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 timedependent SchrÃ¶dinger 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 FeynmanKac formula. This lecture will be selfcontained, although some results will be accepted without proofs.
The lecture notes of Vassili Gelfreich give a beautiful summary of the material. I am currently working on lecture notes, a preliminary draft is
available here.

V. Betz,
2012 lecture notes, unpublished

W.G. Faris,
Outline of Quantum Mechanics, in Entropy and the Quantum, Contemp. Math. 529, 152 (2010)

J. Fröhlich, B. Schubnel,
Do we understand quantum mechanics  finally?, 2012.

V. Gelfreich,
2018 lecture notes, available online

S.J. Gustafson, I.M. Sigal,
Mathematical concepts of quantum mechanics, Springer, 2003.

A. Messiah,
Quantum mechanics, Dover, 1999.