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:

Final exam: Dec. 13, 12 noon (2 1/2 hrs).

Prerequisites: One of PHYS 450, PHYS 402.

Time and location: The class takes place MWF 1 PM - 2 PM in Hebb 10.

Credits: 3.

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

Office hour: My office hour is Tuesdays and Wednesdays, 4-4:30 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).

Past & future lectures

Wednesday, Sept 7. First lecture. Course outline. The Stern-Gerlach experiment.

Friday, Sept 9. TA Training. No lecture.

Monday, Sept 12. Why quantum states live in complex vector spaces: Sequential Stern-Gerlach experiment. Analogy with polarized light.

Wednesday, Sept 14. The axioms of Hilbert space. Matrix representations.

Friday, Sept 16. Measurement in quantum mechanics. Born rule.

Monday, Sept 19. Measurement in quantum mechanics. Compatible and incompatible observables. Uncertainty relation.

Wednesday, Sept 21. (Two lectures) Infinite-dimensional Hilbert spaces. Position and momentum. The wave function. Heisenberg uncertainty relation.

Friday, Sept 23. No class. (Replaced by additional class on Wednesday, Sept 21.)

Monday, Sept 26. No class. (Replaced by additional class on Wednesday, Sept 28.)

Wednesday, Sept 28. (Two lectures). Quantum Dynamics. The Schroedinger Equation. Examples: Nuclear Magnetic Resonance and Medical Imaging. Neutrino oscillations.

Friday, Sept 30. The Schroedinger wave equation. Heisenberg picture and Heisenberg equation of motion.

Monday, Oct 3. Ehrenfest's theorem. The harmonic oscillator.

Wednesday, Oct. 5.Tensor product Hilbert spaces and entangled states. The no-cloning theorem and impossibility of superluminal communication.

Friday, Oct. 7. Density operators -- positivity, hermiticity, and trace. Impossibility of superluminal communication, continued.

Monday, Oct 10. Thanksgiving.--No lecture.

Wednesday, Oct. 12. Start of Sakurai, chapter 3 - Angular momentum theory. Continuous symmetries in physics and Noether's theorem. Angular momentum commutation relations.

Friday, Oct. 14. Spin-1/2 systems revisited. Wave-function acquires phase of (-1) under 2Pi rotations. Experimental verification of the prediction.

Monday, Oct 17. Groups and representations. SU(2) vs. SO(3).

Wednesday, Oct. 19. Eigenvalues and eigenstates of angular momentum.

Friday, Oct. 21. Midterm exam

Monday, Oct 24. Orbital angular momentum.

Wednesday, Oct. 26. No lecture. (Replacement date to be discussed)

Friday, Oct. 28. No lecture. (Replacement date to be discussed)

Monday, Oct 31. Addition of angular momenta.

Wednesday, Nov 2. Lecture I: discrete Symmetries in QM. 1) Parity (space reflection). Lecture II: 2) Discrete translations. Bloch's theorem.

Friday, Nov 4. Lectures I and II: Discrete Symmetries in QM: 3)Time reversal. Time-reversal squared is not the identity on particles with half-integer spin. Kramers degenercy.

Monday, Nov 7. Approximation methods: 1) WKB.

Wednesday, Nov 9. Time-independent perturbation theory, non-degenerate case.

Friday, Nov 11. No lecture (Remembrance Day).

Monday, Nov 14. Time-independent perturbation theory, degenerate case. Example: Stark effect (both degenerate and non-degenerate case).

Wednesday, Nov 16. Time-independent potentials. The interaction picture.

Friday, Nov 18. Time-dependent perturbation theory. The Dyson series. Example: piece-wise constant potential, Fermi's golden rule.

Monday, Nov 21. Identical particles. The symmetrization pastulate. Bosons and fermions. Bose-Einstein condensation. Pauli's exclusion principle.

Wednesday, Nov 23. Identical particles continued. 2-electron systems. Exchange density. Ortho/Parahelium.

Friday, Nov 25. Scattering theory. Scattering amplitude, differential and integral cross section. Lippmann-Schwinger equation.

Monday, Nov 28. Scattering theory continued. The Born approximation. Example: Elastic scattering of protons off a Ca nucleus.

Wednesday, Nov 30. Foundations of quantum mechanics. The Einstein-Podolsky-Rosen paper (1935); local hidden variable models; the CHSH inequality.

Friday, Dec 2. Quantum cryptography: the BB84 protocol. Background: classical cryptography. Security vs practicability; one-time pad and RSA.