The_Spectrum_of_the_Hydrogen_Atom - The Spectrum of the...

The Spectrum of the Hydrogen Atom
XIAOTANG WANG
SOCIAL AND HUMAN SCIENCES
UNIVERSITY OF SOUTHAMPTON
Abstract
In this report, the time-dependent and time-independent Schrödinger equations are stated and
expressed in spherical polar coordinates. The separation of variables technique is review and used for
solving the Schrödinger equation for a hydrogen atom. The general solution is the wave function.
Then, the spectrum and energy levels for the hydrogen atom are discussed.
Introduction
A hydrogen atom consists of a positive charged proton and a negative charged electron. It is the
simplest but the most important atom for many applications in quantum mechanics [1] . The spectral
frequencies and energy levels of the hydrogen atom were first discovered by Niels Bohr in 1913. The
same results were obtained by the Schrödinger equation which is published in 1926. However, the
1

Schrödinger equation gives more details on how the energy of the hydrogen atom changes with time
in its ground state
[2]
.
A study of the energy levels of the hydrogen atom is carried out by solving the time-independent
Schrödinger equation in spherical polar coordinates. The solution of the Schrödinger equation is
separated into two ordinary differential equations, called the radial equation and the angular equation
[3] .
Some useful units and symbols relevant to this report are defined as below,

ħ reduced Planck constant
 m
e
electron mass of a particle
 V central potential
 wave function
E total energy of an moving electron
n principle quantum number = 1, 2, 3,…
l orbital angular momentum quantum number = 0, 1, 2,…
m magnetic quantum number = 0, , ,…,
The Schrödinger Equation
The time-dependent Schrödinger equation is
[4]
where Ĥ, the Hamiltonian, is the total energy operator of the kinetic and potential energies [5] .
In three dimensions, the time-independent Schrödinger equation is [6]
(1.2) where represents the Cartesian coordinates (x, y, z).
By substituting equation (1.2) into (1.1), it takes the form [7] (1.3)
This is known as the Schrödinger equation for a particle in three dimensions.
The Schrödinger Equation for a Hydrogen Atom
A hydrogen atom consists of one electron and one proton. “ The electric potential at a distance r from
a positive charge e of a hydrogen nucleus is e/r. The potential energy of an electron with charge –e, at
this distance, is V= –e (e/r) = –e 2 /r. ” [8] The Schrödinger equation for the hydrogen atom with mass m
e