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2016　Quantum Mechanics III

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Academic unit or major
Physics
Instructor(s)
Takahashi Kazutaka 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Mon1-2(H112)  Thr1-2(H112)  
Group
-
Course number
ZUB.Q313
Credits
2
Academic year
2016
Offered quarter
3Q
Syllabus updated
2017/1/11
Lecture notes updated
-
Language used
Japanese
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Course description and aims

This course covers fundamentals and applications of quantum mechanics. By covering various systems and methods, we study nontrivial quantum effects observed in many-body systems. Systems covered are: nonlinear systems, time-dependent systems, and many-body (Bose, Fermi, and spin) systems. Methods covered are: perturbation theory, field quantization, variational method, the density operator, and so on.
This course will allow students to analyze various systems in quantum mechanics using methods covered in it.

Student learning outcomes

(1) Calculate corrections of energy levels and states of a system with the method of perturbation
(2) Understand field quantization and work with particle creation and annihilation processes in many-body systems
(3) Evaluate many-body effects in various systems

Keywords

Perturbation, identical particles, field quantization, variational method, density operator, quantum entanglement

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills

Class flow

The first class is an overview of the course and the lecture is given with the aid of slides. The contents are divided into three parts: Perturbation theory and its applications (2-8), Descriptions of many-body systems (9-13), and Characteristics of quantum systems (14, 15). Problems related to the contents of this course are provided in Exercises in Quantum Mechanics III.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Overview (approximation methods, indistinguishability 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 degrees of freedom) Explain how the perturbation formula is modified for a degenerate case and calculate corrections in a simple example
Class 5 Time-dependent perturbation (interaction picture, perturbation 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 rules) Calculate Einstein coefficients by applying perturbation theory
Class 7 Time-dependent systems (time-periodic system, dynamical invariant) Find the 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 indistinguishability of identical particles results in the 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 to 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 to 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, Hartree/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

Textbook(s)

Lecture notes distributed on OCW.

Reference books, course materials, etc.

J.J. Sakurai "Modern Quantum Mechanics"

Assessment criteria and methods

Homework (50%) and final exam (50%). Homework is to assess whether students can apply methods discussed in the course to some advanced topics. Understanding of the course material is evaluated with a final exam.

Related courses

  • ZUB.Q204 ： Quantum Mechanics I
  • ZUB.Q206 ： Quantum Mechanics II
  • ZUB.S310 ： Thermodynamics and Statistical Mechanics II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Quantum Mechanics I and II

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