eigenvalues of a quantum-mechanical harmonic oscillator and angular momenta of a
hydrogen-like atom. At the same time, these two physical systems supply us with a
good opportunity to study classical polynomials, e.g., Hermite polynomials, (associated) Legendre polynomials, Laguerre polynomials, Gegenbauer polynomials, and
special functions, more generally. These topics constitute one of the important
branches of mathematical physics. One of the basic concepts of quantum mechanics
is that a physical quantity is represented by a Hermitian operator or matrix. In this
respect, the algebraic approach gives a good opportunity to get familiar with this
concept. We present tangible examples for this. We also emphasize the importance
of the notion of Hermiticity of a differential operator. We often encounter a unitary
operator or unitary transformation alongside the notion of Hermitian operators. We
show several examples of unitary operators in connection with transformation of
vectors and coordinates.
Part II describes Maxwell equations and their applications to various phenomena
of electromagnetic waves. These include their propagation, reflection, and transmission in dielectric media. We restrict ourselves to treating those phenomena in
dielectrics without charge. Yet, we cover a wide range of important topics. In
particular, when two (or more) dielectrics are in contact with each other at a plane
interface, reflection and transmission of light are characterized by various important
parameters such as reflection and transmission coefficients, Brewster angles, and
critical angles. We should have a proper understanding not only from the point of
view of basic study but also to make use of relevant knowledge in optical device
applications such as a waveguide. In contrast to a concept of electromagnetic waves,
light possesses a characteristic of light quanta. We present semiclassical and statistical approaches to blackbody radiation occurring in a simplified system in relation
to Part I. The physical processes are well characterized by a notion of two-level
atoms. In this context, we outline the dipole radiation within the framework of the
classical theory. We briefly describe how the optical processes occurring in a
confined dielectric medium are related to a laser that is of great importance in
fundamental science and its applications. Many of basic equations of physics are
descried as second-order linear differential equations (SOLDEs). Different methods
were developed and proposed to seek their solutions. One of the most important
methods is that of Green’s functions. We present the introductory theory of Green’s
functions accordingly. In this connection, we rethink the Hermiticity of a differential
operator.
In Part III and Part IV, we describe algebraic structures of mathematical physics.
Their understanding is useful to studies of quantum mechanics and electromagnetism whose topics are presented in Part I and Part II. Part III deals with theories of
linear vector spaces. We focus on the discussion of vectors and their transformations
in finite-dimensional vector spaces. Generally, we consider the vector transformations among the vector spaces of different dimensions. In this book, however, we
restrict ourselves to the case of the transformation between the vector spaces of same
dimension, i.e., endomorphism of the space (V
n
! V
n ). This is not only because this
is most often the case with many of physical applications, but because the relevant
operator is represented by a square matrix. Canonical forms of square matrices hold
x
Preface to the First Edition
hydrogen-like atom. At the same time, these two physical systems supply us with a
good opportunity to study classical polynomials, e.g., Hermite polynomials, (associated) Legendre polynomials, Laguerre polynomials, Gegenbauer polynomials, and
special functions, more generally. These topics constitute one of the important
branches of mathematical physics. One of the basic concepts of quantum mechanics
is that a physical quantity is represented by a Hermitian operator or matrix. In this
respect, the algebraic approach gives a good opportunity to get familiar with this
concept. We present tangible examples for this. We also emphasize the importance
of the notion of Hermiticity of a differential operator. We often encounter a unitary
operator or unitary transformation alongside the notion of Hermitian operators. We
show several examples of unitary operators in connection with transformation of
vectors and coordinates.
Part II describes Maxwell equations and their applications to various phenomena
of electromagnetic waves. These include their propagation, reflection, and transmission in dielectric media. We restrict ourselves to treating those phenomena in
dielectrics without charge. Yet, we cover a wide range of important topics. In
particular, when two (or more) dielectrics are in contact with each other at a plane
interface, reflection and transmission of light are characterized by various important
parameters such as reflection and transmission coefficients, Brewster angles, and
critical angles. We should have a proper understanding not only from the point of
view of basic study but also to make use of relevant knowledge in optical device
applications such as a waveguide. In contrast to a concept of electromagnetic waves,
light possesses a characteristic of light quanta. We present semiclassical and statistical approaches to blackbody radiation occurring in a simplified system in relation
to Part I. The physical processes are well characterized by a notion of two-level
atoms. In this context, we outline the dipole radiation within the framework of the
classical theory. We briefly describe how the optical processes occurring in a
confined dielectric medium are related to a laser that is of great importance in
fundamental science and its applications. Many of basic equations of physics are
descried as second-order linear differential equations (SOLDEs). Different methods
were developed and proposed to seek their solutions. One of the most important
methods is that of Green’s functions. We present the introductory theory of Green’s
functions accordingly. In this connection, we rethink the Hermiticity of a differential
operator.
In Part III and Part IV, we describe algebraic structures of mathematical physics.
Their understanding is useful to studies of quantum mechanics and electromagnetism whose topics are presented in Part I and Part II. Part III deals with theories of
linear vector spaces. We focus on the discussion of vectors and their transformations
in finite-dimensional vector spaces. Generally, we consider the vector transformations among the vector spaces of different dimensions. In this book, however, we
restrict ourselves to the case of the transformation between the vector spaces of same
dimension, i.e., endomorphism of the space (V
n
! V
n ). This is not only because this
is most often the case with many of physical applications, but because the relevant
operator is represented by a square matrix. Canonical forms of square matrices hold
x
Preface to the First Edition