Quantum mechanics is one of the most successful and most fundamental scientific theories. It is fundamental in the description of atomic spectra, chemical reactions, electronic properties of condensed matter, superconductivity, etc... This lecture will contain a necessarily brief introduction to some of the fundamental principles of quantum theory: Wave functions in Hilbert space, stationary and time-dependent Schrödinger equations, uncertainty principle, harmonic oscillator and Hydrogen atom.

Mathematically, we will use notions of analysis, PDEs, Fourier analysis, functional analysis, algebra, and probability theory. We will review in particular the spectral theorem for unbounded operators and the Feynman-Kac formula. This lecture will be self-contained, although some results will be accepted without proofs.

Here are lecture notes.