Supplementary notes
Quantum Mechanics
Updated Monday, August 16, 2010 3:53 PM

Copyright © 2006 Dan Dill (
Department of Chemistry, Boston University, Boston MA 02215


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  • Mathematica tips, 5/25/2007, PDF Logo 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).
  • Peering 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.
  • Discussing 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 article.
  • Quantum aspects of light and matter, updated 9/24/2005, PDF Logo 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, PDF Logo 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 one-dimensional potentials.
  • Particle in a box, updated 9/26/2006, PDF Logo 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, PDF Logo 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, PDF Logo 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.
  • I was pleased to see the "news and views" overview, A delayed reaction, in the journal Nature (volume 419, 19 September 2002, page 266) describing the work by Steven Harich et al. (page 281 of the same issue if Nature), in which distortions in wavefunctions are used to unravel the details of the simplest chemical reaction

    H + HD —> H2 + D

    In particular, I think you will find the graphic in the "news and views" to be a nice example of what we have been discussing in the context if the Schrödinger Shooter. The Nature articles are available online at no charge from accounts on the domain.

  • Hermiticity and its consequences, updated October 7, 2006, PDF Logo 243 KB / 13 pages. A key aspect of quantum mechanical operators is that they have the mathematical property of hermiticity. So-called hermitian operators have important consequences: their eigenvalues are always real numbers, different eigenfunctions are orthogonal, and the set of 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, 2004 (PDF: 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 supplement.
  • Angular motion in two-component systems, updated November 30, 2006 (PDF: 18 pages, 756 KB).
  • A 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 integer steps.

  • 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, consisting of a single electron, with 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 with two of its electrons removed, and so on.
  • One-electron atom radial functions, updated November 21, 2003 (PDF: 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, PDF Logo 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 spectroscopic 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 motion—vibrations 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.

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Updated Monday, August 16, 2010 3:53 PM
Dan Dill (