Theoretical Physics V (Advanced Concepts of Quantum Physics)

G. Morigi with E. King, S. Roy, G. Harmon, C. Candeago and F. Mattiotti

Lecture

Attention: Please note that starting from Monday, 13th of April the monday lecture has been moved to 8:30-10:00 (from 12:00-14:00). The lecture on Tuesday remains unchanged.

Regular lecture times:

• Monday 8:30 - 14:00 Uhr, Gebäude E2 6 - Seminarraum E04
• Tuesday 12:00 - 14:00 Uhr, Gebäude E2 6 - Seminarraum E04

First lecture: Tuesday, 07.04.2025, 12:00 Uhr

Exams

The dates for the two exams are:

• Friday, 31.07, from 9:00 to 12:00. Venue: Building E2.6, room E0.4
• Thursday, 17.09, from 14:00 to 17:00. Venue: Building E2.6, room E0.4

Requirements for passing the course

• Pre-exam requirements: At least 50% of points in the exercises and calcuation of some exercises on the blackboard. By ticking an exercise, you bindingly signal your readiness to present your solution on the blackboard. In case of refusal the corresponding points will be subtracted. In case you already acquired admission to the exam in a previous year, this condition is no longer required.
• Passing of one the two exams (the better grade of these will count). 

Prüfungsleistungen

• Prüfungsvorleistung: Mindestens 50% der Votierpunkte und Vorrechnen einiger Aufgaben. Das Ankreuzen einer Aufgabe entspricht einer verbindlichen Bereitschaft, den entsprechenden Lösungsweg an der Tafel zu präsentieren. Bei Verweigerung werden die jeweiligen Punkte aberkannt. Wurde die entsprechende Prüfungszulassung bereits früher erworben, entfallen diese Vorleistungen.
• Bestehen einer der beiden Klausuren (die bessere wird gewertet).

Contents of the lecture

Note: The program of TP5E includes all material till the end of Section 4.

0. Resume

• 0.1 One-particle theories and the continuity equation
  • The continuity equation of non-relativistic quantum mechanics
  • Klein-Gordon equation and a naive attempt to derive a continuity equation
• 0.2 The covariant notation of special relativity

1. The Dirac Equation

• 1.1 Derivation of Dirac Equation and basic properties
• 1.2 Transformations between inertial reference frames
• 1.3 The Dirac equation in covariant notation
• 1.4 Conservation laws for free electrons
• 1.5 Elementary solutions of the Dirac equation
• 1.6 Hole theory and the positron
• 1.7 Charge conjugation
• 1.8 Dirac equation with fields
• 1.9 The Hydrogen atom

2. Scattering theory (1st part)

• 2.1 Dyson series and non-relativistic perturbation theory
  • 2.1.1 First order perturbation theory and the Fermi golen rule
  • 2.1.2 Second order perturbation theory
  • 2.1.3 On the validity of the perturbative expansion to second order
• 2.2 The scattering matrix
• 2.3 Example: non-relativistic bound electron interacting with fields

3. The quantum electromagnetic field

• 3.1 Lagrangian density of the electromagnetic field
  • 3.1.1 Discrete mechanical systems
  • 3.1.2 Continuum systems: the Lagrangian density and Euler-Lagrange equations
  • 3.1.3 Classical scalar fields
  • 3.1.4 The Lagrangian density of the electromagnetic field
• 3.2 Quantization of the electromagnetic field Hamiltonian
  • 3.2.1 The radiation field as a collection of harmonic oscillators
  • 3.2.2 Quantization of the radiation oscillators
• 3.3 Vacuum fluctuations
  • 3.3.1 Frequency cutoff
  • 3.3.2 Casimir effect
• 3.4 The classical limit

4. Photon-Matter interactions

• 4.1 The minimal coupling Hamiltonian in the non-relativistic limit
• 4.2 Matrix elements of the perturbation
  • 4.2.1 Free electron interacting with the emf: fundamental processes
  • 4.2.2 Bound electron interacting with the emf
    • 4.2.2.1 Electric-dipole approximation
    • 4.2.2.2 Matrix elements in electric-dipole approximation
    • 4.2.2.3 Fundamental processes
• 4.3 Spontaneous decay of the excited state of an atom
• 4.4 The Lamb Shift
• 4.5 Scattering processes

5. Second Quantization

• 5.1 Classical fields
  • 5.1.1 Scalar fields
  • 5.1.2 From classical to quantum scalar fields
• 5.2 Second quantization (non-relativistic fields)
  • 5.2.1 change of basis
• 5.3 Relativistic scalar fields
  • 5.3.1 Real scalar field
  • 5.3.2 Complex scalar field

6. Noether's theorem

• 6.1 The principle of stationary action
• 6.2 Noether's theorem
  • 6.2.1 Translational invariance and the canonical energy-momentum tensoe
  • 6.2.2 Gauge invariance

7. Dirac Fields

• 7.1 Lagrangian density
• 7.2 Dirac fields
• 7.3 Positron operator

Literatur

• J. J. Sakurai “Advanced Quantum Mechanics” (Pearson Education, 2006)
• F. J. Dyson "Advanced Quantum Mechanics" (WSPC, 2011)
• K. Huang, "Quantum Field Theory", Wiley and Son Ed.