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2020/2021

Advanced Quantum Mechanics

Code: 100178 ECTS Credits: 6
Degree Type Year Semester
2500097 Physics OT 4 2
The proposed teaching and assessment methodology that appear in the guide may be subject to changes as a result of the restrictions to face-to-face class attendance imposed by the health authorities.

Contact

Name:
Antonio Miguel Pineda Ruiz
Email:
AntonioMiguel.Pineda@uab.cat

Use of Languages

Principal working language:
english (eng)
Some groups entirely in English:
Yes
Some groups entirely in Catalan:
No
Some groups entirely in Spanish:
No

Prerequisites

Recommendation: Quantum physics. Quantum mechanics and theoretical mechanics.

Objectives and Contextualisation

Introduce the most basic concepts (conceptual and mathematical) of quantum field theory. Likewise, the student must acquire the ability to apply with agility the calculation tools to different types of problems.

Competences

  • Apply fundamental principles to the qualitative and quantitative study of various specific areas in physics
  • Be familiar with the bases of certain advanced topics, including current developments on the parameters of physics that one could subsequently develop more fully
  • Carry out academic work independently using bibliography (especially in English), databases and through collaboration with other professionals
  • Communicate complex information in an effective, clear and concise manner, either orally, in writing or through ICTs, and before both specialist and general publics
  • Formulate and address physical problems identifying the most relevant principles and using approximations, if necessary, to reach a solution that must be presented, specifying assumptions and approximations
  • Make changes to methods and processes in the area of knowledge in order to provide innovative responses to society's needs and demands.
  • Take account of social, economic and environmental impacts when operating within one's own area of knowledge.
  • Use critical reasoning, show analytical skills, correctly use technical language and develop logical arguments
  • Use mathematics to describe the physical world, selecting appropriate tools, building appropriate models, interpreting and comparing results critically with experimentation and observation
  • Using appropriate methods, plan and carry out a study or theoretical research and interpret and present the results
  • Work independently, have personal initiative and self-organisational skills in achieving results, in planning and in executing a project
  • Working in groups, assume shared responsibilities and interact professionally and constructively with others, showing absolute respect for their rights.

Learning Outcomes

  1. Analyse the consequences of Dirac’s equation on the nonrelativistic limit.
  2. Analyse the limits of simple high and low energy electromagnetic processes.
  3. Apply gauge invariance for the Lagrangian determination of quantum electrodynamics.
  4. Calculate cross sections for simple electromagnetic processes.
  5. Communicate complex information in an effective, clear and concise manner, either orally, in writing or through ICTs, in front of both specialist and general publics.
  6. Establish the bases for the comprehensive formulation of quantum field theory and its applications.
  7. Establish the phenomenological consequences of relativistic wave equations.
  8. Formulate the bases for the extension to non-Abelian gauge theories.
  9. From a specific initial and final state, structure and develop the strategy and calculation for the cross section of an electromagnetic process.
  10. Identify situations in which a change or improvement is needed.
  11. Identify the social, economic and environmental implications of academic and professional activities within one’s own area of knowledge.
  12. Obtain transitional amplitudes for electromagnetic processes using Feynman’s rules.
  13. Properly handle the algebra of Dirac matrices and the rules for calculating traces.
  14. Study collisions with identical particles.
  15. Use critical reasoning, show analytical skills, correctly use technical language and develop logical arguments
  16. Use phase-space integration correctly.
  17. Work independently, take initiative itself, be able to organize to achieve results and to plan and execute a project.
  18. Working in groups, assume shared responsibilities and interact professionally and constructively with others, showing absolute respect for their rights.
  19.  Carry out academic work independently using bibliography (especially in English), databases and through collaboration with other professionals

Content

1. General motivation

2. Introduction (classical fields)

(a) Motivation for fields: Many body problems. One example

(b) Elements of classical field theory:

• Functional calculus (reminder)
• Lagrangian and Hamiltonian formalism. Euler-Lagrange equations
•Noether theorem (later (5.f))
 

(c) Natural units


3. Non-relativistic Quantum Field Theory. Free fields

  1. (a)  Bosons. Fock space. Number operator (particle interpretation) and statistics. Connection with quantum mechanics

  2. (b)  Fermions. Fock space. Number operator (particle interpretation) and statistics. Connection with quantum mechanics

4. Poincare Group

  1. (a)  Poincare group and Lorentz group.

  2. (b)  Associated Lie algebra. 

  3. (c)  One particle irreducible representation. Wigner method. Little group. Spin, helicity. Massive and massless case

  4. (d)  Discrete symmetries: C, P, T (*)

5. Interaction (scalar case)

  1. (a)  Cross Section and S matrix

    (b)  Interaction picture and S matrix

    (c) Klein-Gordon real field. Propagator and causality

    (d) Motivation for causal (free) fields

    (e) Wick theorem

  2. (f)  Continuous symmetries Noether theorem: associated charges and currents. Energy-momentum tensor

  3. (c)  Klein-Gordon complex field. Charge symmetry. Antiparticle.

6. Quantum Electrodynamics (QED)

  1. (a)  Field for a massless spin-one particle: Electromagnetic field

  2. (b)  Dirac field: construction. Propagator, symmetries, spin: helicity and quirality. Spin-statistics theorem

  3. (c)  Quantization of QED

  4. (d)  S-matrix to O(e2). Feynman diagrams

  5. (e)  Compton scattering to tree level. Feynman diagrams and computational

    techniques: traces, spin, ...

  6. (f)  Generalized Feynman rules

    (g)  About gauge invariance. Example of Ward identity (*)

  7. (h)  Other elementary processes of QED to tree level: e+e− → e+e−, e+e− →

    μ+μ−, ...

    (i)  Decays. Radiative transitions of hydrogen

    (j)  Hydrogen-like atoms in Quantum Field Theory (*)

  8.  

Methodology

There will be teaching lectures where the theory will be explained in detail.

There will be teaching lectures where a selection of the list of exercises will be discussed.

The student should digest at home the theory explained in class, and perform the list of exercises suggested during the lectures.

Activities

Title Hours ECTS Learning Outcomes
Type: Directed      
Problems class 16 0.64 2, 1, 3, 4, 6, 7, 9, 14, 13, 12, 15, 17, 16
Theoretical classes 33 1.32 2, 1, 3, 4, 5, 6, 7, 9, 14, 8, 13, 12, 15, 17, 18, 16
Type: Autonomous      
Discussion, work in groups 22 0.88 2, 1, 3, 4, 5, 6, 7, 9, 14, 19, 8, 13, 12, 15, 17, 18, 16
Problems solved in group or autonomously 30 1.2 2, 1, 3, 4, 5, 6, 7, 9, 14, 19, 8, 13, 12, 15, 17, 18, 16
Study of theoretical foundations 42 1.68 2, 1, 3, 4, 5, 6, 7, 9, 14, 19, 8, 13, 12, 15, 17, 18, 16

Assessment

1st partial exam: 50% of the grade.
2nd Partial exam: 45% of the grade.
Selective delivery of problems: 5% of the grade.

In order to be able to take part in the recovery exam, one should have been previously presented to both exams.
Examination of recovery of the two partials: 95% of the note. There is no minimum mark to be able to opt for the
recovery

Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
Exam 1 50% 2 0.08 2, 1, 3, 4, 5, 6, 7, 9, 14, 8, 10, 13, 12, 15, 17, 16
Exam 2 45% 2 0.08 2, 1, 3, 4, 5, 6, 7, 9, 14, 8, 13, 12, 15, 17, 16
Homework 5% 1 0.04 2, 1, 3, 4, 5, 6, 7, 9, 14, 19, 8, 11, 10, 13, 12, 15, 17, 18, 16
resit exam 95% 2 0.08 2, 1, 3, 4, 5, 6, 7, 9, 14, 8, 13, 12, 15, 17, 16

Bibliography

• D. Lurie, Particles and Fields
• S. Weinberg, The Quantum Theory of Fields
• L.H. Ryder, Quantum Field Theory
• M. Peskin and D. Schroeder, An introduction to Quantum Field Theory
• B. Hatfield, Quantum Field Theory of Point Particles and Strings
• J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model • S. Pokorsky, Gauge Field Theories
• C. Itzykson and J. Zuber, Quantum Field Theory
• F.J. Yndurain, Elements of grup theory. https://arxiv.org/pdf/0710.0468