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Oxford University Press eBooks

The dynamics of microscopic systems

Peter Joshua Atkins, Julio de Paula, David Young Smith

Published Dec 15, 2016• 0 citations
Publisher

Abstract

This chapter discusses the central concepts and equations of quantum theory which demonstrates how particles spread through space like waves and are described mathematically by a wavefunction. It examines how the wavefunction is interpreted and introduces the uncertainty principle, which is one of the most profound departures of quantum mechanics from classical mechanics. It also highlights the dynamical properties of a system that are contained in the wavefunction and are obtained by solving the Schrödinger equation. The chapter reconciles the facts that atoms and molecules can possess only certain energies, that waves exhibit the properties of particles, and that particles exhibit the properties of waves. It reviews the classical concept that have been accommodated by the development of quantum mechanics, in which equations are set up to treat a particle as spread through space like a wave.

DOI
10.1093/hesc/9780198727873.003.0046

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This chapter discusses the central concepts and equations of quantum theory which demonstrates how particles spread through space like waves and are described mathematically by a wavefunction. It examines how the wavefunction is interpreted and introduces the uncertainty principle, which is one of the most profound departures of quantum mechanics from classical mechanics. It also highlights the dynamical properties of a system that are contained in the wavefunction and are obtained by solving the Schrödinger equation. The chapter reconciles the facts that atoms and molecules can possess only certain energies, that waves exhibit the properties of particles, and that particles exhibit the properties of waves. It reviews the classical concept that have been accommodated by the development of quantum mechanics, in which equations are set up to treat a particle as spread through space like a wave.
This chapter lays out the principles of quantum theory. Quantum theory emerged when experimental results on matter and radiation were found to diverge from the predictions of classical mechanics. It was found that particles had the properties of waves, and that waves had the properties of particles. That duality is taken into account by the Schrödinger equation and the introduction of a wavefunction. One of the great upsets of classical physics is embodied in the uncertainty principle, which expresses the impossibility of the simultaneous specification of certain pairs of properties. One consequence of duality is the extension of microscopy and diffraction to beyond the electromagnetic spectrum.
Abstract In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.
Introduction In this chapter we discuss the basic physical concepts and equations related to the behavior of particles in the nanoworld. We introduce the Schrödinger wave equation for particles and determine the ways to calculate observable physical quantities. We find that, in wave mechanics, the motion of a particle confined to a finite volume is always characterized by discrete values of the energy and standing-wave-like wavefunctions, i.e., such a motion is quantized. While motion in an infinite space (i.e., free motion) is not quantized and is described by propagating waves, the energy of the particle is characterized by a continuous range of values. Keeping in mind the diverse variants of nanostructures, by using wave mechanics we analyze some particular examples, which highlight important quantum properties of particles. Many of the examples analyzed can serve as the simplest models of nanostructures and will be exploited in following chapters to understand the fundamentals of processes in nanoelectronics. The Schrödinger wave equation From the previous chapter, we conclude that nanosize physical systems are quantum-mechanical systems, inasmuch as their sizes are comparable to typical de Broglie wavelengths of the particles composing these systems. In dealing with quantum-mechanical systems, one aims at determining the wavefunction of a single particle or of the whole system. As we will demonstrate in the subsequent discussion, knowledge of the wavefunction in quantum mechanics is sufficient to describe completely a particle or even a system of particles.
Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave-like behavior is an important part of the explanation of why atoms have discrete energy levels and make transitions in which electromagnetic radiation is absorbed or emitted at specific frequencies. Important aspects of how quantum systems may be observed and how they develop in time are presented. The Schrödinger equation is introduced and the importance of the wavefunction covered. Mathematical simplifications that arise from considering symmetry are covered. Several examples of simple quantum systems are presented in which the calculation of electronic energy structure based on boundary conditions is illustrated. Sections on wave–particle duality and quantum observables are followed by others on the dynamics of two-level systems, mathematically complete representations of arbitrary states, and standard approaches for handling the temporal development of wavefunctions.
This chapter examines the interpretation of the wavefunction, and specifically what it reveals about the location of a particle. In quantum mechanics, all the properties of a system are expressed in terms of a wavefunction, which is obtained by solving the equation proposed by Erwin Schrödinger. Indeed, wavefunctions provide the essential foundation for understanding the properties of electrons in atoms and molecules, and are central to explanations in chemistry. The chapter then considers how, according to the Born interpretation, the probability density at a point is proportional to the square of the wavefunction at that point. A wavefunction is normalized if the integral over all space of its square modulus is equal to 1. Ultimately, a wavefunction must be single-valued, continuous, not infinite over a finite region of space, and have a continuous slope. The quantization of energy stems from the constraints that an acceptable wavefunction must satisfy.