Hota - Mathematical Physical Chemistry: Practical and Intuitive Methodology

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 this day.

Such an ultimate question is irrelevant to this monograph. Our major aim is to study a standard approach to applying Schrdinger 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 (or operator) 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 (or Hermiticity) 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 Part II.

Springer Nature Singapore Pte Ltd. 2018

Shu Hotta 1

(1)

Quantum mechanics is an indispensable research tool of modern natural science that covers cosmology, atomic physics, molecular science , materials science, and so forth. The basic concept underlying quantum mechanics rests upon Schrdinger equation . The Schrdinger equation is described as a second-order linear differential equation ( SOLDE ). The equation is analytically solved accordingly. Alternatively, equations of the quantum mechanics are often described in terms of operators and matrices, and physical quantities are represented by those operators and matrices. Normally, they are non-commutative . In particular, the quantum-mechanical formalism requires the canonical commutation relation between position and momentum operators. One of the great characteristics of the quantum mechanics is that physical quantities must be Hermitian . This aspect is deeply related to the requirement that these quantities should be described by real numbers. We deal with the Hermiticity from both an analytical point of view ( or coordinate representation ) relevant to the differential equations and an algebraic viewpoint ( or matrix representation ) associated with the operators and matrices. Including these topics, we briefly survey the origin of Schrdinger equation and consider its implications. To get acquainted with the quantum-mechanical formalism, we deal with simple examples of the Schrdinger equation .