# Quantum Mechanics

### PHYSICS764

Angular momentum and symmetries in quantum mechanics from group theory viewpoint; time-independent and time-dependent perturbation theory; path integral formulation; scattering theory; identical particles; applications. 3 units.

#### Possible principal texts:

- Cohen-Tannoudji, Diu and Laloe,
*Quantum Mechanics*- 2 vols. (Wiley-Interscience) - Shankar,
*Principles of Quantum Mechanics*(Springer) - Greiner, Müller,
*Quantum Mechanics: Symmetries*(Springer)

#### Other texts to consider:

- Sakurai,
*Modern Quantum Mechanics*(Addison-Wesley) - Merzbacher,
*Quantum Mechanics*(Wiley, 3^{rd}edition)

#### Prerequisites

The prerequisite is at least one semester of a Quantum Mechanics course at the level of **R. ****Shankar**'s textbook.

In Duke Physics, there is an undergraduate "Quantum Mechanics I" course (PHY 211), with the synopsis: Experimental foundation of quantum mechanics; wave-particle duality; the single-particle Schrodinger equation and the physical meaning of the wave function; methods for studying the single-particle Schrodinger equation; analytical solutions of the harmonic oscillator and hydrogen atom and experimental tests of these solutions; angular momentum and spin systems; and finally the many-particle Schrodinger equation and consequences of identical particles existing in nature.

#### Syllabus

- Time-independent perturbation theory.
- The “real” hydrogen atom.
- Identical particles, exchange interaction, helium atom.
- Many-body states, Slater determinant, Hartree-Fock approximation.
- Variational method: hydrogen molecule, chemical binding.
- Periodic potential, Bloch waves, band structure.
- Time-dependent perturbation theory, Fermi’s Golden Rule.
- Application to two-state system (e.g., spin rotations, NMR).
- Elementary two-state systems: neutral kaons or neutrino oscillations.
- Continuous symmetries, Noether’s theorem, rotation group SO(3).
- Addition of angular momenta, Clebsch-Gordon coefficients.
- Tensor operators, Wigner-Eckart theorem.
- SU(2) and its relationship to SO(3), isospin (weak & strong).
- Path integral formulation of QM: Principles, free particle, semiclassical limit, particle on a circle, Berry’s phase.
- WKB approximation.
- Scattering theory: cross section, S-matrix, T-matrix, unitarity, Born approximation, partial waves, optical theorem.

Choice of special topics: quantum information theory, renormalization group, etc.

*Sample* lecture schedule

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

*Lecture 1:*__Review__of undergrad QM, part 1: Hilbert space, states and observables, measurement, uncertainty relation, identical particles, two-state system.*Lecture 2:*__Review__of undergrad QM, part 2: Schrödinger equation, harmonic oscillator, angular momentum, “ideal” hydrogen atom.*Lecture 3:*Time-independent perturbation theory.*Lecture 4:*The “real” hydrogen atom.*Lecture 5:*Identical particles, exchange interaction, helium atom.*Lecture 6:*Many-body states, Slater determinant, Hartree-Fock approximation.*Lecture 7:*Variational method: hydrogen molecule, chemical binding.*Lecture 8:*Periodic potential, Bloch waves, band structure.*Lecture 9:*Time-dependent perturbation theory, Fermi’s Golden Rule.*Lecture 10:*Application to two-state system (e.g., spin rotations, NMR).*Lecture 11:*Elementary two-state systems: neutral kaons or neutrino oscillations.*Lecture 12:*Continuous symmetries, Noether’s theorem, rotation group SO(3).*Lecture 13:*Addition of angular momenta 1.*Lecture 14:*Addition of angular momenta, Clebsch-Gordon coefficients.*Lecture 15:*Tensor operators, Wigner-Eckart theorem.*Lecture 16:*SU(2) and its relationship to SO(3), isospin (weak & strong).*Lecture 17:*Path integral formulation of QM: Principles, free particle.*Lecture 18:*Path integral – semiclassical limit.*Lecture 19:*WKB approximation.*Lecture 20:*Path integral – example: particle on a circle, Berry’s phase.*Lecture 21:*Scattering theory: cross section, S-matrix, T-matrix, unitarity.*Lecture 22:*Scattering theory: Born approximation.*Lecture 23:*Scattering theory: Partial waves, optical theorem.*Lecture 24:*Special topics.*Lecture 25:*Special topics.