The course treats fundamental and applications of quantum mechanics. By treating various systems and methods, we study nontrivial quantum effects observed in may-body systems. Treated systems are: nonlinear systems, time-dependent systems, and many-body (Bose, Fermi, and spin) systems. Treated methods are: perturbation theory, field quantization, variational method, and density operator, and so on.
This course will allow students to analyze various systems in quantum mechanics using methods treated in it.
(1) Calculate corrections of energy levels and states of a system with the method of perturbation
(2) Understanding of the field quantization and treat particle creation and annihilation processes in many-body systems
(3) Evaluate many-body effects in various systems
Perturbation, identical particles, field quantization, variational method, density operator, quantum entanglement
✔ Specialist skills | Intercultural skills | ✔ Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Problem sheet is dstributed every week. Students are expected to solve problems at home and to present answers on blackboard. Presented answers are discussed.
Course schedule | Required learning | |
---|---|---|
Class 1 | Overview (approximation methods, indistinguishablity of identical particles, field quantization, many-body effects) | Explain the significance of approximation methods and the physical meaning of field quantization |
Class 2 | Perturbation theory of stationary states (1) (perturbative expansion, formula in nondegenerate case, simple examples) | Calculate energy level and wavefunction corrections in a harmonic oscillator system with a perturbation |
Class 3 | Perturbation theory of stationary states (3) (general properties of perturbation, validity of the approximation, level dynamics) | Discuss general properties of perturbation theory |
Class 4 | Perturbation theory of stationary states (3) (formula in degenerate case, examples in systems with many degerees of freedom) | Explain how the perturbation formula is modified in degenerate case and calculate corrections in a simple example |
Class 5 | Time-dependent perturbation (interaction picture, perturabation formula, Fermi golden rule) | Calculate correction terms in a typical time-dependent system and discuss physical meaning of the result |
Class 6 | Absorption and emission of light by atoms (cavity radiation, Planck formula, spontaneous emission, stimulated emission, Einstein coefficients, selection rule) | Calculate Einstein coefficients by applying perturbation theory |
Class 7 | Time-dependent systems (time-periodic system, dynamical invariant) | Find general solution of the Schroedinger equation in time-periodic systems |
Class 8 | Adiabatic approximation (adiabatic state, Landau-Zener transition, geometric phase) | Calculate an approximate solution of the Schroedinger equation in a weakly time-dependent system |
Class 9 | Many-body systems (1) Identical particles (statistics of identical particles, Bose/Fermi statistics, symmetry/antisymmetry of wavefunction) | Explain how the indistinguishablity of the identical particles brings Pauli exclusion principle and Bose-Einstein condensate |
Class 10 | Many-body systems (2) Field quantization (creation and annihilation operators, field quantization) | Explain how the creation and annihilation operators are used to describe physical phenomena |
Class 11 | Many-body systems (3) Applications of Bose/Fermi systems (Helium atom, Bose-Einstein condensate, Hubbard model) | Explain general properties of quantum many-body systems |
Class 12 | Many-body systems (4) Applications of spin systems (magnetic systems, Heisenberg model) | Find the ground state of a simple quantum spin system and explain how it is different from that of the corresponding classical system |
Class 13 | Variational method (general considerations, Hatree/Hartree-Fock approximation, Mean-field theory) | Explain mean-field approximation in many-body systems |
Class 14 | Density operator (Pure and mixed states, von Neumann equation, Bloch vector) | Explain the difference between pure and mixed states |
Class 15 | Quantum entanglement (decomposition of composite systems, reduced density operator, Bell's inequality and its violation) | Explain quantum-specific properties in many-body systems |
unspecified
J.J. Sakurai "Modern Quantum Mechanics"
Homework (50%) and presentation (50%). Homework is to assess whether students can give the correct answer on the problem and the report is written properly. Presentation is checked to see a clear explanation is given and proper replies to questions are made.
Quantum Mechanics I and II