# Graduate Advanced Physics

### PHYSICS765

Dirac equation, canonical field quantization, gauge symmetries, electromagnetic field quantization, identical particles and second quantization, symmetry breaking, gases of interacting bosons and fermions, interaction of quantized radiation field with atoms, Rayleigh scattering, Thomson scattering, nonlinear optical processes, and special topics. 3 units.

#### Possible principal texts:

- E. Merzbacher,
*Quantum Mechanics*, 3^{rd}ed., (1997). - Greiner and Mueller,
*Quantum Mechanics: symmetries*, (2004). - C. A. Brau,
*Modern Problems in Classical Electrodynamics*, (Oxford Univ. Press, 2004). - Pathria,
*Statistical Mechanics*, 2^{nd}ed. (Elsevier, 1996). - C. Cohen-Tannoudji et al.,
*Quantum Mechanics*, 2^{nd}ed. (2006).

#### Other texts to consider:

- J. J. Sakurai,
*Advanced Quantum Mechanics*, (1967). - M. Peskin and D. Schroeder,
*Quantum Field theory*, (1995). - E. M. Lifshitz and L. Pitaevskii,
*Statistical Physics*,*part 2*, from the Landau and Lifshitz series, vol. 9 (1980). - R. Loudon,
*The quantum theory of light*, 3^{rd}ed., (Oxford, 2000). - L. D. Landau and E. Lifschitz,
*Electrodynamics of Continuous Media*(vol. 8).

#### Prerequisites

The prerequisites are **PHYSICS 311** and **PHYSICS 312**.

#### Syllabus

- Canonical field quantization.
- Dirac equation.
- Quantization of EM field in radiation gauge, coherent states.
- Density matrix.
- Second quantization for bosons, weakly interacting Bose gas, low energy excitations.
- Second quantization for fermions, weakly interacting Fermi gas.
- Gauge symmetries: U(1), SU(2), SU(3), symmetry breaking, Goldstone bosons, Higgs mechanism.
- Interaction of radiation with atoms, Rayleigh and Thomson scattering.
- Microscopic theory of dielectric function
- Microscopic theory of magnetism: exchange coupling, Hubbard model, Heisenberg model, spin waves.
- Nonlinear materials: multiphoton processes, Raman scattering.

*Sample* lecture schedule

(based on 25 lectures each of duration 75 minutes).

*Lecture 1:*Canonical field quantization1: scalar field.*Lecture 2:*Canonical field quantization 2: scalar field.*Lecture 3:*Dirac equation 1*Lecture 4:*Dirac equation 2*Lecture 5:*Gauge symmetries 1: U(1)*Lecture 6:*Quantization of EM field in radiation gauge.*Lecture 7:*EM field quantization – coherent states.*Lecture 8:*Density matrix.*Lecture 9:*Second quantization for bosons.*Lecture 10:*Weakly interacting Bose gas, low energy excitations.*Lecture 11:*Second quantization for fermions.*Lecture 12:*Weakly interacting Fermi gas.*Lecture 13:*Gauge symmetries 2: SU(2) and SU(3)*Lecture 14:*Gauge symmetries 3: symmetry breaking, Goldstone bosons, Higgs mechanism.*Lecture 15:*Special topics.*Lecture 16:*Special topics.*Lecture 17:*Interaction of radiation with atoms 1.*Lecture 18:*Interaction of radiation with atoms 2.*Lecture 19:*Rayleigh and Thomson scattering.*Lecture 20:*Microscopic theory of dielectric function*Lecture 21:*Microscopic theory of magnetism 1: exchange coupling, Hubbard model.*Lecture 22:*Microscopic theory of magnetism 2: Heisenberg model, spin waves.*Lecture 23:*Nonlinear materials 1: multiphoton processes.*Lecture 24:*Nonlinear materials 2: Raman scattering.*Lecture 25:*Special topics.