Part I
Quantum Mechanics
Quantum mechanics is clearly distinguished from classical physics whose major
pillars are Newtonian mechanics and electromagnetism established by Maxwell.
Quantum mechanics was first established as a theory of atomic physics that handled
microscopic world. Later on, quantum mechanics was applied to macroscopic world,
i.e., cosmos. A question on how exactly quantum mechanics describes the natural
world and on how far the theory can go remains yet problematic and is in dispute to
thisday.
Such an ultimate question is irrelevant to this monograph. Our major aim is to
study a standard approach to applying Schrödinger equation to selected topics. The
topics include a particle confined within a potential well, a harmonic oscillator, and a
hydrogen-like atoms. Our major task rests on solving eigenvalue problems of these
topics. 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 to
tackle a wide range of quantum-mechanical problems. These problems are usually
posed as an analytical equation (i.e., differential equation) or an algebraic equation.
A Hamiltonian is constructed analytically or algebraically accordingly. Besides
Hamiltonian, physical quantities are expressed as a differential operator or a matrix
operator. In both analytical and algebraic approaches, Hermitian property
(orHermiticity) of an operator and matrix is of crucial importance. This feature
will, therefore, be highlighted not only in this part but also throughout this book
along with a unitary operator and matrix.
Optical transition and associated selection rules are dealt with in relation to the
above topics. Those subjects are closely related to electromagnetic phenomena that
are considered in PartII.
Unlike the eigenvalue problems of the abovementioned topics, it is difficult to get
exact analytical solutions in most cases of quantum-mechanical problems. For this
reason, we need appropriate methods to obtain approximate solutions with respect to
various problems including the eigenvalue problems. In this context, we deal with
approximation techniques of a perturbation method and variational method.
Quantum Mechanics
Quantum mechanics is clearly distinguished from classical physics whose major
pillars are Newtonian mechanics and electromagnetism established by Maxwell.
Quantum mechanics was first established as a theory of atomic physics that handled
microscopic world. Later on, quantum mechanics was applied to macroscopic world,
i.e., cosmos. A question on how exactly quantum mechanics describes the natural
world and on how far the theory can go remains yet problematic and is in dispute to
thisday.
Such an ultimate question is irrelevant to this monograph. Our major aim is to
study a standard approach to applying Schrödinger equation to selected topics. The
topics include a particle confined within a potential well, a harmonic oscillator, and a
hydrogen-like atoms. Our major task rests on solving eigenvalue problems of these
topics. 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 to
tackle a wide range of quantum-mechanical problems. These problems are usually
posed as an analytical equation (i.e., differential equation) or an algebraic equation.
A Hamiltonian is constructed analytically or algebraically accordingly. Besides
Hamiltonian, physical quantities are expressed as a differential operator or a matrix
operator. In both analytical and algebraic approaches, Hermitian property
(orHermiticity) of an operator and matrix is of crucial importance. This feature
will, therefore, be highlighted not only in this part but also throughout this book
along with a unitary operator and matrix.
Optical transition and associated selection rules are dealt with in relation to the
above topics. Those subjects are closely related to electromagnetic phenomena that
are considered in PartII.
Unlike the eigenvalue problems of the abovementioned topics, it is difficult to get
exact analytical solutions in most cases of quantum-mechanical problems. For this
reason, we need appropriate methods to obtain approximate solutions with respect to
various problems including the eigenvalue problems. In this context, we deal with
approximation techniques of a perturbation method and variational method.