Michael MOVSHEV 

MAT 570: Concepts of Quantum Mechanics


Day of:

Material Covered

Suggested Reading

Week 1:

Jan. 28
Lagrangian Mechanics:The principle of the least action, Euler-Lagrange equation.
[Bryant] Lect4
Jan. 30
Symmetries and Noether theorem,Legendre transform
[Bryant] Lect4
Feb. 1
One-dimensional motion.
The motion in a central field and the Kepler problem.
[Takhtajan] Sec.1.5

Week 2:

Feb. 4
Hamiltonian Mechanics. Hamilton’s equations. The action functional in the phase space.
[Bryant] Lect5, Lect6
[Takhtajan] Sec.2.2
Feb. 6
Classical observables and Poisson bracket. Symplectic and Poisson manifolds.
[Takhtajan] Sec2.4-2.7
Feb. 8
Hamilton’s and Liouville’s representations.
[Takhtajan] Sec2.8

Week 3:

Feb. 11
Observables, states and dynamics in Quantum Mechanics.
[Takhtajan] Sec2.1
Feb. 13
Procedure of Quantization. Heisenberg commutation relations. Coordinate and momentum representations. Weyl relations.
[Takhtajan] Sec2.2 Sec 2.3
Feb. 15
Schroedinger Equation. Harmonic oscillator. Hydrogen atom
[Takhtajan] Sec3.1-Sec3.5

Week 4:

Feb. 18
Geometric asymptotics.
[Guillemin, Sternberg] Ch 2
[Takhtajan] Sec3.6
Feb. 20
Geometric asymptotics.
[Guillemin, Sternberg] Ch 2
[Takhtajan] Sec 3.6
Feb. 22
Geometric quantization.
[Woodhouse] Sec 8

Week 5:

Feb. 25
Geometric quantization.
[Woodhouse] Sec 9
Feb. 27
Geometric quantization.
[Woodhouse] Sec 9
Feb. 29
Fedosov's quantization

Week 6:

March 3
Kontsevich's quantization
March 5
Kontsevich's quantization
March 7
Spin and representation theory of SU(2).
Charged spin particle in the magnetic field
[Takhtajan] Ch4.1-Sec 4.2

Week 7:

March 10
Feynman path integral Symbols of the evolution operator and path integrals.
[Grosche] SecII
[Rabin] Lec 6
March 12
Feynman path integral Symbols of the evolution operator and path integrals. Continued
[Grosche] SecII
[Rabin] Lec6
Gaussian path integrals. Regularized determinants of differential operators
[Takhtajan] Sec 5.3Sec 5.5

Week 8:

Spring Recess

Week 9:

March 24
The Harmonic Oscillator via Path Integral
[Rabin] Lect. 7
[Takhtajan] Sec5.4
March 26
Semi-classical asymptotics . Diagrammatic
[Itzykson_Zuber] Sec6.2
March 28
Schwinger-Dyson equation. Semi-classical asymptotics continued
[Takhtajan] Sec5.6

Week 10:

March 31
Introduction to super algebra
Apr. 2
Introduction to super geometry
Apr. 4
Classical mechanics of a super particle, Supersymmetry.
[Deligne Freed]

Week 11:

Apr. 7
Quantization of a super particle. Atiyah-Singer index theorem: Topological prerequisites
Apr. 9
Localization and Duistermaat-Heckman formula,
[Guillemin V.."Moment_maps..."] Sec2
[Atiyah, M. F.,Bott R.] Sec3
Apr. 11
One-loop computations

Week 12:

Apr. 14
Classical systems with constraints. First and second order constraints.
[Henneaux_Teitelboim] Ch1
Apr. 16
Quantization of systems with constraints. BRST and BV
Apr. 18
Symplectic and quantum reductions. Toric manifolds.
[Bryant] Lec7
[Guillemin V.."Moment_maps..."]

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