Chapter 3
Hydrogen-Like Atoms
In a history of quantum mechanics, it was first successfully applied to the motion of
an electron in a hydrogen atom along with a harmonic oscillator. Unlike the case of a
one-dimensional harmonic oscillator we dealt with in Chap. 2, however, with a
hydrogen atom we have to consider three-dimensional motion of an electron.
Accordingly, it takes somewhat elaborate calculations to constitute the Hamiltonian.
The calculation procedures themselves, however, are worth following to understand
underlying basic concepts of the quantum mechanics. At the same time, this chapter
is a treasure of special functions. In Chap. 2, we have already encountered one of
them, i.e., Hermite polynomials. Here we will deal with Legendre polynomials,
associated Legendre polynomials, etc. These special functions arise when we deal
with a physical system having, e.g., the spherical symmetry. In a hydrogen atom, an
electron is moving in a spherically symmetric Coulomb potential field produced by a
proton. This topic provides us with a good opportunity to study various special
functions. The related Schrödinger equation can be separated into an angular part
and a radial part. The solutions of angular parts are characterized by spherical
(surface) harmonics. The (associated) Legendre functions are correlated to them.
The solutions of the radial part are connected to the (associated) Laguerre polynomials. The exact solutions are obtained by the product of the (associated) Legendre functions and (associated) Laguerre polynomials accordingly. Thus, to study the
characteristics of hydrogen-like atoms from the quantum-mechanical perspective is
of fundamental importance.
3.1 Introductory Remarks
The motion of the electron in hydrogen is well known as a two-particle problem
(or two-body problem) in a central force field. In that case, the coordinate system of
the physical system is separated into the relative coordinates and center-of-mass
coordinates. To be more specific, the coordinate separation is true of the case where
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_3
57
Hydrogen-Like Atoms
In a history of quantum mechanics, it was first successfully applied to the motion of
an electron in a hydrogen atom along with a harmonic oscillator. Unlike the case of a
one-dimensional harmonic oscillator we dealt with in Chap. 2, however, with a
hydrogen atom we have to consider three-dimensional motion of an electron.
Accordingly, it takes somewhat elaborate calculations to constitute the Hamiltonian.
The calculation procedures themselves, however, are worth following to understand
underlying basic concepts of the quantum mechanics. At the same time, this chapter
is a treasure of special functions. In Chap. 2, we have already encountered one of
them, i.e., Hermite polynomials. Here we will deal with Legendre polynomials,
associated Legendre polynomials, etc. These special functions arise when we deal
with a physical system having, e.g., the spherical symmetry. In a hydrogen atom, an
electron is moving in a spherically symmetric Coulomb potential field produced by a
proton. This topic provides us with a good opportunity to study various special
functions. The related Schrödinger equation can be separated into an angular part
and a radial part. The solutions of angular parts are characterized by spherical
(surface) harmonics. The (associated) Legendre functions are correlated to them.
The solutions of the radial part are connected to the (associated) Laguerre polynomials. The exact solutions are obtained by the product of the (associated) Legendre functions and (associated) Laguerre polynomials accordingly. Thus, to study the
characteristics of hydrogen-like atoms from the quantum-mechanical perspective is
of fundamental importance.
3.1 Introductory Remarks
The motion of the electron in hydrogen is well known as a two-particle problem
(or two-body problem) in a central force field. In that case, the coordinate system of
the physical system is separated into the relative coordinates and center-of-mass
coordinates. To be more specific, the coordinate separation is true of the case where
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_3
57