Chapter 1
Schrödinger Equation and Its Application
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.
1.1 Early-Stage Quantum Theory
The Schrödinger equation is a direct consequence of discovery of quanta. It stemmed
from the hypothesis of energy quanta propounded by Max Planck (1900). This
hypothesis was further followed by photon (light quantum) hypothesis propounded
by Albert Einstein (1905). He claimed that light is an aggregation of light quanta and
that individual quanta carry an energy E expressed as Planck constant h multiplied
by frequency of light ν, i.e.,
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_1
3
Schrödinger Equation and Its Application
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.
1.1 Early-Stage Quantum Theory
The Schrödinger equation is a direct consequence of discovery of quanta. It stemmed
from the hypothesis of energy quanta propounded by Max Planck (1900). This
hypothesis was further followed by photon (light quantum) hypothesis propounded
by Albert Einstein (1905). He claimed that light is an aggregation of light quanta and
that individual quanta carry an energy E expressed as Planck constant h multiplied
by frequency of light ν, i.e.,
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_1
3