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:

Office hour: My office hour is Wednesdays, 4-5 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. 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
  4. Symmetries in quantum mechanics
    • Continuous symmetries
    • Discrete symmetries: translation (by Delta), parity, time-reversal
  5. Approximation methods
    • The WKB-method
    • Time-independent perturbation theory
    • Time-dependent perturbation theory
  6. Identical particles
    • Permutation symmetry and the symmetrization postulate
    • Two-electron systems
    • The Helium-atom
  7. Quantum information
    • Quantum cryptography - BB84 and Ekert protocol
    • Quantum computation - the circuit model; Universality
    • Grover's data base search
    • Shor's factoring algorithm
    • Adiabatic quantum computation
  8. Foundations of quantum mechanics
    • ``Can quantum mechanics be considered complete?'' (Einstein, Podolsky and Rosen, 1935)
    • The Bell inequalities
    • The Kochen-Specker theorem

Past & future lectures

Wednesday, Sept 4. First lecture. Course outline. The Stern-Gerlach experiment. Sequential Stern-Gerlach experiments. Measurement changes the quantum state. Reading assignment: Sakurai, Section 1.1. Lecture given by Dr. Leon Loveridge.

Monday, Sept 9. Hilbert spaces: Vector spaces, inner product, closedness. The analogy between sequential Stern-Gerlach apparata and polarized light. (Sakurai, Section 1.2)

Wednesday, Sept 11. Hilbert spaces continued: Orthonormal bases, matrix representations. (Sakurai, Section 1.3)

Friday, Sept 13. Measurement in quantum mechanics. Observables, Hermitian operators, eigenvalues and eigenstates. The Born rule.

Monday, Sept 16.Compatible and incompatible observables.

Wednesday, Sept 18. Holiday - no lecture

Friday, Sept 20. Incompatible observables continued. The uncertainty relation.

Monday, Sept 23. Position and momentum. Infinite-dimensional Hilbert spaces. Momentum as generator of translations. The canonical CR between position and momentum.

Wednesday, Sept 25. Commutator vs.Poisson bracket. Meaning of the Jacobi identity. Heisenberg's uncertainty relation and Gaussian wave packets.

Friday, Sept 27. The Schroedinger equation for evolution operators, state vectors and wave functions. Unitarity. Energy-eigenstates. Precession of spin in a magnetic field.

Monday, Sept 30. Application Nuclear magnetic resonance and medical imaging. Reading assignment: Neutrino oscillations (Sakurai & Napolitano)

Wednesday, Oct 2. Density operators. Pure and mixed states. Tensor product Hilbert spaces. Entanglement.

Friday, Oct 4. State purifications, partial trace. Operations on density operators: generalized measurements and completely positive trace preserving maps (CPTP maps). Reading assignment: Continuous interpolation between unitaries and projective measurements, proof of Theorem 2; optional: decoherence. (lecture notes)

Monday, Oct 7. Quantum protocols. (i) NoGos: Superluminal communication, cloning. Goes: Dense coding, teleportation.

Wednesday, Oct 9. The Heisenberg picture, Heisenberg's equation of motion, Ehrenfest's theorem.

Friday, Oct 11.The simple harmonic oscillator--the algebraic method. Energy eigenvalues, time evolution.

Wednesday, Oct 16. Angular momentum theory: Representation of rotations in 3D.

Friday, Oct 18. Angular momentum theory: spin 1/2. Acquiring a phase factor of -1 under 2pi rotations. SO(3) vs SU(2) (started).

Monday, Oct 21. Midterm exam.

Wednesday, Oct 23. Angular momentum theory: SO(3) vs SU(2) (finished), eigenvalues and eigenstates of angular momentum.

Friday, Oct 25. Angular momentum theory: Orbital angular momentum and rotationally invariant potentials.

Monday, Oct 28. Angular momentum theory: addition of angular momenta; Clebsch-Gordan coefficients.

Wednesday, Oct 30. Ch 4: Discrete symmetries: Parity/ space inersion symmetry.

Friday, Nov 1. Lattice translation symmetry: Bloch's theorem.

Monday, Nov 4. Time reversal symmetry: time-reversal is an anti-linear operator; time reversal acting on states and observables. Half-integer spin under double time reversal.

Wednesday, Nov 6. Time-reversal continued: Kramers degeneracy.

Friday, Nov 8. Approximation methods: WKB

Wednesday, Nov 13. Time-independent perturbation theory: non-degenerate and degenerate case. (Lecture given by Dr. Leon Loveridge).

Friday, Nov 15. Perturbation theory continued. (Lecture given by Dr. Leon Loveridge).

Monday, Nov 18. The interaction picture. Two-level system wit a periodic perturbation -- Rabi formula.

Wednesday, Nov 20. Time-dependent perturbation theory. Fermi's golden rule.

Friday, Nov 22. Identical particles. The symmetrization postulate. Bosons and fermions. Pauli exclusion principle. Bose-Einstein condensation.

Monday, Nov 25. Identical particles continued. The Helium atom: Spin-dependence of energy eigenstates, even if the Hamiltonian is not spin-dependent.

Wednesday, Nov 27. Quantum cryptography: Ekert protocol.

Friday, Nov 29. Einstein-Podolsky-Rosen paradox, hidden-variable models, and the Bell inequalities.