Schrödinger Equation and Its Application

Abstract

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 Schrödinger equation. The Schrödinger 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 noncommutative. In particular, the quantum-mechanical formalism requires the canonical commutation relation between position and momentum operators. One of 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 Schrödinger equation and consider its implications. To get acquainted with the quantum-mechanical formalism, we deal with simple examples of the Schrödinger equation.