The Basics of Quantum Mechanics

Abstract

AbstractThis chapter gives a short introduction to quantum mechanics starting from de Broglie’s and Schrödinger’s wave picture. The emphasis is on the mathematical structure of the theory with the aim to form a sound basis for the later study of the electronic Schrödinger equation. The discussion starts in the first two sections with a heuristic derivation of the Schrödinger equation for a single free particle from which, in the third section, the general mathematical framework of quantum mechanics is derived. The fourth section deals with a particular simple quantummechanical system, the harmonic oscillator. The harmonic oscillator serves on one hand as an example of a quantum-mechanical system with completely different properties from the free particle and is ideal to exemplify and illustrate the general concepts of quantum theory. On the other hand the explicit knowledge of its solutions will in later chapters help to develop the mathematical theory further. In the fifth section the weak form of the Schrödinger equation is derived and physically motivated. The equivalence of the weak formulation to the classical operator formulation is shown. In later chapters we will exclusively work with the weak form that is basic for the L2-theory of partial differential equations. The last section is devoted to many-particle systems. The central point here are the symmetry properties of the many-particle wave functions that are not only fundamental for the structure of matter and responsible for many of the strange properties of quantum systems but that will also turn out to be essential for the regularity theory of the electronic Schrödinger equation and for the study of its computational complexity.KeywordsWave FunctionHilbert SpaceQuantum MechanicsHarmonic OscillatorFree ParticleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.