Phys 500: Quantum mechanics I

Course outline

Phys 500 is a required course for all incoming graduate students in Physics, Medical Physics and Astrophysics. Its purpose is two-fold, namely

Material we will cover: Fundamental concepts (pure and mixed quantum states, observables, measurement, uncertainty relations), quantum dynamics, theory of angular momentum, symmetry and conservation laws, perturbation theory, identical particles, quantum mechanics in medicine - medical imaging, quantum mechanics in astrophysics, quantum information and computation, foundations of quantum mechanics.

Schedule and practical information

Special announcements:

Prerequisites: One of PHYS 450, PHYS 402.

Time and location: MWF; 1PM in Hebb 12

Credits: 3.

Grading: Homework: 40%, Midterm: 20%, Final: 40%.

Lecture notes:

Homework assignments:

Office hour: My office hour is Wednesdays, 5-6 PM.

Book: J.J. Sakurai, Modern Quantum mechanics, Addison Wesley (1994, 2010 [with Napolitano]).

Additional source: C.J. Isham, Quantum Theory - Mathematical and Structural Foundations, Imperial College Press (1995).

Topics covered

  1. Fundamental concepts
    • The Stern-Gerlach experiment (1922)
    • Hilbert spaces
    • Quantum meaurement: observables, uncertainty relations
  2. Quantum dynamics
    • The Schroedinger Equation (1926)
    • Schroedinger vs Heisenberg picture
    • Nuclear magnetic resonance and medical imaging
    • The simple harmonic oscillator
    • Density operators
  3. Quantum information
    • Quantum computation - the circuit model; Universality
    • Deutsch-Jozsa algorithm
    • Shor's factoring algorithm
    • Adiabatic quantum computation
  4. The theory of angular momentum
    • Rotations and the angular momentum commutation relations
    • Spin-1/2 systems
    • SO(3) vs SU(2)
    • Addition of angular momenta
  5. Symmetries in quantum mechanics
    • Continuous symmetries
    • Discrete symmetries: translation (by Delta), parity, time-reversal
  6. Approximation methods
    • The WKB-method
    • Time-independent perturbation theory
    • Time-dependent perturbation theory
  7. Identical particles
    • Permutation symmetry and the symmetrization postulate
    • Pauli exclusion principle for fermions, Bose-Eisnstein condensation for bosons, spin-statistics connection
    • Two-electron systems and the Helium-atom
  8. Foundations of quantum mechanics
    • ``Can quantum mechanics be considered complete?'' (Einstein, Podolsky and Rosen, 1935)
    • The Bell inequalities

Past & future lectures

Wednesday, Sept 3. First lecture: Entrance test

Monday, Sept 8. The Stern-Gerlach experiment: Quantization of spin. Sequential Stern-Gerlach (thought) experiments: Measurements are repeatable, measurements disturb the quantum state. Black-body radiation and Planck's formula. Reading assignment: Sections 1.1 (analogy with polarized light) and 1.2 in Sakurai.

Wednesday, Sept 10. Hilbert spaces.

Friday, Sept 12. Measurement in quantum mechanics.

Monday, Sept 15. Measurement in quantum mechanics continued: Compatible and incompatible observables Reading assignment: P. 32 ff ``Incompatible observables'' in Sakurai.

Wednesday, Sept 17. Uncertainty relations. The Heisenberg uncertainty relation (position and momentum). Infinite-dimensional Hilbert spaces: Particle on a line. Reading assignment: (for today and Friday) Section 1.6 in Sakurai (pages 44 - 51).

Friday, Sept 19.Position and momentum. Momentum as generator of translations. Canonical commutaion relations. Momentum operator in the position basis, momentum eigenstates in the position basis.

Monday, Sept 22. The Schroedinger equation, for the time evolution operator, states and wave functions. Unitarity of time evolution.

Wednesday, Sept 24. Example 1: Nuclear magnetic resonance and medical imaging.

Friday, Sept 26. Example 2: Neutrino oscillations.

Monday, Sept 22. The Heisenberg picture. Heisenberg equation of motion, Ehrenfest's theorem. Reading assignment: Ehrenfest's theorem. p. 84ff in Sakurai.

Wednesday, Oct 1. The simple harmonic oscillator (SHO). Algebraic method for solving the SHO: Eigenenergies. Energy eigenfunctions in the position basis, time-evolution of X and P in the Heisenberg picture.

Friday, Oct 3. Density operators.

Monday, Oct 6. Tensor product Hilbert spaces and entanglement. Measurement without paying attention to the outcome: The sequential Stern-Gerlach aparatus revisited. Generalized measurements. Optional reading assignment: Lecture notes, Section 2.2: Weak measurements.

Wednesday, Oct 8. Decoherence in quantum systems.

Friday, Oct 10. Quantum protocols and NoGos. The no-cloning theorem and impossibility of superluminal communication (according to quantum mechanics). Quantum teleportation.

Wednesday, Oct 15. Quantum computation: the circuit model and the adiabatic model.

Friday, Oct 17. Shor's factoring algorithm

Monday, Oct 20. Angular momentum theory: Rotations and angular momentum commutation relations.

Wednesday, Oct 22. The case of spin 1/2. Rotations about an angle of 2*pi can affect a wave function.

Friday, Oct 24. Midterm exam.

Monday, Oct 27. The eigenstates and eigenvalues of angular momentum.

Wednesday, Oct 29.Orbital angular momentum and rotationally invariant problems in position space. Effective potential, spherical harmonics. Example: Hydrogen atom.

Friday, Oct 31. Addition of angular momenta. Clebsch-Gordan coefficients. Reading assignment: Section 3.7 in Sakurai: Go over the recursion relation for Clebsch-Gordan coefficients in detail; it will be needed for HW4.

Monday, Nov 3. Discrete Symmetries: 1. Parity. Action of the parity operator on states and observables. Symmetric double well potential. Laporte's rule for radiative decays of atomic states. The amonia molecule. Parity is not a fundamental symmetry of nature (thy: Lee and Yang, 1956; exp: Wu et al. 1957). Reading assignment: Amonia molecule and parity selection, Laporte's rule; p. 258-260 in Sakurai (p. 276-278 in Sakuarai and Napolitano).

Wednesday, Nov 5. Discrete symmetries: (2) Lattice translation. Particles in a lattice, the tight binding approximation. Joint eigenstates of the Hamiltonian and the translation operator. Bloch's theorem. Example: conductivity in metals. Pre-reading assignment: Section 4.4 in Sakurai: Time-reversal symmetry.

Friday, Nov 7. Discrete symmetries: (3) Time reversal. Time reversal symmetry in classical mechanics, classical electrodynamics and quantum mechanics. The time reversal operator. Anti-unitarity. Transformation of states and observables under time-reversal. Spin-1/2 systems. Kramers degeneracy.

Monday, Nov 10. Time-reversal continued.

Wednesday, Nov 12. Approximation methods: The WKB method.

Friday, Nov 14. Time-independent perturbation theory, non-degenerate case. Eigen-energies and energy eigenstates up to second order. Hydrogen-like atom in an electric field (Stark effect).

Monday, Nov 17. Time-independent perturbation theory, continued. The degenerate case. Linear Stark effect.

Wednesday, Nov 19. Time-dependent potentials. The interaction picture. Rabi formula. Application in NMR and MRI.

Friday, Nov 21. Time-dependent perturbation theory. The interaction picture. The Dyson series. Fermi's golden rule.

Monday, Nov 24. Identical particles. The symmetrization postulate. Fermions and Bosons. The Pauli exclusion principle for Fermions. Bose-Einstein condensation. The spin-statistics connection.

Wednesday, Nov 26. Identical particles continued. The Helium atom.

Friday, Nov 28. The Einstein-Podolsky-Rosen paradox. Hidden-variable models and the Bell inequality (CHSH version).