What's new or updated.
- Mathematica tips, updated May 25, 2007
- Molecular structure: Diatomic molecules in the rigid rotor and
harmonic oscillator approximations, updated November 30, 2006, to compare vibrational and rotational frequencies
- Harmonic oscillator, minor edit on November 30, 2006
- Hermiticity and its consequences, updated October 7, 2006
- Particle in a box, updated September 26, 2006
Many of the following notes are in Adobe Acrobat PDF format. Your web
browser needs to have the (free) Acrobat plugin installed to display the
notes. If you do not already have Acrobat Reader, you can get it from
- Mathematica tips, 5/25/2007, 84 KB / 5 pages. Mathematica is a comprehensive tool for mathematical exposition and analysis. Here are some tips on using Mathematica to document explorations, and an example document fragment. The Mathematica version of this document is MathematicaTipsDefault.nb (Save As... on your computer and then open in Mathematica).
Through the Gates of Time, an interview, by Dennis Overbye of the
New York Times, with physicist John Archibald Wheeler about the "great
smoky dragon" that is the quantum nature of reality. You may need
to register (free) to gain access to the article.
the Nature of Reality, Between Buffets, a report, by Dennis Overbye
of the New York Times, on the March 15, 2002, conference at Princeton
on the "secrets of the universe," organized to honor the 90th
birthday of Dr. John Archibald Wheeler, the Princeton and University
of Texas physicist known for his poetic characterizations of the mysteries
of the universe. You may need to register (free) to gain access to the
- Quantum aspects of light and matter, updated 9/24/2005, 296 KB / 34 pages, introduces key features of the interaction of light with matter and how their interaction is used to sort
out what is going on inside matter.
- Schrödinger's master equation of quantum mechanics, updated 10/5/2004, (PDF: 12 pages, 84 KB)
- Implementing curvature-based solution of the Schrödinger
equation, updated 11/1/2006, 17 KB / 4 pages, details how to carry out the stepwise determination
of a wavefunction given two starting values, how to determine permissible
(quantized) energies, and how to adjust the wavefunctions so they correspond
to unit total probability.
- Sketching wavefunctions (PDF:
12 pages, 114 KB). Try hand sketching wavefunctions for various
- Particle in a box, updated 9/26/2006, 89 KB / 7 pages. A particle confined in a potential-free region by very high potential energy outside the region is a good approximation to what is called a particle in a box. The wavefunctions and energies for such a particle can be determined in several ways.
- Example one-dimensional quantum systems , updated 10/20/2004, 69 KB / 6 pages. Here we will see the effect of changing the potential energy of a particle in a box in various ways. We will discover the energy continuum, photoionization, field ionization, and so how the scanning tunneling microscope (STM) works.
- Wall penetration, updated 10/6/2006, 55 KB / 2 pages. Wavefunctions are able to penetrate into regions
of space where the kinetic energy is negative (so-called forbidden regions).
The penetration is greater for lighter particles and for smaller values
of the kinetic energy in the forbidden region.
- Hermiticity and its consequences, updated October 7, 2006, 243 KB / 13 pages. A key aspect of quantum mechanical operators
is that they have the mathematical property of hermiticity. So-called
operators have important consequences: their eigenvalues are always
real numbers, different eigenfunctions are orthogonal, and the
eigenfunctions can be used to express other functions. Completeness relations in terms of the Dirac delta function are introduced.
- Time dependence in quantum mechanics, updated November 23,
10 pages, 222 KB)
- Linear system of the Schrödinger equation, updated
October 29, 2003 (PDF: 6 pages, 77 KB).
In practical, numerical applications of quantum mechanics in to real
chemical system, chemistry, a powerful method of solving the Schrödinger
equation is to convert it into a linear system. Here is how to do this,
for the example of a particle on a bumpy ring.
- Harmonic oscillator (PDF:
18 pages, 250 KB). A particle confined by a harmonic (parabolic)
potential is a good approximation to the relative motion of atoms
in a molecule and so is one of the key model systems in quantum chemistry.
- Motion of a particle on a ring. This
is an animation probability density of a quantum particle moving on
a ring. The construction of such wavepackets is discussed in the following
- Angular motion in two-component systems, updated November 30, 2006
(PDF: 18 pages, 756
little bit of angular momentum, updated December 3, 2004 (PDF:
4 pages, 53 KB). The quantum aspects of angular momentum are
that it may point on in certain directions (space quantization),
that it may never align along the z axis, and that composite
angular momentum of two angular momenta satisfy the triangle relation
that the possible value of the resulting total angular momentum span
from the difference to the sum of the component angular momenta in
- One-electron atom, updated 2004/10/29 (PDF:
10 pages, 137 KB). The prototype system for
the quantum description of atoms is the so-called one-electron atom,
of a single
charge a, and
an atomic nucleus, with charge a. Examples are the hydrogen atom,
the helium atom with one of its electrons removed, the lithium atom
two of its electrons removed, and so on.
- One-electron atom radial functions, updated November 21,
8 pages, 123 KB). The spatial distribution of orbitals are the
foundation of not only the periodic properties of the elements but
also of chemical bonding.
For this reason it is very useful to understand the distribution in
three dimensions of the orbtials of an electron in a one-electron atom.
The first step in doing this is to become familiar with the qualitative
features of the radial wave functions, shell amplitudes, and shell densities.
- Atlas of spherical harmonics, updated November 23,
2003 (PDF: 12 pages, 3.25 MB - a lot of graphics!). Here are some ways to visualize real spherical harmonics, the angular parts one electron atom wavefunctions.
- Atlas of electrons in atoms, added 11/8/2005, 3.01 MB / 17 pages - a lot of graphics, is a pictorial atlas of wavefunctions, shell amplitudes, and probability densities of electrons in atoms. The file is large because of it three-dimensional graphics. With this atlas you will be able to predict trhe three dimensional structure of electrons wavefunctions in atoms.
- Wavefunction tomography: One-electrom atom. It is a challenge to visualize three dimensional
wavefunctions. One way to do it is to look at the wavefunction values
on a plane, analogous to anatomical slices provided in medical computer
aided tomography. Try your hand at identifying particular s, p or
d wavefunctions of an electron in a hydrogen atom.
- Many-electrom atoms: Fermi holes and
Fermi heaps, updated November 16, 2004 (PDF:
17 pages, 217 KB). Many-electron wavefunctions must change sign
when the labels on any two electrons are interchanged. This property
is called antisymmetry, and its essential consequence is that electrons
either stay out of one another's way, forming what is called a Fermi
hole, or clump together, forming what is called a Fermi heap. Since
electrons repel one another electrically, Fermi holes and Fermi heaps
has drastic effects on the energy of many-electron atoms. The most
profound result is the periodic properties of the elements.
- Animation of carbon atom 1s2py
and 2px2py Fermi holes and Fermi heaps
- Penetration and shielding (PDF:
2 pages, 131 KB) The reason that subshells in many-electron atoms
fill in the order s, p, d, ... is due to the differences in penetration
of s, p, d, ... electron into the region near the nucleus.
- Molecular structure: Separating electronic and nuclear motion,
updated November 22, 2005 (PDF:
4 pages, 50 KB) While electrons are very much lighter than
nuclei, both experience the same electrical forces. The result
is that electrons
move much faster than nuclei. This in turn means that the characteristic
frequencies of electronic motion are much higher than those of
nuclear (vibrational and rotational) motion, and so electronic
transitions occur in more energetic regions of the spectrum (visible,
UV, and X-ray) than do vibrational (IR) and rotational (microwave)
spectroscopic transitions. The way to develop this idea quantitatively
is to use the large mass different between electrons and nuclei
to try to treat their motion separately. It turns out that this
can be done using two key ideas, known as the adiabatic approximation and
the Born-Oppenheimer approximation. The result is separate
Schrödinger equations for the electronic and the nuclear coordinates.
- Molecular structure: Diatomic molecules in the rigid rotor and
harmonic oscillator approximations,
updated November 30, 2006 (PDF:
5 pages, 61 KB) The nuclear Schrödinger equation describes
both the motion of the molecule as a whole through space and relative
and rotations—of the atoms that make up the molecule. The way
to separate the motions through space from the internal motion is to
reexpress the coordinates of each atom with respect to the laboratory
in terms of (1) the three coordinates of the center of mass of the
molecule and (2) the coordinates of each atom with respect to the center
of mass. The Schrödinger equation for the relative motion of the
atoms with respect to the center of mass depends on the details of
the structure of the molecule. The simplest example, which we will
explore here, is a diatomic molecule, AB.
Monday, August 16, 2010 3:53 PM