Quantum Mechanics I

First midterm: Monday, September 9th;

Second midterm: Wednesday, October 23rd.

Third midterm: Wednesday, November 27th.

The grades will be assigned as follows:

A = 8.5 ≤ grade ≤ 10

B = 7 ≤ grade < 8.5

C = 5.5 ≤ grade < 7

Below you can find the preliminary plan for the lectures. It is possible that some details will change along the semester, but not the overall plan.

Books: D = Dirac, S = Sakurai, LB = Le Bellac, G = Gottfried-Yan, F = Feynman-Hibbs, M = Manousakis

- Mathematical structure of QM in finite dimensions 1: Hilbert space, dual space, self-adjoint operators and their properties [D 1, 2; LB 2.1, 2.2; S 1.2, 1.3; G 2.1]
- Mathematical structure of QM in finite dimensions 2: spectral decomposition of hermitian and unitary operators [D 2; LB 2.3]
- Postulates of QM [D 1,2; LB 4.1, 4.2; G 2.4]
- Applications of the postulates 1: ammonia and benzene molecule [LB 5.1]
- Applications of the postulates 2: time-independent perturbation theory [G 3.7a, 3.7b]
- Applications of the postulates 3: time-dependent problems and perturbation theory [M 22]
- The path integral formulation: construction of the path integral, classical limit, the wave function, Schrodinger equation, the path integral for the free particle [F 1, 2, 3, 4]
- Position and momentum operators
- Schrödinger equation for systems in d=1: general properties and simple examples
- Schrödinger equation for systems in d=1: harmonic oscillator (Frobenius and algebraic resolution)
- Schrödinger equation for systems in d=1: harmonic oscillator (path integral resolution and coherent states)
- Quantum Mechanics in 3 dimensions, Ehrenfest theorem, parity as a first example of symmetry
- Symmetries I: canonical quantization, symmetries from canonical quantization, action of symmetries on states, Wigner theorem, projective representations, Lie groups and Lie algebras
- Symmetries II: space and time translations, symmetries of the Hamiltonian, rotations in N dimensions, structure constants, rotations in 3 dimensions
- Symmetries III: construction of the unitary representations of SO(3), examples in 2 and 3 dimensions
- Symmetries IV: more on projective representations, SU(2) as the covering group of SO(3)
- Sum of angular momenta, Clebsch-Gordan coefficients [G3.5, LB10.6.2], and connection with group representations
- Orbital angular momentum and spherical harmonics
- Schrödinger equation for systems in d=3: free particle in spherical coordinates [G3.4], two body problem [G3.6]
- Schrödinger equation for systems in d=3: central potential and radial equation [G3.6b, LB10.4.1], hydrogen atom [G3.6c, LB10.4.2]
- Wigner-Eckart theorem and spherical tensors [see Robert Littlejohn lectures]
- ...

- List 1 (Lectures 2 and 3)
- List 2 (Lectures 4 and 5)
**updated 2019** - List 3 (Lectures 6 and 7)
**updated 2019** - List 4 (Lectures 8 and 9 plus more exercises on perturbation theory in finite dimensions
- List 5 (Position and momentum representations of QM)
- List 6 (Systems in 1 dimension and harmonic oscillator)
- List 7 (Symmetries)
**updated 2019** - List 8 (Orbital angular momentum and spin)
- List 9 (Hydrogen atom and angular momenta composition)
- List 10 (Wigner-Eckart, fine and hyperfine structure)

- E. Manousakis,
*"Practical Quantum Mechanics"* - M. Le Bellac,
*"Quantum Physics"* - K. Gottfried, T. Yan,
*"Quantum Mechanics: Fundamentals"* - J.J. Sakurai,
*"Modern Quantum Mechanics"* - R. Shankar,
*"Principles of Quantum Mechanics"* - S. Weinberg,
*"Lectures on Quantum Mechanics"* - R. Feynman, A. Hibbs,
*"Path Integrals and Quantum Mechanics"* - K. Konishi, G. Paffuti,
*"Quantum Mechanics: A new introduction"*

- Online notes by Robert Littlejohn
- Lectures on Path Integral in Quantum physics by R. Rosenfelder (arXiv:1209.1315)
- Advanced Quantum Mechanics by Alexander Altland
- Notes on tensor products by Hitoshi Murayama