Physics 511-512
Quantum Mechanics I, II

Luming Duan

Fall 2012 - Winter 2013

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: Tuesday, Thursday, 11:30AM -1:00PM
 Location: 455 Dennison

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): Paul Bierdz (Randall 4409) Email: paopao@umich.edu
                                         

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

 
  • J.J. Sakurai, Modern Quantum Mechanics (1982)
  • E. Merzbacher, Quantum Mechanics (1998)
  • P.A.M. Dirac, The Principles of Quantum Mechanics 
  • David J. Griffiths Introduction to Quantum Mechanics.

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: Non-relativistic Theory
  • J. Preskill, Lecture Notes on quantum information 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 2012)

  • Introduction
    • History of quantum mechanics
      1. Old quantum theory
      2. Establishment of quantum mechanics
      3. Development of quantum mechanics
    • Relation of quantum mechanics to current physics frontiers
    • General structure of this course
  • Fundamentals (formalism) of quantum mechanics
    • Overview
    • Wave-particle duality (Albers:p19-23; Sakurai:p2-9)
      1. What is wave-particle duality
      2. The double-slit experiment
      3. The Stern-Gelach experiment
    • Quantum states and vectors in the Hilbert space (Albers:p24-34; Sakurai:p10-35)
      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

    • Observables and operators in the Hilbert space (Albers:p24-34; Sakurai:p10-35)
      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

    • Measurements, Probability interpretation, and Von Neumann's projectors (Albers:p24-34; Sakurai:p10-35)
      1. Formalism for quantum measurements
      2. Applications: distinguishing quantum states and quantum cryptography
    • Examples of Hilbert space: finite-dimensional systems (outside of textbook, further reading: J. Preskill, Lecture Notes on quantum information and computation)
      1. Physical examples of two-dimensional (qubit) systems
      2. Description of qubit systems
      3. Extension to d-dimensions (qudit systems)
    • Examples of Hilbert space: continuous-variable systems  (Albers:p35-38; Sakurai:p41-59)
      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
    • Uncertainty relation (Albers:p47-49; Sakurai:p23-35)

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

    • Quantum dynamics (Albers:p44-53; Sakurai:p68-108)
      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 (outside of textbook, further reading: J. Preskill, Lecture Notes on quantum information and computation)
      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 (Albers:p193-195, Griffiths:p420-427, further reading: J. Preskill, Lecture Notes on quantum information and computation)
      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 (outside of textbook, further reading: J. Preskill, Lecture Notes on quantum information and computation; Zurek, arxiv, Physics Today article)
      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 (Albers 63-65; Further reading: see review articles on atomic clocks from Rev. Mod.Phys. )
      •  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 (Albers: p185-190)
    3. 1D continuous-variable systems: sectionally-constant and delta potentials (Albers 66-68)
      • General methods and boundary conditions
      • General concepts from these potentials: quantization, tunneling, and scattering
    4. The harmonic oscillators (Albers p68-73, Sakuri:p89-96, Griffiths: p40-58)
      • 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 (Albers p74-84; Griffiths:p131-171)
      • 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 (Albers p84-90; Griffiths:p145-171)
      • Series expansion method, energy spectrum, general discussion about applications
    3. Charged particles in magnetic fields: the Aharonov-Bohm phase and the Landau levels (part of material in Albers p324-341; Sakurai:p109-142)
      • 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 (Albers:p102-112, Sakurai:p152-173)
      • 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 (Alber: p113-119; Sakurai:p168-173, p217-222)
      • 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 (Albers:p120-130; Sakurai: p203-216)
      • 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 (Alber:p138-143; Sakurai:p232-242)
      • 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 (Alber:p144-150; Sakurai:p232-242)
      • 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 (Alber:p151-163; Sakurai:p251-281)
      • 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 2013)

  • 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 (Albers:p202-205; Sakurai:p285-303; Griffiths: p249-265)
      • 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 (Albers:p206-211; Sakurai:p285-303;)
      • 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 (Albers:p288-291; Sakurai:p316-340; Griffiths: p340-367)
      • 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) (Albers:p313-314; Griffiths: p431-434)
    1. The variational method (Albers:p223-228; Sakurai:p313-316; Griffiths: p293-314)
      • 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 (Albers:p225-227; Griffiths: p299)
      • The band-gap structure for atoms in the optical lattice (outside of textbook)
      • Quantum phase transitions in quantum magnetism: mean-field theory for the anisotropic Heisenberg model (outside of textbook, further reading: any book on ¡°Quantum Magnetism¡±)
    3. The semi-classical (WKB) method (Albers:p237-243; Griffiths: p315-339)
      • The WKB approximation
      • Turning points and the connection formula
    4. Example applications of the WKB method (Albers:p243-247; Griffiths: p315-339)
      • 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 (Albers:p341-349; Griffiths: p368-393)
      • 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 (out of textbook, see arxiv research articles on STIRAP)
      • The adiabatic quantum algorithm (out of textbook, see arxiv research articles on adiabatic quantum computing)
      • 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 (Albers:p212-216; Sakurai: p304-312, Griffiths: p266-276)
      • 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 (Albers:p218-220; Griffiths:p283-292)
      • 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 (Albers:p217-218; Sakurai: p304-312, Griffiths: p277-282)
      • Atomic magnetic moment
      • Zeeman effects within Fine structure
      • Zeeman effects within hyperfine structure
    2. The molecular structure (Albers:p229-236; Griffiths: p304-314)
      • 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 (Albers:p264-269,291-307; Sakurai: p379-385,424-428; part in Griffiths: p394-408)
      • 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 (Albers:p264-270; Sakurai: p386-389, Griffiths: p408-419)
      • The Born series
      • Some applications with spherically symmetric potentials
    4. The transition matrix and the optical theorem (Albers:p291-307; Sakurai: p390-391)
      • 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 (Albers:p270-272; Sakurai: p399-409, Griffiths: p399-407)
      • 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 (Sakurai: p410-420)
      • The low-energy scattering
      • The resonance scattering
      • Feshbach resonance and its applications in ultracold atomic physics (outside of textbook)
  • Quantum many particle systems

(Outside of the textbooks. Some further readings for this chapter: A. Fetter and J. Walecka, ¡°Quantum Theory of Many-Particle Systems¡±, Ch. 1, p3-50; Abrikosov, Gorkov, Dzyaloshinski, ¡°Methods of Quantum Field Theory in Statistical Physics¡±, Ch. 1,p1-40; Dirac, ¡°Principles of Quantum Mechanics¡±, p207-252.)

    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

Homework

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