Quantum Mechanics Course - Polytechnic of Milan

Repo for 'Mathematics of Quantum Mechanics' course at the Polytechnic of Milan, Master Course in Mathematical Engineering (AY 2023-2024). This repository contains homework assignments provided by the course instructor and my solutions to those assignments.

(Final grade: 30 cum laude / 30).

Course Overview

Topics Covered

1. Crisis of Classical Physics

2. Postulates of Quantum Mechanics

3. Hilbert Spaces and Operator Theory

   • Fourier transform, Sobolev spaces and distributions; linear operators: adjoint and inverse operator, symmetric, self-adjoint and unitary operators, projectors; Heisenberg uncertainty relation; self-adjointness criterion and self-adjoint extensions; spectrum and resolvent; spectral theorem and probabilistic interpretation of quantum mechanics; spectral decomposition; compact operators; mixed states and density matrices; Stone theorem and existence of the dynamics.

4. Free Particle and Harmonic Oscillator

   • time evolution of wave packets, dispersion and interference, semiclassical limit; harmonic oscillator: spectrum, creation and annihilation operators, coherent states.

5. Hydrogen Atom

   • Self-adjointness, spectrum, angular momentum, and eigenfunctions.

Repository Contents

Homeworks

• Homework 1

  • Shift Operators and Spectral Properties.

• Homework 2

  • Kinetic Energy on the interval, self-adjoint extension and perturbation theory.

• Homework 3

  • Perturbed Magnetic Laplacian on the whole space, bounds for the ground state.

Seminar

• A variational approach to energy minimization for pertubed Hamiltonians.

For more information about the course, visit the Polytechnic of Milan website.