
10.2 Stimulated Emission and Lasers
10.1: Molecular Bonding and Spectra
The Coulomb force is the only one to bind atoms.
The combination of attractive and repulsive forces creates a
stable

Lorenzo Menculini
The Dirac equation in an external
magnetic field in the context of
minimal length theories
Tesi di laurea triennale
Universit degli Studi di Perugia
a
Settembre 2012
`
Universita degli Studi di Perugia
`
Facolta di Scienze Matematiche, F

6. Second Quantization and Quantum Field Theory
6.0.
Preliminary
6.1.
The Occupation Number Representation
6.2.
Field Operators and Observables
6.3.
Equation of Motion and Lagrangian Formalism for Field
Operators
6.0.
Preliminary
Systems with variable num

Chapter 2
Quantisation of the Electromagnetic Field
Abstract The study of the quantum features of light requires the quantisation of
the electromagnetic eld. In this chapter we quantise the eld and introduce three
possible sets of basis states, namely, th

Chapter 2
Solutions of the Dirac Equation in an External
Electromagnetic Field
In this chapter, the solutions of the Dirac equation for a fermion in an external
electromagnetic eld are presented for the cases of a pure magnetic eld of arbitrary
strength,

56
CHAPTER 2. SECOND QUANTISATION
2.4.2
Answers
1. (a) Making use of the commutation relations for bosons, one nds
a aa = a(a a 1),
a aa = a (1 + a a)
from which the results follow. Using these results, one nds that, providing a| = 0,
a a a| = a(a a 1)| =

Chapter 5
Broken Symmetry and Collective
Phenomena
Previously, we have seen how the eld integral method can be deployed in many-particle
theories. In the following chapter, we will learn how elements of perturbation theory can be
formulated eciently by st

Chapter 3
Feynman Path Integral
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as
developing the general construction scheme, particular emphasis is placed on establishing
the interconnections between the quantum

Chapter 6
Tripos Questions
The following problems are taken from recent Part III Natural Science Tripos papers.
Note that the questions are each intended to take around 45 minutes to complete.
|cfw_
6.1 Questions
Part III Physics Questions 1998
1. Accordi

184 CHAPTER 5. BROKEN SYMMETRY AND COLLECTIVE PHENOMENA
5.5
Problem Set
5.5.1
Questions on the Functional Field Integral
1. In chapter 4., the connection between the coherent state path integral and the
Feynman path integral for a Harmonic oscillator was

142
CHAPTER 4. FUNCTIONAL FIELD INTEGRAL
4.4
Questions on the Field Integral Method
1. Exercises on Fermion Coherent States: To practice the coherent state method,
we begin with a few simple exercises on the fermionic coherent state which complements the

3.5. QUESTIONS ON THE PATH INTEGRAL
3.5
115
Questions on the Path Integral
1. Quantum Harmonic Oscillator: As emphasized in lectures, the quantum harmonic oscillator provides a valuable arena in which to explore the Feynman path
integral and methods of fu

Advanced Quantum Mechanics
Problem set 4
Eliran Peretz 021907936
Question 1:
a) The Lagrangian density of the system formed by the electromagnetic field and the sources is
composed of two terms:
L L0 + Lin
Where
L0
1
E 2 B2
8
Is the Lagrangian density of

7 Grassmann integration
In this lecture we introduce Grassmann variables and introduce integration over Grassmann
variables. The basic objects that we will consider are vectors consisting out of m commuting
components and n anti-commuting components. Such

Advansed Quantum Mechanics
Problem set 3
Eliran Peretz 021907936
Question 1:
(a) The particle density operator x
x x is in first-quantized form. When a
general one-body operator is written in first-quantized form J
J x , the second
quantized form is:

Lecture XXIV
70
Lecture XXIV: Superconductivity and Gauge Invariance
To establish origin of perfect diamagnetism and zero resistance,
one must accommodate electromagnetic eld in Ginzburg-Landau Action
Inclusion of electromagnetic eld into BCS action: p p

Lecture XX
58
Lecture XX: Superuidity
Previously, we have seen that, when treated in a mean-eld or saddle-point approximation,
the eld theory of the weakly interacting Bose gas shows a transition to a Bose-Einstein
condensed phase when = 0 where the order

Lecture XIX
55
Lecture XIX: Bose-Einstein Condensation
Previously, we have seen how the functional eld integral technique can be developed to
explore the impact of the electronic degrees of freedom on the eective Coulomb interaction
in a metal. However, o

Advanced Quantum Mechanics
Problem Set 2
Eliran Peretz 021907936
Question 1:
(a)
This problem is very easy to solve using Mathematica. The wave function is given by
j k 2 K 2 r
R k, r
j kr cos n kr sin
rR
rR
We require the continuity of the logarithmic

Lecture IX
25
Lecture IX: Bogoliubov Theory of weakly interacting Bose gas
Although strong interaction eects can lead to the formation of novel ground states of the
electron system, the properties of the weakly interacting system mirror closely the trivia

Advanced Quantum Mechanics
Eliran Peretz 021907936
Question 1:
Taking the x representation and inserting a complete set of x eigenstates, the LippmannSchwinger equation reads
x x dx ' x
1
x ' V x ' x ' .
E H 0 i
Following the notes we insert a complete s

Lecture XV
43
Lecture XV: Many-body (Coherent State) Path Integral
Could formulate many-body propagator (Green function), but here, convenient to focus on
partition function.
Quantum partition function
Z=
cfw_nFock Space
n|e(HN ) |n ,
= 1/kB T,
chemica

Chapter 4
Functional Field Integral
In this chapter, the concept of path integration is generalized to integration over quantum elds. Specically we will develop an approach to quantum eld theory that takes as
its starting point an integration over all con

Lecture V
12
Lecture V: Second Quantised Representation of Operators
So far we have developed an operator-based formulation of many-particle states. However,
for this representation to be useful, we have to understand how the action of rst quantised
opera

Lecture IV
9
Lecture IV: Second Quantisation
We have seen how the elementary excitations of the quantum chain can be presented
in terms of new elementary quasi-particles by the ladder operator formalism. Can this
approach be generalised to accommodate oth