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 bra-formalism 59
5 Schrodinger equation and Liouville equation 73
6 Quantum mechanics of the harmonic oscillator 83
7 Green's functions 105
8 Time-independent 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 quantum-classical crossover as phase transition 753
29 Summarizing remarks 773
A Properties of Jacobian elliptic functions 775