The third volume of W. Thirring's famous textbook on mathematical physics is devoted to quantum mechanics and especially to its applications to atomic systems. The author builds the theory on an axiomatic basis and uses the functional analysis of unbounded operators to bridge the gap to physics and its experiments. Mathematics is shown to explain the axioms in depth and to provide the right tool for testing numerical data in experiments. The topics covered are Hilbert spaces, the theory of unbounded operators, scattering theory, atoms and molecules.