After a consideration of basic quantum mechanics, this introduction aims at a side by side treatment of fundamental applications of the Schroedinger equation on the one hand and the applications of the path integral on the other. Different from traditional texts and using a systematic perturbation method, the solution of Schroedinger equations includes also those with anharmonic oscillator potentials, periodic potentials, screened Coulomb potentials and a typical singular potential, as well as the investigation of the large order behavior of the perturbation series. On the path integral side, after introduction of the basic ideas, the expansion around classical configurations in Euclidean time, such as instantons, is considered, and the method is applied in particular to anharmonic oscillator and periodic potentials. Numerous other aspects are treated on the way, thus providing the reader an instructive overview over diverse quantum mechanical phenomena, e.g. many other potentials, Green's functions, comparison with WKB, calculation of lifetimes and sojourn times, derivation of generating functions, the Coulomb problem in various coordinates, etc. All calculations are given in detail, so that the reader can follow every step.
Hamiltonian Mechanics; Mathematical Foundations of Quantum Mechanics; Dirac's Ket- and Bra-Formalism; Schrodinger Equation and Liouville Equation; Quantum Mechanics of the Harmonic Oscillator; Green's Functions; Time-Independent Perturbation Theory; The Density Matrix and Polarization Phenomena; Quantum Theory: The General Formalism; The Coulomb Interaction; Quantum Mechanical Tunneling; Linear Potentials; Classical Limit and WKB Method; Power Potentials; Screened Coulomb Potentials; Periodic Potentials; Anharmonic Oscillator Potentials; Singular Potentials; Large Order Behavior of Perturbation Expansions; The Path Integral Formalism; Classical Field Configurations; Path Integrals and Instantons; Path Integrals and Bounces on a Line; Periodic Classical Configurations; Path Integrals and Periodic Classical Configurations; Quantization of Systems with Constraints; The Quantum-Classical Crossover as Phase Transition.