Physics 511-512
Quantum Mechanics I, II

Luming Duan

Fall 2008 - Winter 2009

Welcome to the homepage of the course for Quantum Theory and Atomic Physics! This is a two-semester graduate-level course.

Course Description

This course will focus on the explanation of fundamental concepts, mathematical structure, and calculation methods of quantum mechanics. The course covers the following topics: fundamental concepts of quantum mechanics and its mathematical structure, exactly solvable quantum systems, symmetries in quantum mechanics, approximation methods, atomic and Molecular structure, scattering theory, quantum many-particle systems, relativistic wave equations.

Class meetings

Time: Wednesday, Friday, 10:00AM -11:30AM
 Location: 4404 Randall

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/QM1-pub.html

Teaching assistant (Grader): ???, Email: ???
                                         

Course Requirements

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

Prerequisites

The basic mathematical prerequisites are linear algebra and calculus. It would be useful to have a previous course on introductory quantum mechanics at the undergraduate level, but that is not an essential requirement. Some results from group theory will be used for discussion on symmetries, but I will explain the results before I use them.

Books and References

Main Reference

Books

 Either of them

Other
recommended
books

 
  • P.A.M. Dirac, The Principles of Quantum Mechanics 
  • E. Merzbacher, Quantum Mechanics (1998)

Further Readings

 
  • C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (1997)
  • R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals
  • 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

First term (Fall 2008)

  • Introduction

1.    History of quantum mechanics

      1. Old quantum theory
      2. Establishment of quantum mechanics
      3. Development of quantum mechanics

2.    Relation of quantum mechanics to current physics frontiers

3.    General structure of this course

  • Fundamentals (formalism) of quantum mechanics
    1. Overview
    2. Wave-parcticle duality
      1. What is wave-parcticle duality
      2. The double-slit experiment
      3. The Stern-Gelach experiment
    1. Quantum states and vectors in the Hilbert space
      1. Why vectors
      2. Mathematical review about vectors

A complex vector space, adjoint vectors, inner product and norm, basis of a vector space, expansion with a basis, Hilbert space

    1. Observables and operators in the Hilbert space
      1. How to describe observables
      2. Mathematical review about operators

Linear operators, representation of operators, functions of operators, Eigenvalue and eigenvectors, Hermitean and unitary operators, spectral decomposition, some theorems about operators

    1. Measurements, Probability interpretation, and Von Neuman's projectors
      1. Formalism for quantum measurements
      2. Applications: distinguishing quantum states and quantum cryptography
    2. Examples of Hilbert space: finite-dimensional systems
      1. Physical examples of two-dimensional (qubit) systems
      2. Description of qubit systems
      3. Extension to d-dimensions (qudit systems)
    3. Examples of Hilbert space: continuous-variable systems
      1. Normalization of continuous-variable states and Dirac's delta-function
      2. Coordinate-basis representation
      3. Fourier transform theorem
      4. Momentum-basis representation
      5. Transfer between coordinate and momentum bases
      6. Cononical commutation relation
    4. Uncertainty relation

         Proof of uncertainty relation, minimum uncertainty state (coherent state)

    • Quantum dynamics
      1. Evolution operator
      2. Schrodinger equation
      3. Representations of Schrodinger equation in different bases
      4. Schrodinger picture and Heisenberg picture
      5. Applications to free particles and harmonic oscillators, quantum dispersion
    • Multiparticle systems, density matrix, and quantum entanglement
      1. Hilbert space of multi-particles, distinguishable vs. indistinguishable particles
      2. Density matrix and its properties
      3. Quantum entanglement, characterization, and von Neumann's entropy
    • Quantum nonlocality and Bell's inequalities
      1. Background: EPR paradox and local hidden variable theories
      2. The Bell-CHSH inequality
      3. Experimental detection of the CHSH inequality
    • Open quantum systems, decoherence, and interpretation of measurements
      1. Evolution of open quantum systems, Kraus representation
      2. Master equation and Lindblad form
      3. Decoherence
      4. Interpretation of wave-function collapse for measurements
  • Exactly solvable quantum systems
    1. Overview
    2. Evolution of finite-dimensional systems and applications to atomic clocks and neutrino oscillation
      •  Evolution of 2D (qubit) systems: time-independent Hamiltonian
      • Time-dependent Hamiltonian and the interaction picture
      • Resonant Rabi oscillation
      • Atomic clocks and Ramsey methods
      • Neutrino interference and oscillation
    3. 1D continuous-variable systems: sectionally-constant and delta potentials
      • General methods and boundary conditions
      • General concepts from these potentials: quantization, tunneling, and scattering
    4. The harmonic oscillators
      • The annilation and creation operators
      • Energy spectrum
      • Energy eigenstates and their representation in the coordinate basis
      • Generalization of the harmonic potentials

    3-dimensions, n  decoupled and coupled harmonic oscillators, examples with trapped ions, phonons in solids, and photons from the EM field

    1. Spherically symmetric potentials and angular momentum
      • Angular momentum and its commutation relations
      • Eigenvalues and co-eigenstates of the angular momentum operators
      • Reduction of 3D spherically symmetric potentials to 1D potentials
    2. Hydrogen-like atoms
      • Series expansion method, energy spectrum, general discussion about applications
    3. Charged particles in magnetic fields: the Aharonov-Bohm phase and the Landau levels
      • Brief review of classical EM
      • Gauge invariance: classical and quantum
      • The Aharonov-Bohm effect explained from the gauge transformation
      • Charged particles in a constant magnetic field and the Landau levels
      • Brief discussion of the quantum Hall effects
  • Symmetries in quantum mechanics
    1. Brief overview about symmetries and their applications
    2. Mathematical review: groups and algebra
      • Groups: finite and continuous (Lie) groups
      • Typical Lie groups: O(n), SO(n), U(n), SU(n)
      • Representation of groups: reducible and irreducible representations
      • Representation of Lie group and Lie algebra in quantum mechanics
      • Translation and rotation groups and algebra, Euler angles
    3. Wigner's d-functions and Schwinger's representation of angular momentum operators
      • Representation of angular momentum operators and rotation groups, Wigner's d-functions
      • Schwinger's representation of angular momentum operators
      • Calculation of Wigner's d-functions from Schwinger's representation
    4. Addition of angular momenta and Clebsch-Gordan coefficients
      • Why addition of angular momenta
      • Addition of angular momenta: procedure

Note its relation with irreducible decomposition of tensor product representation of rotation groups (algebra)

      • Clebsch-Gordan (CG) coefficients and their properties
      • Calculation of C-G coefficients: intuitive picture and the recursion relations
      • Addition of more-than-two angular momenta
    1. Rotation transformations and selection rules for scalar and vector observables
      • Symmetry transformations on operators: Heisenberg versus Schrodinger picture
      • Scalar and vector observables under rotation, spherical components
      • Selection rules for vector observables
    2. Rotation transformations and selection rules for tensor observables, Wigner-Eckart theorem
      • Tensor observables
      • Decomposition of tensor observables into spherical components

Note its relation with addition of angular momenta, both corresponding to irreducible decomposition of tensor product representation of rotation groups (algebra)

      • Wigner-Eckart theorem and its proof
      • Selection rules for tensor observables
    1. Discrete symmetries
      • Examples of discrete symmetries: space and time reversion, translation and rotation in lattices
      • Space reversion and parity
      • Time reversion, anti-unitary operators, and Kramers degeneracy

Second term (Winter 2009)

  • Brief introduction and review

        Review of the course structure and the first-semester contents

  • Approximation methods in quantum mechanics
    1. Overview of approximation methods in QM
    2. Time-independent perturbation methods
      • The general method and classification of perturbation theory
      • Bound-state non-degenerate perturbation

Basic recursion relations for arbitrary order perturbation, explicit formula for 1st and 2nd order perturbation

      • The degenerate perturbation method
      • Convergence, asymptotic series, and limit of perturbation theory
    1. Example applications of the time-independent perturbation
      • d.c. Stark shift (from a static electric field) of the ground state of the hydrogen-like atoms (non-degenerate perturbation)
      • d.c. Stark shift of excited states of the hydrogen-like atoms (degenerate perturbation)
      • Atom's polarizability and atom's trap
      • a.c. Stark shift (from a laser) of the ground state of the hydrogen-like atoms
      • Design of optical lattice from a.c. Stark shift
    2. Time-dependent perturbation, transition, and Fermi's golden rule
      • Time-dependent perturbation (formalism)
      • Time-energy uncertainty relation
      • Fermi's golden rule for transition (discrete spectrum to continuous spectrum)
      • Atomic transition through broadband incoherent (thermal) light

Stimulated emission and absorption, explain of spontaneous emission

      • Quantum Zeno effect (coherent vs incoherent transitions)
    1. The variational method
      • The general idea
      • The variational principle for the stationary Schrodinger equation
      • The variational principle for the dynamical Schrodinger equation
      • A simple illustrative example: Harmonic potential
    2. Example applications of the variational method
      • The ground state of the Helium-like atoms
      • The band-gap structure for atoms in the optical lattice
      • Quantum phase transitions in quantum magnetism: mean-field theory for the anisotropic Heisenberg model
    3. The semi-classical (WKB) method
      • The WKB approximation
      • Turning points and the connection formula
    4. Example applications of the WKB method
      • Quantization condition for a single-minimum potential
      • Quantization condition and the eigen-energies of the double-well potential
      • Tunneling through any potential barrier
    5. 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

Proof the adiabatic theorem, dynamical and Berry's (geometric) phase, Estimation of the transition probability

      • The adiabatic passage with counter-intuitive pulses
      • The adiabatic quantum algorithm
      • The Berry's phase for a two-level system
  • Structure of atoms and molecules
    1. Overview
      • Overview of the atomic and molecular structure
      • About the units
    2. Fine structure of the hydrogen-like atoms
      • Spin-orbital coupling
      • Splitting of the energy levels due to the spin-orbital coupling
      • The relativistic correction to eigen-energies and the whole fine structure
    3. The hyperfine structure of the hydorgen-like atoms
      • The basic picture
      • The hyperfine coupling Hamiltonian

N, L coupling, N,S coupling (contact vs. dipole terms)

      • The hyperfine splitting of the ground state

Level splitting, applications in radio astronomy

    1. Influence of magnetic fields on atomic structure: Zeeman effects
      • Atomic magnetic moment
      • Zeeman effects within Fine structure
      • Zeeman effects within hyperfine structure
    2. The molecular structure
      • The Born-Oppenheimer approximation
      • The electronic structure of the molecular hydrogen ion
      • Vibrational and rotational levels
  • Scattering theory
    1. Overview
      • motivation and general picture of scattering
    2. Formulation of scattering and the Lippmann-Schwinger equation
      • The general problem and the Green's operator (propagator)
      • Derivation of the Lippmann-Schwinger equation
      • The Lippmann-Schwinger (LP) equation in the coordinate basis
      • The cross section from the LP equation
    3. The Born approximation and its applications
      • The Born series
      • Some applications with spherically symmetric potentials
    4. The transition matrix and the optical theorem
      • The dressed propagator and the transition matrix (T-matrix)
      • The cross section with the T-matrix
      • The optical theorem
    5. Method of partial-wave expansion
      • Partial-wave expansion
      • Unitarity requirement and the collision phase shift
      • Solve the collision phase shift
    6. The low-energy scattering and the the resonance scattering
      • The low-energy scattering
      • The resonance scattering
      • Feshbach resonance and its applications in ultracold atomic physics
  • Quantum many particle systems
    1. Overview
    2. First-quantization representation of identical particles
      • Identical particles and different quantum statistics
      • The first-quantization representation of states for bosons and fermions
    3. Second-quantization representation of identical particles
      • Bosons
      • Fermions
    4. Non-interacting bosonic and fermionic systems
      • Bose condensation
      • Fermi surface
    5. Interacting bosonic systems
      • Mean-field theory and the Gross-Pitaevskii equation
    6. Interacting fermionic systems
      • Mean-field theory and the Hartree-Fock equation

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