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

- To give all attending students an overview over fundamental concepts and modern developments in quantum mechanics and its applications.
- To prepare those students whose graduate work will involve quantum mechanics for the more advanced courses on the subject.

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.

**Special announcements:**

- Midterm: Fri, Oct 21 (in class).

**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).

**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.