After a consideration of basic quantum mechanics, this introduction aims at a side by side treatment of fundamental applications of the Schr?inger 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 Schr?inger 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.
TABLE OF CONTENTS 1 Introduction 1 2 Hamiltonian mechanics 23 3 Mathematical foundations of quantum mechanics 41 4 Dirac's ket and braformalism 59 5 Schrodinger equation and Liouville equation 73 6 Quantum mechanics of the harmonic oscillator 83 7 Green's functions 105 8 Timeindependent perturbation theory 129 9 The density matrix and polarization phenomena 161 10 Quantum theory : the general formalism 169 11 The Coulomb interaction 199 12 Quantum mechanical tunneling 249 13 Linear potentials 265 14 Classical limit and WKB method 281 15 Power potentials 307 16 Screened Coulomb potentials 319 17 Periodic potentials 339 18 Anharmonic oscillator potentials 379 19 Singular potentials 435 20 Large order behaviour of perturbation expansions 471 21 The path integral formalism 503 22 Classical field configurations 537 23 Path integrals and instantons 583 24 Path integrals and bounces on a line 619 25 Periodic classical configurations 649 26 Path integrals and periodic classical configurations 675 27 Quantization of systems with constraints 715 28 The quantumclassical crossover as phase transition 753 29 Summarizing remarks 773 A Properties of Jacobian elliptic functions 775
