Physics 460
Quantum Mechanics II

Luming Duan

 Winter 2007

Welcome to the course homepage of the physics 460! This is the second-semester undergraduate quantum mechanics.

Course Description

This course will focus on the explanation of concepts and methods of quantum mechanics and their applications. The course will cover the following topics: review of formalisim of quantum mechanics, finite-dimensional quantum dynamics and its applications, perturbation theory and its applications, atomic and molecular structure, variational method and its applications, the WKB approximation and the adiabatic theorem, scattering theory.

Class meetings

 Time: Tuesday, Thursday, 11:30AM -1:00PM
 Location: 335WH (West Hall)

Instructors

Luming Duan
4219 Randall Hall
Telephone: (734) 763-3179
Email: lmduan@umich.edu
Web: http://www-personal.umich.edu/~lmduan/
Course website: http://www-personal.umich.edu/~lmduan/QM-under460-pub-07.html

Office hours: I am in office most of the time, you may just stop by, or you can send an email for an appointment.

Teaching assistant (Grader): 

Course Requirements

There will be regularly assigned problem sets. Your letter grade will be based on two exams (Midterm 20% dates???; Final: 40%, dates???), homework (30%), and class participation (10%).

Prerequisites

The basic mathematical prerequisites are linear algebra and calculus. I also assume that you have taken quantum mechanics I (physics 453 or equivalent).

Books and References

Textbook		David J. Griffiths Introduction to Quantum Mechanics.
Other recommended textbooks		J.J. Sakurai, Modern Quantum Mechanics (1982) C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (1997) S. Gasiorowicz, Quantum mechanics E. Merzbacher, Quantum Mechanics (1998)
Further Readings recommended		P.A.M. Dirac, The Principles of Quantum Mechanics  R. P. Feynman, the Feynman lectures on physics III M. Born, Atomic physics L. Landau and E. Lifshitz, Quantum Mechanics: Nonrelativistic Theory J. Preskill, Lecture Notes on quantum inforamtion and computation (the first four chapters, which are general quantum mechanics), see http://www.theory.caltech.edu/people/preskill/ph229/#lecture

Course Outline

• History and perspective of quantum mechanics
1. History and development of quantum mechanics
2. Relation of quantum mechanics to current physics frontiers
3. Structure of this course


• Review of formalism of quantum mechanics
1. states, evolution, and measurements
• States in QM
• Observables in QM
• Evolution in QM
  
• Example application: quantum no-cloning theorem
• Measurements in QM
  
• Example application: Quantum cryptography
  
2. Finite-dimension vs. continuous-variable quantum systems
• Examples of finite-dimension physical systems
• Description of 2-dimensional (qubit) systems
• Continuous-variable systems
• Coordinate and momentum basis representations
  
3. Uncertainty relation, Schrodinger versus Heisenberg pictures
• Uncertainty relation
• Evolution in the Schrodinger picture
• Evolution in the Heisenberg picture
• Example application of the Heisenberg equation: quantum dispersion
4. Quantum dynamics of two-dimensional systems and its applications
• Evolution of 2-dimensional (qubit) systems: Rabi oscillation
• Atomic clocks and the Ramsey method
• Neutrino oscillation
1. Multi-partite quantum systems, density matrix, and quantum entanglement
• Multi-partite quantum systems
• Density matrix/operator
• Properties of the density matrix
• Quantum entanglement and von Neumann entropy

• Perturbation theory
1. Overview: classification of perturbation theory
2. Time-independent perturbation methods
• The general method
• Non-degenerate perturbation (1st and 2nd order)
• The degenerate perturbation method
• Combination of degenerate and non-degenerate perturbation, examples
  
3. Stark effect and optical lattice
   
• Brief review of the hydrogen structure
• d.c. Stark shift
  
• a.c. Stark shift and optical lattice
  
4. Time-dependent perturbation theory and Fermi's golden rule
• Time-dependent perturbation
  
• Fermi's golden rule
  
5. Atomic transitions, selection rules, and quantum Zeno effect
   
• Atomic transitions through incoherent light
• Selection rules
  
• Quantum Zeno effect (coherent vs. incoherent evolution)
  

 
• Variational  method and its applications
1. Overview
2. The variational method
• The general idea
• The variational principle for the Schrodinger equation
• An example: Harmonic potential
3. Applications of the variational method
• The Helium-like atoms
• Quantum phase transitions in quantum magnetism: mean-field theory


• The semiclassical (WKB)  method and the adiabatic theorem
1. The semi-classical (WKB) method
• The WKB approximation
• Quantization condition under infinite wells
  
• Turning points and the connection formula
2. Applications of the WKB method
• Quantization condition for a general potential
• Tunneling through a potential barrier
• Example: theory of alpha decay
  
3. The adiabatic approximation and the Berry's phase
• The fast and slow evolution: sudden versus adiabatic approximation
• The adiabatic theorem and the Berry's phase
4. Applications of the adiabatic approximation
• <>The adiabatic passage for quantum control
• <>Adiabatic quantum computation<>
• <><>The Berry's phase for a two-level system


• Structure of atoms and molecules
1. Overview
   
2. Fine structure of the hydrogen-like atoms
• The relativistic correction
• Spin-orbital coupling
• Energy correction due to the spin-orbital coupling
3. The hyperfine structure of the hydorgen-like atoms
• The basic picture of coupling
  
• The hyperfine splitting of the ground state
4. Atomic structure under magnetic fields: Zeeman effects
• Hamiltonian under a magnetic field
  
• Zeeman effects
5. The molecular structure
• The general picture and the Born-Oppenheimer approximation
• The molecular hydrogen ion
• The electronic structure
  
• Vibrational and rotational levels

• Scattering theory
1. Partial-wave expansion
• General picture of scattering
• Partial-wave expansion
• Unitarity and the collision phase shift
• How to solve the collision phase shift
2. The Lippmann-Schwinger equation and the Born approximation
   
• The Green's function
• The Lippmann-Schwinger (LP) equation in the coordinate basis
• The cross section from the LP equation
• The Born approximation
• Applications of the Born approximation
  

Notices

Midterm exam: date???
Final exam: ???

Problem Sets

• Problem Set 1 [PS] [PDF] (due Tuesday, Jan. 24).
• Problem Set 2 [PS] [PDF] (due Tuesday, Feb. 7).
• Problem Set 3 [PS] [PDF] (due Thursday, Feb. 16).

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Links

• Physics department of the University of Michigan.
• Luming Duan's Homepage