Introduction to Quantum Mechanics

Schrödinger Equation and Path Integral

This text on quantum mechanics begins by covering all the main topics of an introduction to the subject. It then concentrates on newer developments. In particular it continues with the perturbative solution of the Schrödinger equation for various potentials and thereafter with the introduction and evaluation of their path integral counterparts. Considerations of the large order behavior of the perturbation expansions show that in most applications these are asymptotic expansions. The parallel consideration of path integrals requires the evaluation of these around periodic classical configurations, the fluctuation equations about which lead back to specific wave equations. The period of the classical configurations is related to temperature, and permits transitions to the thermal domain to be classified as phase transitions.

In this second edition of the text important applications and numerous examples have been added. In particular, the chapter on the Coulomb potential has been extended to include an introduction to chemical bonds, the chapter on periodic potentials has been supplemented by a section on the band theory of metals and semiconductors, and in the chapter on large order behavior a section has been added illustrating the success of converging factors in the evaluation of asymptotic expansions. Detailed calculations permit the reader to follow every step.

Contents:

• Introduction
• Hamiltonian Mechanics
• Mathematical Foundations of Quantum Mechanics
• Dirac's Ket- and Bra-Formalism
• Schrödinger 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
• Summarizing Remarks

Readership: Undergraduates and graduate students in physics; researchers in mathematical and particle physics.
Key Features:

• Different from other texts on the subject, this text shows that perturbation theory is much more useful than is often believed
• Dealing with the complementary path integrals, it is shown that these can largely be evaluated with the help of elliptic integrals (another specific aspect of this text)
• This new edition contains many additional examples and applications, specifically on chemical bonds and metals