NTMF112 - Quantum theory - selected applications

This lecture is supposed to be complement of Quantum theory I and II NOFY076, NOFY079, taught by Martin Čížek and Pavel Cejnar. The lecture takes place in room T8 on Thursdays 14:00-15:30

Actual informations:

26.5.2023: I slightly improved links and cleared some missing links. Notable addition is link to lecture notes of Wolfgang Domcke and better version of my notes on tight binding models. I am afraid that I will not be able to improve the materials much at the moment. If you are missing something or missing material for some topic of your interest in the language that you understand, please contact me. The EXAM dates are available through SIS or you can contact me.

19.5.2023: There should be reasonable selection of homeworks available now in the syllabus below. Pick one of them and solve it before exam. In the exam we will discuss the topic related to the homework. The other topics may be discussed briefly, but the mark will reflect your knowledge on the selected topic. The links to literature are still not complete. Especially the material in English is missing for some topics. I still work on this issue.

2.5.2023: I actualized sylabus. Homeworks and notes links still missing.

16.2.2023: This is preliminary web page, to be filled with information in the course of the lecture.

Syllabus:

Tight-binding model and its applications
- Motivation: LCAO method, Huckel model for organic molecules, atom chains, discretization of continuous problems.
- Simple infinite chain. Trasnlation symmetry and hamiltonian. Stationary states and band structure. Normalization.
- Other observables. Position and velocity operator, Fermi velocity. Effective mass approximaion.
- Modifications of the model. Chain with point perturbation, bound states.
- Generalization to more dimensions and model of graphene.
[Notes_tight binding ], [Poznamky_graphene in Czech ]
Homeworks:
- H1a: Tight-binding chain with two types of atoms TB.alkene.
- H1b: Graphene ribbon TB.ribbon.
- H1c: Tight-binding half chain and surface state. (contact me if you are interested)
Particle in MG-field, Landau levels
- Classical motion of charged particle in magnetic field.
- Hamiltonian, and its form for homogeneous magnetic field.
- Spectrum and solution by commutation relations. Solution in coordinate representation.
- Observables: radius and energy, coordinates of center of motion.
- Calibration transformation and Aharonov-Bohm effect.
[1][2] [Notes-in Czech], [Cambridge course on Quantum Hall Effect] Covid white-boards: [PartI] [PartII]
Homeworks:
- H2a: Particle fixed on circle in MG field. MG.circle
- H2b: Landau levels using axial symmetry. MG.sym.gauge
Rotation of rigid body:
- Overview of classical physics of rotating body, lab frame, body frame, hamiltonian.
- Hilbert space and hamiltonian on it. Angular momentum operators (lab and body frame).
- Spherical and symetric top. Wigner functions as solution of the problem.
[Wiki on Wigner D-functions], [3][Few notes also here]
Homeworks:
- H3a: Derivation of commutation relations for angular momentum operators. RT.commutation
- H3b: General assymetric top. RT.top
Born-Oppenheimer approximation, Landau-Zener model:
- Adaiabatic and diabatic represetation.
- Coupling terms.
- Landau-Zener model in diabatic representation.
I partially used [4] chapters II and VII
Homeworks:
- H4a: Numerical solution of Landau-Zener model. (contact me if you are interested)
- H4b: Jahn-Teller model and its numerical solution. (contact me if you are interested)
Small vibrations of molecules
- Translation, rotation and internal coordinates.
- Kinetic and potential energy in approximation of small vibrations, FG-matrixes.
- Vibrational eigenmodes.
- Few notes on symmetry.
This was prepared using [4] chapters III and IV
Homeworks:
- H5a: Eigenmodes of simple model for CO2 molecule. (contact me if you are interested)
- H5b: Eigenmodes of three particles on ring. (contact me if you are interested)
Method of selfconsistent field , Hatree fock equations, atoms and molecules.
- Hamiltonian of many-electron system in second quantization.
- Slater determinant, variational principle, derivation of Hartree-Fock equations.
- Single electron solutions (orbitals) and energies, total energy.
- Application to atoms and molecules.
[1][2] [Lecture notes QMII - in czech ] [Poznamky-in czech ]
Homeworks:
- H6a: Huckel model for benzene molecule and Hubbard-like terms. MF.benzene
- H6b: Gross-Pitaevsky model, derivation and numerical solution. MF.BEC
Toy models in scattering
- Review of scattering theory (mostly in 1D).
- Schrodinger equation versus integral equation approach. Green's functions. Boundary conditions.
- Delta-potential model. Scattering wavefunction, T-matrix, S-matrix, transition and reflection.
- Separable potential model and its generalization. Formula for scattering wavefunction and T-matrix.
- Scattering in atomic chain (model of defect (impurity) in crystal) - formulation.
[Lecture notes QMII - in czech ] [Brief notes - English ]
Homeworks:
- H7a: Scattering in double-delta potential, resonances. SC.2delta
- H7b: Numerical solution of potential scattering in 1D. SC.separable
- H7c: Scattering in atomic chain - solution. (contact me for details if you are interested)

Introduction to Floquet theory

Floquet and Bloch theorem.
Splitting of hamiltonian.
Coupled channel problem.

[Lecture notes Viebahn [Lecture notes Santoro ]
Homeworks:

H8a: ???. (it could be interesting to solve some toy model, contact me if you are interested)

Literature:

[1]	Pavel Cejnar - A Condensed Course of Quantum Mechanics (Karolinum 2013).
[2]	L.E. Ballentine - Quantum Mechanics. A Modern Development (World Scientific, Singapore, 1998)
[3]	A.R. Edmonds - Angular momentum in quantum mechanics. (Princeton 1957)
[4]	W. Domcke - Theorie der Molekulschwingungen un der vibronischen Wechselwirkung. (Lecture notes in German, HHU Duesseldorf 1999, avalable on this link)