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.

### Chapter

## The dynamics of microscopic systems

### Chapter

## Introduction and orientation

This chapter introduces two approaches to quantum mechanics, one is to follow the historical development of the theory from the first indications that the whole fabric of classical mechanics and electrodynamics should be held in doubt to the resolution of the problem. The other approach is to stand back at a point late in the development of the theory and see its underlying theoretical structure. The chapter shows that in the first approach the theory is seen gradually emerging from confusion and dilemma, while the second is a more formal approach that can still be exciting and compelling in a different sense. The chapter highlights the logic and elegance in a scheme that starts from only a few postulates, yet reveals a rich, experimentally verifiable structure. It traces some of the most fundamental concepts of the nature of matter and its behaviour that were overthrown and replaced by a puzzling but powerful new description.

### Chapter

## The quantum mechanics of motion

This chapter pays particular attention to the quantum mechanics of motion. It is not possible to explain the structures and properties of atoms and molecules, where the role of electrons is central, without using quantum mechanics. Many of the processes of biochemistry involve the transfer of electrons and protons from a donor to an acceptor, and this migration can be understood only in terms of quantum theory. Here, the chapter introduces three fundamental types of motion — translation, rotation, and vibration — along with their characteristic quantum mechanical behaviours. It shows how various unexpected properties emerge, such as the ability of particles to penetrate into classically forbidden regions, the quantization of energy and angular momentum, and the existence of energy that cannot be removed from the system.

### Chapter

## 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.

### Book

### Peter Atkins and Ronald Friedman

Molecular Quantum Mechanics shows how the subject of quantum mechanics embraces the behaviour of all known forms of matter, including the atoms and molecules from which we, and all living organisms, are composed. The book leads us through this subject, exploring the fundamental physical principles that explain how all matter behaves. The text takes us from the foundations of quantum mechanics, through quantum models of atomic, molecular, and electronic structure, and on to discussions of spectroscopy, and the electronic and magnetic properties of molecules.

### Chapter

## Rotational motion and the hydrogen atom

This chapter considers the quantum mechanics of translation and vibration. Both types of motion can be solved in certain cases, and both are important not only in their own right but also because they form a basis for the description of more complicated types of motion. It mentions translational motion, which has the advantage of introducing in a simple way many of the striking features of quantum mechanics. The chapter describes certain features of wavefunctions that are common to all the problems. The combination of features of wavefunctions with the solution of the Schrödinger equation results in one of the most characteristic features of quantum mechanics, the quantization of energy.

### Chapter

## Linear motion and the harmonic oscillator

This chapter expresses the whole of quantum mechanics in terms of a small set of postulates, which embraces the behaviour of all known forms of matter. It introduces the postulates and illustrates how they are used, including how to apply them to problems of chemical interest, such as atomic and molecular structure and the properties of molecules. It also establishes the full significance of the Hψ=Eψequation and provides a foundation for its application. The chapter includes the solutions of the Schrödinger equation for a particle and the operator, which is a symbol for an instruction to carry out some action on a function. The principal mathematical difference between classical mechanics and quantum mechanics is that whereas in the former physical observables are represented by functions, in quantum mechanics they are represented by mathematical operators.

### Chapter

## Angular momentum

This chapter considers rotational motion as the second class of motion of an object around a fixed point and angular momentum, which is one of the most important topics in quantum mechanics. It discusses the rotational motion and angular momentum in terms of solutions of the Schrödinger equation and see how its properties emerge from the operators for angular momentum and their commutation relations. It also examines the quantum mechanical description of a particle travelling on a circular ring, which applies to the motion of a bead on a circle of wire and any body rotating in a plane. The chapter describes the moment of inertia the property, which determines the characteristics of the rotational motion of a body. It cites a particle of mass m travelling on a circle of radius r in the xy-plane and its potential energy that is constant and taken to be zero.

### Chapter

## Atomic spectra and atomic structure

This chapter explores the point at which the hope of finding exact solutions is set aside, starting with looking for methods of approximation. It cites the problems of quantum chemistry that cannot be solved exactly, such as the problem for which the Schrödinger equation can be solved exactly. As soon as the shape of the potential is distorted from the forms already considered, or more than two particles interact with one another, the equation cannot be solved exactly. The chapter highlights several ways of making progress, such as being guided to a form for the wavefunction by using principles of classical mechanics and taking note of the small magnitude of Planck's constant. Another way is to guess the shape of the wavefunction of the system and make use of self-consistent field procedures, which is an iterative method for solving the Schrödinger equation for systems.

### Chapter

## Techniques of approximation

This chapter assesses the mathematical theory of symmetry, which is one of the most remarkable in quantum mechanics. Not only does it simplify calculations, but it also reveals unexpected connections between apparently disparate phenomena. It considers angular momentum as a part of group theory and as properties of the harmonic oscillator. The chapter talks about the conservation of energy and of momentum that can be discussed in terms of group theory, which is used to classify the fundamental particles and the selection rules that govern what spectroscopic transitions are allowed. Integrals occur throughout quantum chemistry, for they include expectation values, overlap integrals, and matrix elements.

### Chapter

## Translation

This chapter describes a particle confined to a finite region of space that can possess
only certain discrete energies. It also looks at the corresponding wavefunctions. It
analyzes the quantization of energy that emerges as a natural consequence of solving the
Schrödinger equation and the conditions imposed on it. It also introduces a striking
non-classical feature of small particles, which describes their ability to penetrate into
and through regions where classical physics would forbid them to be found. The chapter talks
about the translational energy levels of a particle which are confined to a finite region of
space and are quantized. It explores how quantum mechanics reveal the ability of a particle
to penetrate into and through regions where classical physics would forbid it to be
found.

### Book

### T. P. Softley

Atomic Spectra starts off by looking at quantum mechanics and the relationship of quantum mechanics with light. The next chapter considers the structure and spectrum of the hydrogen atoms. The text also covers the spectrum of the helium atom. Finally, the text examines the spectra of many-electron atoms.

### Book

### H. Grant Guy and Richards W. Graham

Computational Chemistry starts by arguing that the uses of computers in chemistry are many and varied. This ranges from the modelling of solid state systems to the design of complex molecules which can be used as drugs. This text introduces the many methods currently used by practising computational chemists and shows the value of computers in modern chemical research. The text describes the various computational techniques available and explains how they can be applied to single molecules, to assemblies of molecules, and to molecules undergoing reaction. An introductory chapter outlines the hardware and software available, and looks at some applications and developments. Subsequent chapters cover quantum mechanics, molecular mechanics, statistical mechanics, the modelling of biomolecules, and drug design. Whilst emphasizing the use of computers to model biological systems, the chapters explain how the methods can be applied to a whole range of chemical problems.

### Book

### N. J. B. Green

Quantum Mechanics 1 starts from the basis that quantum mechanics is of central importance in chemistry. To understand matter and its chemical transformation it is necessary to take a microscopic view connecting experimental observation to the properties of constituent molecules. However, at this microscopic level, atoms and sub-atomic particles do not obey the classical laws of mechanics that pertain to the everyday macroscopic world. They obey the laws of quantum mechanics. This text explains the fundamentals of quantum mechanics from the point of view of chemistry; describes areas of chemistry where quantum mechanics is most important; and shows how quantum mechanics is applied to chemical problems. To this end, the book is divided into two parts: the first deals with the foundations of quantum mechanics (concentrating on exactly soluble problems and the reduction of complicated problems to simple foundations) and the second is a tool kit for applying quantum mechanics to chemical problems (thus concentrating on approximate methods).

### Chapter

## Introduction

This introductory chapter provides an overview of quantum mechanics, which is a subject of central importance in chemistry, and is by nature a mathematical theory. The chemist's approach to the understanding of matter and its chemical transformations is to take a microscopic view, connecting experimental observation with the properties of constituent molecules. However, at the microscopic level, atoms, molecules, and their constituent particles do not obey the familiar laws of mechanics that pertain to the everyday macroscopic world. For example, the existence of molecular line spectra indicates that molecules can be found only at certain well-defined energies, unlike objects at our own scale whose energy appears to be continuously variable. This book aims to explain the fundamentals of quantum mechanics from the point of view of chemistry; describe the areas of chemistry where quantum mechanics is most important; and show how quantum mechanics is applied to chemical problems.

### Book

### N. J. B. Green

Quantum Mechanics 2 opens by stating that the chemist's approach to the understanding of matter and its chemical transformations is to take a microscopic view, connecting experimental observation with the properties of the constituent molecules. Atoms and sub-atomic particles do not obey the classical laws of mechanics but conform rather to the laws of quantum mechanics. Quantum mechanics is thus of central importance in chemistry. In order to understand the behaviour of molecules and their constituent particles it is necessary to have a thorough grounding in the principles and applications of quantum mechanics. This text provides a toolkit for applying quantum mechanics to chemical problems, introducing more advanced approaches using approximate methods. It describes areas of chemistry where quantum mechanics is important, and shows how quantum mechanics can be applied to chemical problems.

### Chapter

## Translational motion

This chapter evaluates translational motion, motion through space, which is one of
the fundamental types of motion treated by quantum mechanics. According to quantum
theory, a particle constrained to move in a finite region of space is described by
only certain wavefunctions and can possess only certain energies. That is,
quantization emerges as a natural consequence of solving the Schrödinger equation
and the conditions imposed on it. The solutions also expose a number of
non-classical features of particles, especially their ability to tunnel into and
through regions where classical physics would forbid them to be found. Light
particles are more able to tunnel through barriers than heavy ones. The chapter also
considers the occurrence of degeneracy, which is a consequence of the symmetry of
the system.

### Chapter

## Operators and observables

This chapter explains that a central feature of quantum theory is its
representation of observables by ‘operators’, which act on the wavefunction and
extract the information it contains. It shows how operators are constructed and
used. Observables are represented by hermitian operators, which have real
eigenvalues and orthogonal eigenfunctions. When the system is not described by a
single eigenfunction of an operator, it may be expressed as a superposition of such
eigenfunctions. One consequence of the use of operators is the ‘uncertainty
principle’, one of the most profound departures of quantum mechanics from classical
mechanics. The uncertainty principle restricts the precision with which
complementary observables may be specified and measured simultaneously.

### Chapter

## Quantum Mechanics and Spectroscopy

This chapter reviews the basic quantum mechanics of spectroscopy, analyzing forms of spectroscopy based on the measurement of energy as an atom or molecule undergoes a change in quantum states. It looks at different aspects of quantum mechanics in order to understand instrumental techniques and analysis involving the interaction of light and matter. It also looks at how spectroscopy explores particular energy level differences between quantum states in molecules and atoms. The chapter stresses the importance of quantum mechanics in the development of spectroscopic techniques used to investigate atomic composition and chemical structures. It cites an example that outlines the quantum mechanic events that led to the line spectrum of hydrogen in a gas in a discharge tube.

### Chapter

## Quantum Chemistry

This chapter discusses quantum chemistry. Pretty much all of computational chemistry relies on quantum mechanics in the sense that molecular systems follow the laws of quantum mechanics. But ‘quantum chemistry‘ has a more specific meaning: it is the study of chemistry through the use of approximate solutions to the electronic Schrödinger equation. The chapter describes the background and principles of quantum chemistry, with a focus on the conceptually most important approximate approach, which is Hartree–Fock theory. In principle, the time-dependent Schrödinger equation could be used to predict what happens in chemical reactions. The basic approximation in Hartree–Fock theory is an assumption that electrons move independently of one another throughout the molecular system. The chapter then looks at the calculation of a Hartree–Fock wavefunction.