Objectives and Goals
This course is designed to be the third in the sequence of
required graduate quantum mechanics courses. Prerequisites are
the successful completion of PHY 211 and PHY 212 or passing a
test on the material covered in PHY 211 and PHY 212. PHY 315
builds upon the core formalism, concepts, and techniques of
quantum mechanics. While both angular momentum and symmetries in
quantum mechanics have been previously explored, they are
unified in their treatment here. Scattering theory, first
encountered in PHY 212, is treated with more depth and
sophistication here. Many body quantum mechanics, quantization
of the electromagnetic field, and the relativistic treatment of
spin-1/2 particles are presented. These concepts are important
for research in both the theoretical and experimental realms of
particle physics, nuclear physics, quantum optics, quantum dots,
quantum fabrication, etc.
The course provides exposure to the following topics:
- The rotation group and its irreducible unitary
representations;
- Group theory perspective of angular momentum
addition;
- Scattering Theory and the S-Matrix Formalism;
- Bound-states and Resonances;
- Identical Particles and Fock Space formalism;
- Many Body Quantum Mechanics and Applications;
- Quantization of relativistic particles;
- Quantum Mechanics of Phonons, Photons and Electrons;
- Photo-Electric Effect and Sponteneous Emission;
This course also serves to develop the deep conceptual
understanding of quantum mechanics using an appropriate level of
mathematical rigor which is necessary for doctoral research.
Methods and Approach
- Format:
-
This course is taught through 50-minute lectures (3 per week)
and weekly homework sets. Depending on the pace of the course
and the topics covered, lectures may continue until the last
day of the undergraduate classes. Lectures generally involve
blackboard presentations by the professor, but student
participation is encouraged. Homework sets generally consist
of 7-8 problems designed to take approximately 12 hours of
concentrated effort to complete. Students are encouraged to
discuss the homework with their peers, but are required to
write solutions independently. Students are responsible for
all material covered in the lectures and in the homework
problems. Some concepts and applications that are important
are covered only in the homework.
- Texts:
-
The primary texts for the course are the latest editions of
Shankar's "Principles of Quantum Mechanics" and Merzbacher's
"Quantum Mechanics". Students are encouraged to consult
additional texts. Some of these that can be quite useful
are:
- Weinberg, "The Quantum Theory of Fields", Vol I.
- Negele, "Quantum Many Particle Systems"
- Greiner, " Quantum Mechanics Special Chapters"
- Sakurai, "Advanced Quantum Mechanics"
- Exams and Grades:
-
There are two midterm exams and one final exam for the
course. Exams are designed to test each student's grasp of the
fundamental concepts. Exams will ingeneral test not only the
material covered in the class but also a basic understanding
of quantum mechanics that the student is expected to have
obtained from previous courses such as PHY211 and
PHY212. Grades for the course are determined by homework
(30%), midterm(s) (30%), and the final exam (40%).
Sample Syllabus
(Each bullet represents one week.)
- Lecture 1: Symmetries and Group Theory
Lecture 2: Basics of Group Theory
Lecture 3: Representations: Reducible vs.
Irreducible- Lecture 4: Discrete Groups: Permutation Group,
CPT
Lecture 5: Continuous Groups: SU(N), Lie
Algebras
Lecture 6: SU(2), Rotations and its Irreducible
Representations- Lecture 7: Addition of Angular Momentum
Lecture 8: Tensor Operators
Lecture 9: Wigner Eckart Theorem- Lecture 10: Classical Scattering
Lecture 11: Quantum Mechanical Scattering: Time
Dependent vs. Time Independent Descriptions
Lecture 12: In and Out States: S-Matrix- Lecture 13: Lippmann Schwinger Equation and the
Greens function solution
Lecture 14: Warm up: Applications in 1d
Lecture 15: 3-d Scattering and Partial Wave
Expansion- Lecture 16: Example: Spherical Well
Lecture 17: Exam 1
Lecture 18: Perturbation Theory: Born
Approximation- Lecture 19: Scattering cross section Bound States
and Resonances
Lecture 20: WKB Approximation
Lecture 21: Application of WKB to Bound States- Lecture 22: Application of WKB to Tunneling and
Spontaneous Symmetry Breaking
Lecture 23: Identical Particles and Spin
Statistics
Lecture 24: Fock Space Formalism- Lecture 25: Field Operators and Schrodingers
Equation
Lecture 26: Free Particles and Quantum
Statistics
Lecture 27: Interactions and Perturbation Theory- Lecture 28: One-dimensional example with
delta-function interaction
Lecture 29: The Hartree Fock Approximation
Lecture 30: Quasi Particles and Fermi Liquids- Lecture 31: Exam 2
Lecture 32: Classical Description of Lattice
Vibrations
Lecture 33: Quantizing Lattice Vibrations:
Phonons- Lecture 34: Classical E-M Fields: Maxwells
Equations
Lecture 35: Quantizing the E-M Field: Photons
Lecture 36: Interaction of photons with Atom:- Lecture 37: Photo-Electric-Effect.
Lecture 38: Vacuum Fluctuations and Sponteneous
Emission
Lecture 39: The Dirac Hamiltonian, Fock Space
Description- Lecture 40: Negative Energy States, Dirac Sea and
Positrons
Lecture 41: Non-Relativistic Approximation and
Spin