Relativistic Quantum Physics

Relativistic Quantum Physics

Code: 51545
ECTS: 5.0
Lecturers in charge: prof. dr. sc. Dubravko Klabučar - Lectures
Lecturers: Clay James Grewcoe - Exercises
Take exam: Studomat

1. komponenta

Lecture typeTotal
Lectures 30
Exercises 15
* Load is given in academic hour (1 academic hour = 45 minutes)
COURSE GOALS: The course Relativistic Quantum Physics aims at expanding and perfecting the knowledge and skills acquired at the Quantum Physics and Classical Electrodynamics courses, by formulating the quantum theory in agreement with the special theory of relativity. In the first part of the course, students learn the basic concepts of relativistic quantum physics and techniques of solving Dirac and Klein-Gordon equation. In the second part, the transition is made from relativistic quantum mechanics with a fixed number of particles to quantum electrodynamics (QED), which serves as an example of a quantum field theory (QFT), using the Feynman propagator formalism. Students are thereby expected to master heuristic and intuitive aspects of free and perturbative QFT and the simplest examples of scattering in QED through calculating tree Feynman diagrams. This leads to better understanding of fundamental laws of physics and of the nature of space-time, as well as explanation of some fundamental notions such as spin, which could be introduced only ad hoc previously, in the courses held in the earlier years of study. In addition to that, this course gives practical support to the course on Elementary Particle Physics and provides competences necessary for the course on Quantum Field Theory 1. It also contributes to better understanding of other courses such as Nuclear Physics and Atomic and Molecular Physics, e.g., by clarifying under which conditions the effects of relativity and field quantization are essential, or can be safely neglected.

1.2 demonstrate a thorough knowledge of advanced methods of theoretical physics including classical mechanics, classical electrodynamics, statistical physics and quantum physics
1.3 demonstrate a thorough knowledge of the most important physics theories (logical and mathematical structure, experimental support, described physical phenomena) 1.4 describe the state of the art in - at least- one of the presently active physics specialities

2.1 identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models 2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
2.4 adapt available models to new experimental data
3.2 develop a personal sense of responsibility, given the free choice of elective/optional courses


4.2 present one's own research or literature search results to professional as well as to lay audiences
4.3 develop the written and oral English language communication skills that are essential for pursuing a career in physics


5.1 search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of technical English is required) 5.2 remain informed of new developments and methods and provide professional advice on their possible range and applications
5.3 carry out research by undertaking a PhD
After completing the course Relativistic Quantum Physics (and passing the exam), a student will be able to:
* Explain how the reasoning yielding the non-relativistic, Schrödinger equation can be generalized to the relativistic case, yielding thereby the Klein-Gordon (KG) equation. Obtain, by its generalization, the Dirac equation in the Hamiltonian form, and then rewrite it in the Lorentz-covariant form. Demonstrate that the Dirac equation yields, in the non-relativistic limit, the Pauli equation with the correct giromagnetic factor relating the spin and magnetic moment of a fermion.
* Solve the KG and Dirac equations for a free particle, expalin the differences between scalar solutions of the KG equation and spinor solutions of the Dirac equation, construct general wave packets out of plane waves, and explain the meaning of bilinear covariants of Dirac spinors, such as the 4-vector current density.
* Demonstrate non-conservation of spin and orbital angular momentum, as well as conservation of total angular momentum and helicity. Express four-component Dirac spinors trough Pauli central-field spinors. Explain the relativistic spectrum obtained by solving the Dirac equation for hydrogen and hydrogen-like systems with the Coulomb interaction.
* Explain how Dirac's hole theory, modeling vacuum as the filled "sea" of fermions, leads to quantum many-body theory, or rather, quantum field theory (QFT), where the number of quantum particles is not fixed any more. Explain the prediction of positrons and other anti-particles in Dirac's theory, and transformations of charge conjugation, and space and time reflections.

* Explain the motivations for QFT because of the Lamb shift in the hydrogen atom spectrum and radiative corrections of the giromagnetic factor of electron and muon. Estimate when one should take into account creation and annihilation of particles and anti-particles, i.e., the effects of QFT ("the second quantization"), and when one can, in a good approximation, stay with the theoretical concepts of the "ordinary" quantum physics with fixed number of particles, such as wave functions, potentials, classical fields, classical currents, etc.
* Explain the concept of propagators in the context of non-relativistic quantum mechanics, and extend it to relativistic quantum physics utilizing the heuristic propagator approach of Feynman and Stückelberg.
* Apply the Feynman rules to calculations of the simplest scattering and processes in QED (tree Feynman diagrams in the lowest order in coupling).

Division in 15 lectures:
1. The need for formulating of relativistic quantum theory. Transition from the non-relativistic to relativistic quantum theory.
2. Relativistic quantum-mechanical equation for spin 0 bosons: Klein-Gordon equation, some of its applications and some problems.
3. Relativistic quantum-mechanical equation for spin 1 fermions: the Dirac equation for 4-component spinors. Non-relativistic limes, Pauli equation, spin, the electron magnetic moment and giromagnetic factor 2.
4. Lorentz covariance of the Dirac equation, gamma-matrices and their properties. Bilinear covarijants of Dirac spinors.
5. Solutions of the free Dirac equation. Projection operators of energy and spin.
6. Wave packets as superpositions of plane Dirac waves. Non-conservation of spin and orbital angular momentum, conservation of the total angular momentum and helicity.
7. Dirac particle in a central field. Four-component Dirac spinors expressed through Pauli spinors of central field.
8. Solving the Dirac equation for hydogen and hydrogen-like systems with the Coulomb interaction. Relativistic energy spectrum of hydrogen.
9. On some motivations for QFT: Lamb shift and radiative corrections of the giromagnetic factor.
10. Dirac theory of positrons and other antiparticles. Charge conjugation, parity and time reversal.
11. The propagator theory and description of scattering.
12. The Feynmanov propagator for fermions and antifermions of spina 1/2. The photon propagator.
13. Scattering in QED. The Feynman rules.
14. Relationship with the cannonical quantization of electromagnetic and fermion fields.
15. Calculation of some processes in QED in the lowest order of coupling, tree Feynman diagrams.

Obligatory attendance of lectures and exercises.

The exam consists of written and oral parts.
    W. Greiner, Relativistic Quantum Mechanics: Wave Equations, Springer-Verlag; 3rd edition (2000).
    W. Greiner, J. Reinhardt, Quantum Electrodynamics, Springer-Verlag; 4th edition (2009).
    F. Mandl, G. Shaw, Quantum Field Theory, John Wiley & Sons, revised edition (1993).
Prerequisit for:
Enrollment :
Attended : Classical Electrodynamics
Attended : Quantum Physics
7. semester
Izborni predmeti - Mandatory studij - Physics

8. semester Not active
Izborni predmeti - Mandatory studij - Physics
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