Quantum Physics III

Readings

Course Texts

• Griffiths, D. J. Introduction to Quantum Mechanics. 2nd ed. Pearson, 2014. ISBN: 9789332535015. (Required)
• Cohen-Tannoudji, C. Quantum Mechanics. Vol. 2. Wiley-VCH, 1991. ISBN: 9780471164357. (Strongly Recommended)
• Shankar, R. Principles of Quantum Mechanics. Springer, 2013. ISBN: 9781461576754. (Strongly Recommended)
• Sakurai, J. J. Modern Quantum Mechanics. Addison Wesley, 1993. ISBN: 9780201539295.
• Feynman, R. The Feynman Lectures on Physics. Vol. 3. Paperback, 2003. ISBN: 9788131792131.
• Ohanian, H. Principles of Quantum Mechanics. Prentice Hall, 1989. ISBN: 9780137127955.

Schedule and Reading List

The first third of the course covers review material from 8.05 Quantum Mechanics II, including wave mechanics, energy eigenstates, the variational principle, Stern Gerlach, spin 1/2, operators and spin states, vector spaces and operators, Dirac’s bra-ket notation, x and p basis states, the uncertainty principle and compatible operators, the quantum harmonic oscillator, coherent states, two state systems, multiparticle states and tensor products, angular momentum, central potentials, and addition of angular momentum. The second two thirds of the class is summarized below.

SUBJECTS	READINGS	SUBTOPICS
Time-independent perturbation theory	Lecture Notes, Chapter 1 [Griffiths] Chapter 6 [Cohen-Tannoudji] Chapter XI(including Complements A-D) [Cohen-Tannoudji] Chapter XII	Time-independent perturbation theory for degenerate states: Diagonalizing perturbations and lifting degeneracies Time-independent perturbation theory for nondegenerate states: Energy and wavefunction perturbations through second order Degeneracy reconsidered Simple examples: Perturbing a two-state system, a simple harmonic oscillator, and a bead on a ring The fine structure of hydrogen, revisited: Relativistic and spin-orbital effects The hydrogen atom in a magnetic field, revisited: The Zeeman effect The hydrogen atom in a electric field: The Stark effect Van der Waals interaction between neutral atoms
The Semi-classical (or WKB) approximation	Lecture Notes, Chapter 1 [Griffiths] Chapter 8	Form of wave functions in classically allowed and classically forbidden regions Handling turning points: Connection formulae Tunnelling Semiclassical approximation to bound state energies
The adiabatic approximation and Berry’s phase	Lecture Notes, Chapter 2 [Griffiths] Chapter 10	The Born-Oppenheimer approximation and the rotation and vibration of molecules The adiabatic theorem Application to spin in a time-varying magnetic field Berry’s phase, and the Aharonov-Bohm effect revisited Resonant adiabatic transitions and The Mikheyev-Smirnov-Wolfenstein solution to the solar neutrino problem
Time-dependent perturbation theory	Lecture Notes, Chapter 2 [Griffiths] Chapter 9 [Cohen-Tannoudji] Chapter XIII	General expression for transition probability; Adiabatic theorem revisited Sinusoidal perturbations; Transition rate Emission and absorption of light; Transition rate due to incoherent light; Fermi’s Golden Rule Spontaneous emission; Einstein’s A and B coefficients; How excited states of atoms decay; Laser
Scattering	Lecture Notes, Chapter 2 [Griffiths] Chapter 11 [Cohen-Tannoudji] Chapter VIII	Definition of cross-section \(\sigma\); and differential cross section \(\sigma/ \Omega\); General form of scattering solutions to the Schrodinger equation, the definition of scattering amplitude \(f\), and the relation of \(f\) to \(d\sigma/d\Omega\); Optical theorem The Born approximation: Derivation of Born approximation to \(f\); Application to scattering from several spherically symmetric potentials, including Yukawa and Coulomb; Scattering from a charge distribution Low energy scattering: The method of partial waves; Definition of phase shifts; Relation of scattering amplitude and cross section to phase shifts; Calculation of phase shifts; Behavior at low energies; Scattering length; Bound states at threshold; Ramsauer-Townsend effect; Resonances.
Density Operators	Lecture Notes, Chapter 3 [Sakurai] Chapter 3.4 [Cohen-Tannoudji] Complements EIII and FIV	Pure and mixed states Spin-\(1/2\) density operators Partial trace Generalized measurements and quantum operations Thermal states Decoherence
Introduction to the quantum mechanics of identical particles	Lecture Notes, Chapter 4 [Griffiths] Chapter 5.1, 5.2 [Cohen-Tannoudji] Chapter XIV	N-particle systems: Identical particles are indistinguishable Exchange operator, symmetrization and antisymmetrization Exchange symmetry postulate: Bosons and fermions Pauli exclusion principle: Slater determinants; Non-interacting fermions in a common potential well Exchange force and a first look at hydrogen molecules and helium atoms
Degenerate Fermi systems	Lecture Notes, Chapter 4 [Griffiths] Chapter 5.3 [Cohen-Tannoudji] Chapter XI Complement F	Fermions in a box at zero temperature: Density of states; energy; degeneracy pressure White dwarf stars: Equation of state at \(T = 0\); Chandrasekhar limit; neutron stars Electrons in metals: Periodic potentials; Bloch waves; introduction to band structure; metals vs. insulators
Charged particles in a magnetic field	Supplementary notes [Griffiths] Section 10.2.3 (Aharonov-Bohm effect) [Cohen-Tannoudji] Chapter VI Complement E	Canonical quantization The classical Hamiltonian for a particle in a static magnetic field The Schrodinger equation for a charged particle in a magnetic field, via canonical quantization Gauge invariance Landau level wave functions. Counting the states in a Landau level De Haas-Van Alphen effect Integer Quantum Hall Effect: Introduction to the ordinary Hall effect; Quantum mechanical problem of a particle in crossed magnetic and electric fields; Calculation of Hall current due to a single filled Landau level; From this idealized calculation to real systems: The role of impurities. The Aharonov-Bohm effect
Quantum Computing and quantum information	Lecture Notes, Chapter 5	Using many two-state systems as a quantum computer Grover algorithm Simon’s algorithm