Shu Hotta

Mathematical Physical ChemistryPractical and Intuitive Methodology

Second Edition

Mathematical Physical Chemistry

Shu Hotta

Mathematical PhysicalChemistryPractical and Intuitive Methodology

Second Edition

Shu HottaTakatsuki, Osaka, Japan

ISBN 978-981-15-2224-6 ISBN 978-981-15-2225-3 (eBook)https://doi.org/10.1007/978-981-15-2225-3

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To my wife KazueandTo the memory of Roro

Preface to the Second Edition

This book is the second edition ofMathematical Physical Chemistry. Mathematics isa common language of natural science including physics, chemistry, and biology.Although the words mathematical physics and physical chemistry (or chemicalphysics) are commonly used, mathematical physical chemistry sounds rather uncom-mon. Therefore, it might well be reworded as the mathematical physics for chemists.The book title could have been, for instance, “The Mathematics of Physics andChemistry” accordingly, in tribute to the famous book that was written three-quartersof a century ago by H. Margenau and G. M. Murphy. Yet, the word mathematicalphysical chemistry is expected to be granted citizenship, considering that chemistryand related interdisciplinary fields such as materials science and molecular scienceare becoming increasingly mathematical.

The main concept and main theme remain unchanged, but this book’s secondedition contains the theory of analytic functions and the theory of continuous groups.Both the theories are counted as one of the most elegant theories of mathematics. Themathematics of these topics is of a somewhat advanced level and something like a“sufficient condition” for chemists, whereas that of the first edition may be aprerequisite (or a necessary condition) for them. Therefore, chemists (or may bephysicists as well) can creatively use the two editions. In association with thesemajor additions to the second edition, the author has disposed the mathematicaltopics (the theory of analytic functions, Green’s functions, exponential functions ofmatrices, and the theory of continuous groups) at the last chapter of individual parts(Part I through Part IV).

At the same time, the author has also made several specific revisions including theintroductory discussion on the perturbation method and variational method, both ofwhich can be effectively used for gaining approximate solutions of various quantum-mechanical problems. As another topic, the author has presented the recent progresson organic lasers. This topic is expected to help develop high-performance light-emitting devices, one of the important fields of materials science. As in the case ofthe first edition, readers benefit from going freely back and forth across the wholetopics of this book.

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Once again, the author wishes to thank many students for valuable discussionsand Dr. Shin’ichi Koizumi at Springer for giving him an opportunity to writethis book.

Takatsuki, Japan Shu HottaOctober 2019

viii Preface to the Second Edition

Preface to the First Edition

The contents of this book are based upon manuscripts prepared for both undergrad-uate courses of Kyoto Institute of Technology by the author entitled “PolymerNanomaterials Engineering” and “Photonics Physical Chemistry” and a master’scourse lecture of Kyoto Institute of Technology by the author entitled “Solid-StatePolymers Engineering.”

This book is intended for graduate and undergraduate students, especially thosewho major in chemistry and, at the same time, wish to study mathematical physics.Readers are supposed to have a basic knowledge of analysis and linear algebra.However, they are not supposed to be familiar with the theory of analytic functions(i.e., complex analysis), even though it is desirable to have relevant knowledgeabout it.

At the beginning, mathematical physics looks daunting to chemists, as used to bethe case with myself as a chemist. The book introduces the basic concepts ofmathematical physics to chemists. Unlike other books related to mathematicalphysics, this book makes a reasonable selection of material so that students majoringin chemistry can readily understand the contents in spontaneity. In particular, westress the importance of practical and intuitive methodology. We also expect engi-neers and physicists to benefit from reading this book.

In Part I and Part II, the book describes quantum mechanics and electromagne-tism. Relevance between the two is well considered. Although quantum mechanicscovers the broad field of modern physics, in Part I we focus on a harmonic oscillatorand a hydrogen (like) atom. This is because we can study and deal with many offundamental concepts of quantum mechanics within these restricted topics. More-over, knowledge acquired from the study of the topics can readily be extended topractical investigation of, e.g., electronic states and vibration (or vibronic) states ofmolecular systems. We describe these topics by both analytic method (that usesdifferential equations) and operator approach (using matrix calculations). We believethat the basic concepts of quantum mechanics can be best understood by contrastingthe analytical and algebraic approaches. For this reason, we give matrix representa-tions of physical quantities whenever possible. Examples include energy

ix

eigenvalues of a quantum-mechanical harmonic oscillator and angular momenta of ahydrogen-like atom. At the same time, these two physical systems supply us with agood opportunity to study classical polynomials, e.g., Hermite polynomials, (asso-ciated) Legendre polynomials, Laguerre polynomials, Gegenbauer polynomials, andspecial functions, more generally. These topics constitute one of the importantbranches of mathematical physics. One of the basic concepts of quantum mechanicsis that a physical quantity is represented by a Hermitian operator or matrix. In thisrespect, the algebraic approach gives a good opportunity to get familiar with thisconcept. We present tangible examples for this. We also emphasize the importanceof the notion of Hermiticity of a differential operator. We often encounter a unitaryoperator or unitary transformation alongside the notion of Hermitian operators. Weshow several examples of unitary operators in connection with transformation ofvectors and coordinates.

Part II describes Maxwell equations and their applications to various phenomenaof electromagnetic waves. These include their propagation, reflection, and transmis-sion in dielectric media. We restrict ourselves to treating those phenomena indielectrics without charge. Yet, we cover a wide range of important topics. Inparticular, when two (or more) dielectrics are in contact with each other at a planeinterface, reflection and transmission of light are characterized by various importantparameters such as reflection and transmission coefficients, Brewster angles, andcritical angles. We should have a proper understanding not only from the point ofview of basic study but also to make use of relevant knowledge in optical deviceapplications such as a waveguide. In contrast to a concept of electromagnetic waves,light possesses a characteristic of light quanta. We present semiclassical and statis-tical approaches to blackbody radiation occurring in a simplified system in relationto Part I. The physical processes are well characterized by a notion of two-levelatoms. In this context, we outline the dipole radiation within the framework of theclassical theory. We briefly describe how the optical processes occurring in aconfined dielectric medium are related to a laser that is of great importance infundamental science and its applications. Many of basic equations of physics aredescried as second-order linear differential equations (SOLDEs). Different methodswere developed and proposed to seek their solutions. One of the most importantmethods is that of Green’s functions. We present the introductory theory of Green’sfunctions accordingly. In this connection, we rethink the Hermiticity of a differentialoperator.

In Part III and Part IV, we describe algebraic structures of mathematical physics.Their understanding is useful to studies of quantum mechanics and electromagne-tism whose topics are presented in Part I and Part II. Part III deals with theories oflinear vector spaces. We focus on the discussion of vectors and their transformationsin finite-dimensional vector spaces. Generally, we consider the vector transforma-tions among the vector spaces of different dimensions. In this book, however, werestrict ourselves to the case of the transformation between the vector spaces of samedimension, i.e., endomorphism of the space (Vn ! Vn). This is not only because thisis most often the case with many of physical applications, but because the relevantoperator is represented by a square matrix. Canonical forms of square matrices hold

x Preface to the First Edition

an important position in algebra. These include a triangle matrix, diagonalizablematrix as well as a nilpotent matrix and idempotent matrix. The most general formwill be Jordan canonical form. We present its essential parts in detail taking atangible example. Next to the general discussion, we deal with an inner productspace. Once an inner product is defined between any couple of vectors, the vectorspace is given a fruitful structure. An example is a norm (i.e., “length”) of a vector.Also, we gain a clear relationship between Part III and Part I. We define variousoperators or matrices that are important in physical applications. Examples includenormal operators (or matrices) such as Hermitian operators, projection operators, andunitary operators. Once again, we emphasize the importance of Hermitian operators.In particular, two commutable Hermitian matrices share simultaneous eigenvectors(or eigenstates) and, in this respect, such two matrices occupy a special position inquantum mechanics.

Finally, Part IV describes the essence of group theory and its chemical applica-tions. Group theory has a broad range of applications in solid-state physics, solid-state chemistry, molecular science, etc. Nonetheless, the knowledge of group theorydoes not seem to have fully prevailed among chemists. We can discover an adequatereason for this in a preface to the first edition of Chemical Applications of GroupTheory written by F. A. Cotton. It might well be natural that definition and statementof abstract algebra, especially group theory, sound somewhat pretentious for chem-ists, even though the definition of group is quite simple. Therefore, we presentvarious examples for readers to get used to notions of group theory. The notion ofmapping is important as in the case of the linear vector spaces. Aside from beingadditive with calculation for a vector space and multiplicative for a group, thefundamentals of calculation regulations are pretty much the same regarding thevector space and group. We describe the characteristics of symmetry groups in detailpartly because related knowledge is useful for molecular orbital (MO) calculationsthat are presented in the last section of the book. Representation theory is probablyone of the most daunting notions for chemists. Practically, however, the representa-tion is just homomorphism that corresponds to a linear transformation in a vectorspace. In this context, the representation is merely denoted by a number or a matrix.Basis functions of representation correspond to basis vectors in a vector space.Grand orthogonality theorem (GOT) is a “nursery bed” of the representation theory.Therefore, readers are encouraged to understand its essence apart from the rigorousproof of the theorem. In conjunction with Part III, we present a variety of projectionoperators. These are very useful to practical applications in, e.g., quantum mechanicsand molecular science. The final parts of the book are devoted to applications ofgroup theory to problems of physical chemistry, especially those of quantumchemistry, more specifically molecular orbital calculations. We see how symmetryconsideration, particularly the use of projection operators, saves us a lot of labor.Examples include aromatic hydrocarbons and methane.

The previous sections sum up the contents of this book. Readers may start withany part and go freely back and forth. This is because contents of many parts areinterrelated. For example, we emphasize the importance of Hermiticity of differen-tial operators and matrices. Also projection operators and nilpotent matrices appear

Preface to the First Edition xi

in many parts along with their tangible applications to individual topics. Hence,readers are recommended to carefully examine and compare the related contentsthroughout the book. We believe that readers, especially chemists, benefit from thewriting style of this book, since it is suited to chemists who are good at intuitiveunderstanding.

The author would like to thank many students for their valuable suggestions anddiscussions at the lectures. The author also wishes to thank many students forvaluable discussions and Dr. Shin’ichi Koizumi at Springer for giving him anopportunity to write this book.

Kyoto, Japan Shu HottaOctober 2017

xii Preface to the First Edition

Contents

Part I Quantum Mechanics

1 Schrödinger Equation and Its Application . . . . . . . . . . . . . . . . . . . 31.1 Early-Stage Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Simple Applications of Schrödinger Equation . . . . . . . . . . . . . . 141.4 Quantum-Mechanical Operators and Matrices . . . . . . . . . . . . . . 211.5 Commutator and Canonical Commutation Relation . . . . . . . . . . 27Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Quantum-Mechanical Harmonic Oscillator . . . . . . . . . . . . . . . . . . . 312.1 Classical Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Formulation Based on an Operator Method . . . . . . . . . . . . . . . . 332.3 Matrix Representation of Physical Quantities . . . . . . . . . . . . . . 412.4 Coordinate Representation of Schrödinger Equation . . . . . . . . . 442.5 Variance and Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . 51References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Hydrogen-Like Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Constitution of Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4 Generalized Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 723.5 Orbital Angular Momentum: Operator Approach . . . . . . . . . . . . 773.6 Orbital Angular Momentum: Analytic Approach . . . . . . . . . . . . 91

3.6.1 Spherical Surface Harmonics and AssociatedLegendre Differential Equation . . . . . . . . . . . . . . . . . . 92

3.6.2 Orthogonality of Associated Legendre Functions . . . . . 1033.7 Radial Wave Functions of Hydrogen-Like Atoms . . . . . . . . . . . 107

3.7.1 Operator Approach to Radial Wave Functions . . . . . . . 1073.7.2 Normalization of Radial Wave Functions . . . . . . . . . . . 1123.7.3 Associated Laguerre Polynomials . . . . . . . . . . . . . . . . 116

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3.8 Total Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4 Optical Transition and Selection Rules . . . . . . . . . . . . . . . . . . . . . . 1254.1 Electric Dipole Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2 One-Dimensional System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3 Three-Dimensional System . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.4 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424.5 Angular Momentum of Radiation . . . . . . . . . . . . . . . . . . . . . . . 147References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5 Approximation Methods of Quantum Mechanics . . . . . . . . . . . . . . 1515.1 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.1.1 Quantum State and Energy Level Shift Caused byPerturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.1.2 Several Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.2 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6 Theory of Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.1 Set and Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.1.1 Basic Notions and Notations . . . . . . . . . . . . . . . . . . . . 1826.1.2 Topological Spaces and Their Building Blocks . . . . . . . 1856.1.3 T1-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.1.4 Complex Numbers and Complex Plane . . . . . . . . . . . . 197

6.2 Analytic Functions of a Complex Variable . . . . . . . . . . . . . . . . 1996.3 Integration of Analytic Functions: Cauchy’s Integral Formula . . . . 2076.4 Taylor’s Series and Laurent’s Series . . . . . . . . . . . . . . . . . . . . . 2166.5 Zeros and Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2236.6 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2256.7 Calculus of Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276.8 Examples of Real Definite Integrals . . . . . . . . . . . . . . . . . . . . . 2306.9 Multivalued Functions and Riemann Surfaces . . . . . . . . . . . . . . 248

6.9.1 Brief Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2486.9.2 Examples of Multivalued Functions . . . . . . . . . . . . . . . 256

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Part II Electromagnetism

7 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2697.1 Maxwell’s Equations and Their Characteristics . . . . . . . . . . . . . 2697.2 Equation of Wave Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2767.3 Polarized Characteristics of Electromagnetic Waves . . . . . . . . . 2807.4 Superposition of Two Electromagnetic Waves . . . . . . . . . . . . . 285References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

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8 Reflection and Transmission of Electromagnetic Wavesin Dielectric Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2958.1 Electromagnetic Fields at an Interface . . . . . . . . . . . . . . . . . . . 2958.2 Basic Concepts Underlying Phenomena . . . . . . . . . . . . . . . . . . 2978.3 Transverse Electric (TE) Waves and Transverse

Magnetic (TM) Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3038.4 Energy Transport by Electromagnetic Waves . . . . . . . . . . . . . . 3088.5 Brewster Angles and Critical Angles . . . . . . . . . . . . . . . . . . . . 3128.6 Total Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3158.7 Waveguide Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

8.7.1 TE and TM Waves in a Waveguide . . . . . . . . . . . . . . . 3208.7.2 Total Internal Reflection and Evanescent Waves . . . . . . 327

8.8 Stationary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

9 Light Quanta: Radiation and Absorption . . . . . . . . . . . . . . . . . . . . 3399.1 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3399.2 Planck’s Law of Radiation and Mode Density of

Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3419.3 Two-Level Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3459.4 Dipole Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3499.5 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

9.5.1 Brief Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3549.5.2 Organic Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

9.6 Mechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

10 Introductory Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37910.1 Second-Order Linear Differential Equations (SOLDEs) . . . . . . . 37910.2 First-Order Linear Differential Equations (FOLDEs) . . . . . . . . . 38410.3 Second-Order Differential Operators . . . . . . . . . . . . . . . . . . . . 38910.4 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39410.5 Construction of Green’s Functions . . . . . . . . . . . . . . . . . . . . . . 40110.6 Initial Value Problems (IVPs) . . . . . . . . . . . . . . . . . . . . . . . . . 408

10.6.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40810.6.2 Green’s Functions for IVPs . . . . . . . . . . . . . . . . . . . . . 41110.6.3 Estimation of Surface Terms . . . . . . . . . . . . . . . . . . . . 41410.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

10.7 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

Part III Linear Vector Spaces

11 Vectors and Their Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 43311.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43311.2 Linear Transformations of Vectors . . . . . . . . . . . . . . . . . . . . . . 43811.3 Inverse Matrices and Determinants . . . . . . . . . . . . . . . . . . . . . . 448

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11.4 Basis Vectors and Their Transformations . . . . . . . . . . . . . . . . . 452Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

12 Canonical Forms of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45912.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 45912.2 Eigenspaces and Invariant Subspaces . . . . . . . . . . . . . . . . . . . . 46812.3 Generalized Eigenvectors and Nilpotent Matrices . . . . . . . . . . . 47312.4 Idempotent Matrices and Generalized Eigenspaces . . . . . . . . . . 47812.5 Decomposition of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48512.6 Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

12.6.1 Canonical Form of Nilpotent Matrix . . . . . . . . . . . . . . 48812.6.2 Jordan Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49312.6.3 Example of Jordan Canonical Form . . . . . . . . . . . . . . . 501

12.7 Diagonalizable Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

13 Inner Product Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52313.1 Inner Product and Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52313.2 Gram Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52613.3 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53513.4 Orthonormal Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

14 Hermitian Operators and Unitary Operators . . . . . . . . . . . . . . . . . 54714.1 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54714.2 Normal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55414.3 Unitary Diagonalization of Matrices . . . . . . . . . . . . . . . . . . . . . 55614.4 Hermitian Matrices and Unitary Matrices . . . . . . . . . . . . . . . . . 56414.5 Hermitian Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 56814.6 Simultaneous Eigenstates and Diagonalization . . . . . . . . . . . . . 574References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

15 Exponential Functions of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 58115.1 Functions of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58115.2 Exponential Functions of Matrices and Their

Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58515.3 System of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 592

15.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59215.3.2 System of Differential Equations in a Matrix

Form: Resolvent Matrix . . . . . . . . . . . . . . . . . . . . . . . 59515.3.3 Several Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

15.4 Motion of a Charged Particle in Polarized Electromagnetic Wave 611References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

xvi Contents

Part IV Group Theory and Its Chemical Applications

16 Introductory Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62116.1 Definition of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62116.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62416.3 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62516.4 Isomorphism and Homomorphism . . . . . . . . . . . . . . . . . . . . . . 62716.5 Direct-Product Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

17 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63517.1 A Variety of Symmetry Operations . . . . . . . . . . . . . . . . . . . . . 63517.2 Successive Symmetry Operations . . . . . . . . . . . . . . . . . . . . . . . 64317.3 O and Td Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65417.4 Special Orthogonal Group SO(3) . . . . . . . . . . . . . . . . . . . . . . . 663

17.4.1 Rotation Axis and Rotation Matrix . . . . . . . . . . . . . . . 66417.4.2 Euler Angles and Related Topics . . . . . . . . . . . . . . . . . 669

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

18 Representation Theory of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 67918.1 Definition of Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 67918.2 Basis Functions of Representation . . . . . . . . . . . . . . . . . . . . . . 68318.3 Schur’s Lemmas and Grand Orthogonality Theorem (GOT) . . . . 69218.4 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69718.5 Regular Representation and Group Algebra . . . . . . . . . . . . . . . 70018.6 Classes and Irreducible Representations . . . . . . . . . . . . . . . . . . 70718.7 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71018.8 Direct-Product Representation . . . . . . . . . . . . . . . . . . . . . . . . . 72018.9 Symmetric Representation and Antisymmetric

Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

19 Applications of Group Theory to Physical Chemistry . . . . . . . . . . . 72919.1 Transformation of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 72919.2 Method of Molecular Orbitals (MOs) . . . . . . . . . . . . . . . . . . . . 73419.3 Calculation Procedures of Molecular Orbitals (MOs) . . . . . . . . . 74219.4 MO Calculations Based on π-Electron Approximation . . . . . . . . 747

19.4.1 Ethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74719.4.2 Cyclopropenyl Radical . . . . . . . . . . . . . . . . . . . . . . . . 75719.4.3 Benzene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76419.4.4 Allyl Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770

19.5 MO Calculations of Methane . . . . . . . . . . . . . . . . . . . . . . . . . . 777References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798

20 Theory of Continuous Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79920.1 Introduction: Operators of Rotation and Infinitesimal

Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799

Contents xvii

20.2 Rotation Groups: SU(2) and SO(3) . . . . . . . . . . . . . . . . . . . . . . 80620.2.1 Construction of SU(2) Matrices . . . . . . . . . . . . . . . . . . 80820.2.2 SU(2) Representation Matrices: Wigner Formula . . . . . 81220.2.3 SO(3) Representation Matrices and Spherical

Surface Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . 81420.2.4 Irreducible Representations of SU(2) and SO(3) . . . . . . 82320.2.5 Parameter Space of SO(3) . . . . . . . . . . . . . . . . . . . . . . 83120.2.6 Irreducible Characters of SO(3) and Their Orthogonality 837

20.3 Clebsch�Gordan Coefficients of Rotation Groups . . . . . . . . . . . 84320.3.1 Direct-Product of SU(2) and Clebsch�Gordan

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84320.3.2 Calculation Procedures of Clebsch�Gordan Coefficients 84920.3.3 Examples of Calculation of Clebsch�Gordan

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85820.4 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 864

20.4.1 Definition of Lie Groups and Lie Algebras:One-Parameter Groups . . . . . . . . . . . . . . . . . . . . . . . . 865

20.4.2 Properties of Lie Algebras . . . . . . . . . . . . . . . . . . . . . 86820.4.3 Adjoint Representation of Lie Groups . . . . . . . . . . . . . 874

20.5 Connectedness of Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . 88520.5.1 Several Definitions and Examples . . . . . . . . . . . . . . . . 88520.5.2 O(3) and SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88820.5.3 Simply Connected Lie Groups: Local Properties

and Global Properties . . . . . . . . . . . . . . . . . . . . . . . . . 890References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899

xviii Contents

Part IQuantum Mechanics

Quantum mechanics is clearly distinguished from classical physics whose majorpillars are Newtonian mechanics and electromagnetism established by Maxwell.Quantum mechanics was first established as a theory of atomic physics that handledmicroscopic world. Later on, quantum mechanics was applied to macroscopic world,i.e., cosmos. A question on how exactly quantum mechanics describes the naturalworld and on how far the theory can go remains yet problematic and is in dispute tothisday.

Such an ultimate question is irrelevant to this monograph. Our major aim is tostudy a standard approach to applying Schrödinger equation to selected topics. Thetopics include a particle confined within a potential well, a harmonic oscillator, and ahydrogen-like atoms. Our major task rests on solving eigenvalue problems of thesetopics. To this end, we describe both an analytical method and algebraic (oroperator)method. Focusing on these topics, we will be able to acquire various methods totackle a wide range of quantum-mechanical problems. These problems are usuallyposed as an analytical equation (i.e., differential equation) or an algebraic equation.A Hamiltonian is constructed analytically or algebraically accordingly. BesidesHamiltonian, physical quantities are expressed as a differential operator or a matrixoperator. In both analytical and algebraic approaches, Hermitian property(orHermiticity) of an operator and matrix is of crucial importance. This featurewill, therefore, be highlighted not only in this part but also throughout this bookalong with a unitary operator and matrix.

Optical transition and associated selection rules are dealt with in relation to theabove topics. Those subjects are closely related to electromagnetic phenomena thatare considered in PartII.

Unlike the eigenvalue problems of the abovementioned topics, it is difficult to getexact analytical solutions in most cases of quantum-mechanical problems. For thisreason, we need appropriate methods to obtain approximate solutions with respect tovarious problems including the eigenvalue problems. In this context, we deal withapproximation techniques of a perturbation method and variational method.

In the last part, we study the theory of analytic functions, one of the most eleganttheories of mathematics. This approach not only helps cultivate a broad view of puremathematics, but also leads to the acquisition of practical methodology. The last partdeals with the introductory set theory and topology aswell.

2 Part I Quantum Mechanics

Chapter 1Schrödinger Equation and Its Application

Quantum mechanics is an indispensable research tool of modern natural science thatcovers cosmology, atomic physics, molecular science, materials science, and soforth. The basic concept underlying quantum mechanics rests upon Schrödingerequation. The Schrödinger equation is described as a second-order linear differentialequation (SOLDE). The equation is analytically solved accordingly. Alternatively,equations of the quantum mechanics are often described in terms of operators andmatrices and physical quantities are represented by those operators and matrices.Normally, they are noncommutative. In particular, the quantum-mechanical formal-ism requires the canonical commutation relation between position and momentumoperators. One of great characteristics of the quantum mechanics is that physicalquantities must be Hermitian. This aspect is deeply related to the requirement thatthese quantities should be described by real numbers. We deal with the Hermiticityfrom both an analytical point of view (or coordinate representation) relevant to thedifferential equations and an algebraic viewpoint (or matrix representation) associ-ated with the operators and matrices. Including these topics, we briefly survey theorigin of Schrödinger equation and consider its implications. To get acquaintedwith the quantum-mechanical formalism, we deal with simple examples of theSchrödinger equation.

1.1 Early-Stage Quantum Theory

The Schrödinger equation is a direct consequence of discovery of quanta. It stemmedfrom the hypothesis of energy quanta propounded by Max Planck (1900). Thishypothesis was further followed by photon (light quantum) hypothesis propoundedby Albert Einstein (1905). He claimed that light is an aggregation of light quanta andthat individual quanta carry an energy E expressed as Planck constant h multipliedby frequency of light ν, i.e.,

© Springer Nature Singapore Pte Ltd. 2020S. Hotta, Mathematical Physical Chemistry,https://doi.org/10.1007/978-981-15-2225-3_1

3

E ¼ hν ¼ ħω, ð1:1Þ

where ħ � h/2π and ω ¼ 2πν. The quantity ω is called angular frequency with νbeing frequency. The quantity ħ is said to be a reduced Planck constant.

Also Einstein (1917) concluded that momentum of light quantum p is identical tothe energy of light quantum divided by light velocity in vacuum c. That is, we have

p ¼ E=c ¼ ħω=c ¼ ħk, ð1:2Þ

where k � 2π/λ (λ is wavelength of light in vacuum) and k is called wavenumber.Using vector notation, we have

p= ħk, ð1:3Þ

where k � 2πλ n (n: a unit vector in the direction of propagation of light) is said to be a

wavenumber vector.Meanwhile, Arthur Compton (1923) conducted various experiments where he

investigated how an incident X-ray beam was scattered by matter (e.g., graphite,copper). As a result, Compton found out a systematical redshift in X-ray wave-lengths as a function of scattering angles of the X-ray beam (Compton effect).Moreover he found that the shift in wavelengths depended only on the scatteringangle regardless of quality of material of a scatterer. The results can be summarizedin a simple equation described as

Δλ ¼ hmec

1� cos θð Þ, ð1:4Þ

where Δλ denotes a shift in wavelength of the scattered beam; me is a rest mass of anelectron; θ is a scattering angle of the X-ray beam (see Fig. 1.1). A quantity h

mechas a

dimension of length and denoted by λe. That is,

recoiled electron ( )

rest electronincident X-ray (ℏ )

scattered X-ray ( )

(a)

(b)

ℏ ′

−ℏ

ℏ ′

Fig. 1.1 Scattering of an X-ray beam by an electron. (a) θ denotes a scattering angle of the X-raybeam. (b) Conservation of momentum

4 1 Schrödinger Equation and Its Application

λe � h=mec: ð1:5Þ

In other words, λe is equal to the maximum shift in the wavelength of the scatteredbeam; this shift is obtained when θ ¼ π/2. The quantity λe is called an electronCompton wavelength and has an approximate value of 2.426 � 10�12 [m].

Let us derive (1.4) on the basis of conservation of energy and momentum. To thisend, in Fig. 1.1 we assume that an electron is originally at rest. An X-ray beam isincident to the electron. Then the X-ray is scattered and the electron recoils as shown.The energy conservation reads as

ħωþ mec2 ¼ ħω0 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2c2 þ me

2c4p

, ð1:6Þ

where ω and ω0 are initial and final angular frequencies of the X-ray; the second termof RHS is an energy of the electron in which p is a magnitude of momentum afterrecoil. Meanwhile, conservation of the momentum as a vector quantity reads as

ħk= ħk0 þ p, ð1:7Þ

where k and k0 are wavenumber vectors of the X-ray before and after being scattered;p is a momentum of the electron after recoil. Note that an initial momentum of theelectron is zero since the electron is originally at rest. Here p is defined as

p � mu, ð1:8Þ

where u is a velocity of an electron and m is given by [1].

m ¼ me=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� uj j2=c2

q: ð1:9Þ

Figure 1.1 shows that �ħk, ħk0, and p form a closed triangle.From (1.6), we have

mec2 þ ħ ω� ω0ð Þ� �2 ¼ p2c2 þ me

2c4: ð1:10Þ

Hence, we get

2mec2ħ ω� ω0ð Þ þ ħ2 ω� ω0ð Þ2 ¼ p2c2: ð1:11Þ

From (1.7), we have

p2 ¼ ħ2 k� k0ð Þ2 ¼ ħ2 k2 þ k02 � 2kk0 cos θ� �

¼ ħ2

c2ω2 þ ω02 � 2ωω0 cos θ

� �, ð1:12Þ

1.1 Early-Stage Quantum Theory 5

where we used the relations ω ¼ ck and ω0 ¼ ck0 with the third equality. Therefore,we get

p2c2 ¼ ħ2 ω2 þ ω02 � 2ωω0 cos θ� �

: ð1:13Þ

From (1.11) and (1.13), we have

2mec2ħ ω� ω0ð Þ þ ħ2 ω� ω0ð Þ2 ¼ ħ2 ω2 þ ω02 � 2ωω0 cos θ

� �: ð1:14Þ

Equation (1.14) is simplified to the following:

2mec2ħ ω� ω0ð Þ � 2ħ2ωω0 ¼ �2ħ2ωω0 cos θ:

That is,

mec2 ω� ω0ð Þ ¼ ħωω0 1� cos θð Þ: ð1:15Þ

Thus, we get

ω� ω0

ωω0 ¼ 1ω0 �

1ω¼ 1

2πcλ0 � λð Þ ¼ ħ

mec21� cos θð Þ, ð1:16Þ

where λ and λ0 are wavelengths of the initial and final X-ray beams, respectively.Since λ0 � λ ¼ Δλ, we have (1.4) from (1.16) accordingly.

We have to mention another important person, Louis-Victor de Broglie (1924) inthe development of quantum mechanics. Encouraged by the success of Einstein andCompton, he propounded the concept of matter wave, which was referred to as thede Broglie wave afterward. Namely, de Broglie reversed the relationship of (1.1) and(1.2) such that

ω ¼ E=ħ, ð1:17Þ

and

k ¼ pħ

or λ ¼ h=p, ð1:18Þ

where p equals jpj and λ is a wavelength of a corpuscular beam. This is said to be thede Broglie wavelength. In (1.18), de Broglie thought that a particle carrying anenergy E and momentum p is accompanied by a wave that is characterized by anangular frequency ω and wavenumber k (or a wavelength λ¼ 2π/k). Equation (1.18)implies that if we are able to determine the wavelength of the corpuscular beamexperimentally, we can decide a magnitude of momentum accordingly.

6 1 Schrödinger Equation and Its Application

In turn, from squares of both sides of (1.8) and (1.9) we get

u ¼ p

me

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ p=mecð Þ2

q : ð1:19Þ

This relation represents a velocity of particles of the corpuscular beam. If we aredealing with an electron beam, (1.19) gives the velocity of the electron beam. As anonrelativistic approximation (i.e., p/mec � 1), we have

p � meu:

We used a relativistic relation in the second term of RHS of (1.6), where anenergy of an electron Ee is expressed by

Ee ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2c2 þ me

2c4p

: ð1:20Þ

In the meantime, deleting u2 from (1.8) and (1.9) we have

mc2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2c2 þ me

2c4p

:

Namely, we get [1].

Ee ¼ mc2: ð1:21Þ

The relation (1.21) is due to Einstein (1905, 1907) and is said to be the equivalencetheorem of mass and energy.

If an electron is accompanied by a matter wave, that wave should be propagatedwith a certain phase velocity vp and a group velocity vg. Thus, using (1.17) and (1.18)we have

vp ¼ ω=k ¼ Ee=p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2c2 þ me

2c4p

=p > c,

vg ¼ ∂ω=∂k ¼ ∂Ee=∂p ¼ c2p=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2c2 þ me

2c4p

< c,

vpvg ¼ c2:

ð1:22Þ

Notice that in the above expressions, we replaced E of (1.17) with Ee of (1.20). Thegroup velocity is thought to be a velocity of a wave packet and, hence, a propagationvelocity of a matter wave should be identical to vg. Thus, vg is considered as aparticle velocity as well. In fact, vg given by (1.22) is identical to u expressed in(1.19). Therefore, a particle velocity must not exceed c. As for photons (or lightquanta), vp ¼ vg ¼ c and, hence, once again we get vpvg ¼ c2. We will encounter thelast relation of (1.22) in Part II as well.

The above discussion is a brief historical outlook of early-stage quantum theorybefore Erwin Schrödinger (1926) propounded his equation.

1.1 Early-Stage Quantum Theory 7

1.2 Schrödinger Equation

First we introduce a wave equation expressed by

∇2ψ ¼ 1v2

∂2ψ

∂t2, ð1:23Þ

where ψ is an arbitrary function of a physical quantity relevant to propagation of awave; v is a phase velocity of wave; ∇2 called Laplacian is defined below

∇2 � ∂2

∂x2þ ∂2

∂y2þ ∂2

∂z2: ð1:24Þ

One of special solutions for (1.24) called a plane wave is well studied and expressedas

ψ ¼ ψ0ei k ∙ x�ωtð Þ: ð1:25Þ

In (1.25), x denotes a position vector of a three-dimensional Cartesian coordinateand is described as

x= e1 e2 e3ð Þx

y

z

0B@

1CA, ð1:26Þ

where e1, e2, and e3 denote basis vectors of an orthonormal base pointing to positivedirections of x-, y-, and z-axes. Here we make it a rule to represent basis vectors by arow vector and represent a coordinate or a component of a vector by a column vector;see Sect. 9.1.

The other way around, now we wish to seek a basic equation whose solution isdescribed as (1.25). Taking account of (1.1)–(1.3) as well as (1.17) and (1.18), werewrite (1.25) as

ψ ¼ ψ0ei p

ħ ∙ x�Eħtð Þ, ð1:27Þ

where we redefine p= e1 e2 e3ð Þpxpypz

0B@

1CA and E as quantities associated with those of

matter (electron) wave. Taking partial differentiation of (1.27) with respect to x, weobtain

8 1 Schrödinger Equation and Its Application

∂ψ∂x

¼ iħpxψ0e

i pħ ∙ x�E

ħtð Þ ¼ iħpxψ : ð1:28Þ

Rewriting (1.28), we have

ħi∂ψ∂x

¼ pxψ : ð1:29Þ

Similarly we have

ħi∂ψ∂y

¼ pyψ andħi∂ψ∂z

¼ pzψ : ð1:30Þ

Comparing both sides of (1.29), we notice that we may relate a differential operatorħi

∂∂x to px. From (1.30), similar relationship holds with the y and z components. That

is, we have the following relations:

ħi∂∂x

$ px,ħi∂∂y

$ py,ħi∂∂z

$ pz: ð1:31Þ

Taking partial differentiation of (1.28) once more,

∂2ψ∂x2

¼ iħpx

� �2

ψ0ei p

ħ ∙ x�Eħtð Þ ¼ � 1

ħ2p2xψ : ð1:32Þ

Hence,

�ħ2∂2ψ∂x2

¼ p2xψ : ð1:33Þ

Similarly we have

�ħ2∂2ψ∂y2

¼ p2yψ and � ħ2∂2ψ∂z2

¼ p2zψ : ð1:34Þ

As in the above cases, we have

�ħ2∂2

∂x2$ p2x , � ħ2

∂2

∂y2$ p2y , � ħ2

∂2

∂z2$ p2z : ð1:35Þ

Summing both sides of (1.33) and (1.34) and then dividing by 2m, we have

1.2 Schrödinger Equation 9

� ħ2

2m∇2ψ ¼ p2

2mψ ð1:36Þ

and the following correspondence

� ħ2

2m∇2 $ p2

2m, ð1:37Þ

where m is the mass of a particle.Meanwhile, taking partial differentiation of (1.27) with respect to t, we obtain

∂ψ∂t

¼ � iħEψ0e

i pħ ∙ x�E

ħtð Þ ¼ � iħEψ : ð1:38Þ

That is,

iħ∂ψ∂t

¼ Eψ : ð1:39Þ

As the above, we get the following relationship:

iħ∂∂t

$ E: ð1:40Þ

Thus, we have relationships between c-numbers (classical numbers) and q-numbers(quantum numbers, namely, operators) in (1.35) and (1.40). Subtracting (1.36) from(1.39), we get

iħ∂ψ∂t

þ ħ2

2m∇2ψ ¼ E � p2

2m

� �ψ : ð1:41Þ

Invoking the relationship on energy

Total energyð Þ ¼ Kinetic energyð Þ þ Potential energyð Þ, ð1:42Þ

we have

E ¼ p2

2mþ V , ð1:43Þ

where V is a potential energy. Thus, (1.41) reads as

10 1 Schrödinger Equation and Its Application

iħ∂ψ∂t

þ ħ2

2m∇2ψ ¼ Vψ : ð1:44Þ

Rearranging (1.44), we finally get

� ħ2

2m∇2 þ V

� �ψ ¼ iħ

∂ψ∂t

: ð1:45Þ

This is the Schrödinger equation, a fundamental equation of quantum mechanics. In(1.45), we define a following Hamiltonian operator H as

H � � ħ2

2m∇2 þ V : ð1:46Þ

Then we have a shorthand representation such that

Hψ ¼ iħ∂ψ∂t

: ð1:47Þ

On going from (1.25) to (1.27), we realize that quantities k and ω pertinent to afield have been converted to quantities p and E related to a particle. At the same time,whereas x and t represent a whole space-time in (1.25), those in (1.27) are charac-terized as localized quantities.

From a historical point of view, we have to mention a great achievementaccomplished by Werner Heisenberg (1925) who propounded matrix mechanics.The matrix mechanics is often contrasted with the wave mechanics Schrödingerinitiated. Schrödinger and Pau Dirac (1926) demonstrated that wave mechanics andmatrix mechanics are mathematically equivalent. Note that the Schrödinger equationis described as a nonrelativistic expression based on (1.43). In fact, kinetic energyK of a particle is given by [1].

K ¼ mec2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� u=cð Þ2

q � mec2:

As a nonrelativistic approximation, we get

K � mec2 1þ 1

2uc

� �2

� mec2 ¼ 1

2meu

2 � p2

2me,

where we used p � meu again as a nonrelativistic approximation; also, we used

1.2 Schrödinger Equation 11

1ffiffiffiffiffiffiffiffiffiffiffi1� x

p � 1þ 12x

when x (>0) corresponding to uc

� �2is enough small than 1. This implies that in the

above case the group velocity u of a particle is supposed to be well below lightvelocity c. Dirac (1928) formulated an equation that describes relativistic quantummechanics (the Dirac equation).

In (1.45) ψ varies as a function of x and t. Suppose, however, that a potentialV depends only upon x. Then we have

� ħ2

2m∇2 þ V xð Þ

ψ x, tð Þ ¼ iħ

∂ψ x, tð Þ∂t

: ð1:48Þ

Now, let us assume that separation of variables can be done with (1.48) such that

ψ x, tð Þ ¼ ϕ xð Þξ tð Þ: ð1:49Þ

Then, we have

� ħ2

2m∇2 þ V xð Þ

ϕ xð Þξ tð Þ ¼ iħ

∂ϕ xð Þξ tð Þ∂t

: ð1:50Þ

Accordingly, (1.50) can be recast as

� ħ2

2m∇2 þ V xð Þ

ϕ xð Þ=ϕ xð Þ ¼ iħ

∂ξ tð Þ∂t

=ξ tð Þ: ð1:51Þ

For (1.51) to hold, we must equate both sides to a constant E. That is, for a certainfixed point x0 we have

� ħ2

2m∇2 þ V x0ð Þ

ϕ x0ð Þ=ϕ x0ð Þ ¼ iħ

∂ξ tð Þ∂t

=ξ tð Þ, ð1:52Þ

where ϕ(x0) of a numerator should be evaluated after operating ∇2, while with ϕ(x0)in a denominator, ϕ(x0) is evaluated simply replacing x in ϕ(x) with x0. Now, let usdefine a function Φ(x) such that

Φ xð Þ � � ħ2

2m∇2 þ V xð Þ

ϕ xð Þ=ϕ xð Þ: ð1:53Þ

Then, we have

12 1 Schrödinger Equation and Its Application

Φ x0ð Þ ¼ iħ∂ξ tð Þ∂t

=ξ tð Þ: ð1:54Þ

If RHS of (1.54) varied depending on t, Φ(x0) would be allowed to have variousvalues, but this must not be the case with our present investigation. Thus, RHS of(1.54) should take a constant value E. For the same reason, LHS of (1.51) shouldtake a constant.

Thus, (1.48) or (1.51) should be separated into the following equations:

Hϕ xð Þ ¼ Eϕ xð Þ, ð1:55Þ

iħ∂ξ tð Þ∂t

¼ Eξ tð Þ: ð1:56Þ

Equation (1.56) can readily be solved. Since (1.56) depends on a sole variable t, wehave

dξ tð Þξ tð Þ ¼ E

iħdt or d ln ξ tð Þ ¼ E

iħdt: ð1:57Þ

Integrating (1.57) from zero to t, we get

lnξ tð Þξ 0ð Þ ¼

Etiħ

: ð1:58Þ

That is,

ξ tð Þ ¼ ξ 0ð Þ exp �iEt=ħð Þ: ð1:59Þ

Comparing (1.59) with (1.38), we find that the constant E in (1.55) and (1.56)represents an energy of a particle (electron).

Thus, the next task we want to do is to solve an eigenvalue equation of (1.55).After solving the problem, we get a solution

ψ x, tð Þ ¼ ϕ xð Þ exp �iEt=ħð Þ, ð1:60Þ

where the constant ξ(0) has been absorbed in ϕ(x). Normally, ϕ(x) is to be normal-ized after determining the functional form (vide infra).

1.2 Schrödinger Equation 13