TEKNISKA HOGSKOLAN I LINKOPING

Institut for fysik, kemi och biologi

Iryna Yakymenko

tel.: 288947

e-mail:irina@ifm.liu.se

## QUANTUM DYNAMICS, TFYA28, 6 points

### Aim:

The main aim of the course is to give an exposition of the advanced concepts of quantum
quantum mechanics such as time-dependent perturbation theory,
theory of many-particle systems, scattering theory, quantization of electromagnetic field and
relativistic quantum theory.

### Course Organisation:

The course contains 32 seminars with the theory and solution of problems following a special plan. Course will be given by Iryna Yakymenko. Course language is English.

### Course Literature

I.I. Yakimenko. Lecture Notes in Quantum Dynamics.
For the supplementary reading the following books are recommended:
- Bransden B.H., C.J. Joachain. Quantum mechanics, Second edition, Pearson Education, 2000.
- E. Merzbacher. Quantum mechanics, 3rd edition, John Wiley
and Sons, 1998.
- Landau L.D. and Lifshits E.M. Quantum mechanics (Non-relativistic Theory), Oxford, New York: Pergamon Press,1977.
- Dirac P.A.M. Principles of quantum mechanics. 4th ed, Oxford University Press, 1958 (reprinted in 1999).
- Messiah A. Quantum Mechanics, Two Volumes Bound as One. New York: Dover (2000).

### Examination

Solution of the home problems and oral presentation.
### Plan of lectures

The theory will be expounded following the book
I.I. Yakimenko. Lecture Notes in Quantum Dynamics.
The examples are from the Exercises in Quantum Dynamics (will be hand-out during seminars).

**Lecture 1: INTRODUCTION TO QUANTUM MECHANICS: p.1-19.**
Vectors in Abstract Vector Space.
Operators in Abstract Vector Space.
Eigenvalues and Eigevectors of Operators.
Matrix Representation of Operators.

**Lecture 2: DIRAC FORMULATION OF QUANTUM MECHANICS: p.20-27.**
Vectors in Abstract Vector Space.
Operators in Abstract Vector Space.
Eigenvalues and Eigevectors of Operators.
Matrix Representation of Operators.

**Lecture 3: LINEAR HARMONIC OSCILLATOR: p.28-34.**
Eigenvalue Problem.
Eigenvalue Problem for Number Operator.
Unitary Operators.

**Lecture 4: PAULI SPIN MATRICES AND SYSTEMS OF PARTICLES WITH SPIN ONE-HALF: p.35-39.**
Introduction.
Wave Mechanics with Spin. Pauli Spin Operators.

**Lecture 5: PRINCIPLES OF QUANTUM DYNAMICS: p.42-51.**
Evolution of Probability Amplitude and Time Dependent Operator.
Schrodinger, Heisenberg and Dirac Representation.
Propagator in Coordinate Representation.
Feynman's Path integral Formulation of Quantum Dynamics.

**Lecture 6: MEASUREMENT AND DENSITY MATRIX: p. 53-63.**
Heisenberg Uncertainty Principle.
Complete Set of Commuting Operators.
Measurement, Mixed State, Density Operator or Density Matrix. Entropy. Quantum Dynamics of Spin Systems.

** Seminar 1. DENSITY MATRIX FOR PURE AND MIXED STATES.**
Solution of problems 1-6 from the Exercise Set.

**Lecture 7: TIME-DEPENDENT PERTURBATION THEORY: p. 64-73.**
Introduction.Perturbation Method.
Fermi Golden Rule. Periodic Perturbation. Asymptotic Behaviour of Transition Probabilities.

**Seminar 2. PAULI MATRICES, TIME-DEPENDENT PERTURBATION THEORY.**
Solution of problems 7-11 from the Exercise Set.

**Lecture 8. QUANTUM STATISTICS: p. 75-89.**
Physical Quantities in Quantum Statistics.
Quantum Mechanical Ensembles.
Applications to Single-Particle Systems.
Systems of Non-Interacting Particles.
Photon Gas and Planck's Law.
Ideal Gas.
Bose-Einstein Gases: Bose-Einstein Condensation.
Fermi-Dirac Gases.

**Lecture 9. THEORY OF MANY-PARTICLE SYSTEMS: p. 90-103.**
Variational Approach.
Many-Particle System: Wave function and Hamiltonian. Hartree Equations. Hartree-Fock Equations.
Jellium and the Hartree-Fock Approximation.

**Lecture 10. THOMAS-FERMI APPROXIMATION AND DENSITY-FUNCTIONAL THEORY: p. 104-112.**
Thomas-Fermi Approximation.
Principles of Density-Functional theory.
Kohn-Sham Equations.
Spin-Dependent Density-Functional Theory.

**Lecture 11. TRANSPORT IN TWO-DIMENSIONAL SEMICONDUCTOR HETEROSTRUCTURES: p. 113-127.**
Introduction. Semiconductor Heterostructure and Artificial Atoms. Modulation-Doped Heterostructure.
Electronic Properties of Modulation-Doped Heterostructure.
Counting States.
Landauer Formula: Physical Foundation.
Landauer Formula: Devivation.
Quantization of Conductance in Nanostructures. 0.7 Conductance Anomaly.

**Lecture 12. ADDITION OF ANGULAR MOMENTA: p. 128-137.**
Total Spin for Two Electrons.
Algebraic Approach.
Total Angular Momentum, Clebsh-Gordon coefficients.
Spectral Line Splitting.

**Seminar 3. SPIN IN MAGNETIC FIELDS, ANGULAR MOMENTA.**
Solution of problems 12-14 from the Exercise Set.

**Lecture 13. SECOND QUANTIZATION OF FERMION AND BOSON SYSTEMS: p. 138-148.**
Introduction.
Creation and Annihilation Operators.
Algebra of Creation and Annihilation Operators.
Quantum Dynamics of Identical Particles.
Quantum Field Operators and Second Quantization Formalism.

**Lecture 14. TIGHT-BINDING MODEL: p. 149-151.**
Motion in Periodic Potentials. Tight-binding Model

** Seminar 4. THEORY OF MANY-PARTICLE SYSTEMS, SECOND QUANTIZATION FORMALISM.**
Solution of problems 15-17 from the Exercise Set.

** Lecture 15. SCATTERING: p.152-162.**
Introduction.
Cross Section. Green's Function Approach. Born Approximation.

**Lecture 16. SCATTERING: PARTIAL WAVES, PHASE SHIFTS AND RESONANCES: p. 163-171.**
Introduction.
Partial Wave Analysis. Phase Shifts and Resonances.

**Seminar 5. SCATTERING. Solution of problems 18-19 from the Exercise Set.**

**Lecture 17. QUANTIZATION OF ELECTROMAGNETIC FIELD: p. 172-182.**
Basic Idea.
Quantization of Electromagnetic Field. Photon States.

** Lecture 18. INTERACTION OF LIGHT WITH MATTER: p. 183-194.**
Emission and Absorption of Photons by Atoms.
Spontaneous Emission. Planck's Formula in Quantum Theory of Radiation.
Reyleigh, Thomson and Raman Scattering. Resonance Fluorescence.

**Seminar 6. RELATIVISTIC ELECTRON.**
Solution of problems 20-23 from the Exercise Set.

** Lecture 20. EINSTEIN-ROSEN-PODOLSKI PARADOX AND BELL'S INEQUALITIES: p. 204-207.**