Our newly designed and implemented course of introductory quantum mechanics, heavily relying on high-end graphic workstations, may serve as an innovative model for the teaching of such courses. Based on this successful educational experience, we now strive to make our programs available on the Internet, using Java and VRML, to enhance accessibility to the students and to promote collaborative work outside the conventional classroom.
It is our suggestion to derive a general model for the instruction of these courses through a newly designed sequencing of the course - where learning of each physical principle is achieved in a three step process:
We have selected the instruction of the course "Introduction to Chemical Bonding" as a test case. This is an undergraduate chemistry course, which exhibits the aforementioned symptoms which we would like to cure.
Our research focuses on ways to overcome this mathematical barrier. The major effort is to place mathematical models in their proper place in the scientific hierarchy - derived from physical reality and used to explain and predict it. To achieve that, we provide for each subject a physical context in which to teach the relevant mathematics, and from which to draw intuitively the required properties of the mathematical objects. Once the physical phenomenon has been presented and discussed, a basic mathematical model is presented. Following this, students can actively manipulate relevant mathematical objects in a physical context, using the computer. This process, actively carried out by the student with the aid of the computer, should present the student with enough proficiency and mathematical insight to understand further manipulation of the model - up to the ability to analyze and apply. As an example, when teaching the subject of a rigid rotor we first discuss the phenomenon of discrete splitting of a molecular beam passing through a magnetic field, and we describe the model of an angular wavefunction. Following that, the students participate in a computer lab session, investigating the static and dynamic properties of this angular wavefunction. Once they incorporate the relations between the angular wavenumber and the angular frequency of a wavepacket, they can see why this model predicts the quantization of angular momentum which is evident in the experiment.
The computer plays a major role in this scheme, as it can be used to broaden the range of physical phenomena discussed. Since most real-life phenomena don't have a simple mathematical derivation, most real-life phenomena were not presented to the student in the traditional courses. But the computer isn't limited by such restrictions - for example, it is hard to solve the equations for an anharmonic oscillator on the blackboard, but it's a trivial computation for the computer. With the use of a computer much more complex manipulations and simulations can take place behind the scenes, and a much wider range of physical information maybe drawn upon toward the derivation of the mathematical model (e.g. the concept of an oscillating dipole moment, which is usually neglected in the discussion of electronic spectroscopy in favor of a stationery description). The student is able to distinguish the parts of reality exactly described by the model, the parts which are just approximations, and these which fall outside the scope of the derivation. It is also possible to use the computer to simulate an experiment which is too complicated or expensive to be demonstrated in the classroom.
The second use of the computer is to compensate for the poor mathematical ability exhibited by many students. The student doesn't have to know a whole lot of mathematics to use the computer - just a quick preview of the relevant objects and operations (e.g. complex waves and the superposition principle). In depth understanding of the mathematics is achieved through problem solving, aided by manipulation of the objects and their parameters in a graphical fashion . The student is offered a simple visual substitute for complex mathematical objects using three-dimensional graphics. The 3D graphic representation is more intuitive than the traditional 2D representation, and poses less obstacles than an equation. The graphic user interface presents an opportunity to perform simple interactive manipulations which amount to complex mathematical operations (e.g. a Fourier transform), and thus simplifies their teaching.
In the last academic year, for the first time, the course was structured differently. Based on the general proposed model, the five hours a week course was divided into three sessions. In the first (a one hour lecture), a physical phenomenon was demonstrated using computer graphics, computer animation or video projected on a screen, together with the basic mathematical objects and manipulations needed to construct a model (again, by a direct projection from the computer). The second session (two hours) took place in a computer lab, where the students interacted themselves independently with the computer programs. The students answered a series of pre-defined questions that explored the mathematical model. The last session (two hours) was again in a lecture form, in which more complicated physical systems were discussed, based on the knowledge gained at the computer lab.
From a computational point of view, the following issues were addressed:
It is our goal, in this academic year, to establish a firmer interface between the intuitive-graphical and formal-symbolic approaches. We will encourage students to use the computer outside of the lab sessions, by opening a web site dedicated to the course. Having full access to the programs and materials presented during the lectures and labs (including pre-lab preparation instructions and quizzes), students will be able to use the computer independently as an aid to solve their self-study assignments, to review the course subject material, and to expand their knowledge to subjects outside the scope of the course through imbedded URL links. We will also use materials from the lab sessions in the lectures, by projecting images from the web site in the class. This will leed to tighter integration of the computer graphical language into the formal parts of the course (lectures and self-study assignments). We shall examine if this integration produces the desired ability, to use both complementing representations to examine and explain a quantum phenomena.
To achieve this, we need to change our platform dependent (SGI) programs to an open architecture (platform-independent), to allow free distribution of the software and access for many students simultaneously (both on campus and at home). The World Wide Web (WWW) was chosen initially as the basic interface for its modularity (cross linking of text pages via the HTML protocol). The recently added value following the development of Java (which brings interactivity into the World Wide Web) and of VRML 2.0 (which communicates 3D graphics and animation over the Internet), makes this virtual platform ideal for educational purpose. It is also platform independent, and naturally distributed over the Internet to allow full and free access to all the students.
Extensive effort has been made so far to migrate the SGI code (e.g. the superposition program) to Java (e.g. the superposition applet). We currently investigate the
abilities of the External Authoring Interface (EAI) as a mechanism to link Java
interactivity to VRML graphics, with satisfying results, but not without
hardships. The versatility and ease of use of this interface are shadowed by
inconsistencies in implementation across platforms. This is still an evolving
technology, and we hope it will mature before the beginning of the next
semester...
A question which is still open, and should be examined in the
near future, is that of Java performance. A straight forward translation to Java
of the algorithms that were used last year, may result in very slow code. This
can be a major problem when doing three dimensional calculations (e.g. the Hydrogen atom and dihydrogen molecule
ion), and will require careful analysis and improvement of the algorithms.
Two complex harmonic
waves and their superposition are displayed, propagating in time and space. The
colors represent the coordinate-dependent phase, where red stands for positive
real, peach for positive imaginary, blue for negative real and plum for negative
imaginary. The student can change the amplitude, global phase, wave number and
angular frequency of each component, and see the change in the display in real
time. This program is used for teaching the basic concepts of wave mechanics -
the spatial and temporal periodicity of a harmonic wave, and the superposition
principle. 
The superposition
of up to 33 wave components is displayed, and propagated in time. The student
can control the amplitude and the phase of each component, and also determine
the dispersion relation between angular frequency and wave number. This program
is used to teach the concept of wavepackets and Fourier couples. At a basic
level, it is used to show wave-particle duality, coordinate- and momentum-space
representations, the difference between phase and group velocities, and between
the propagation of light waves and the propagation of matter waves. At a higher
level, it can be used to teach about Discrete Fourier Transforms, for
computational quantum mechanics. 
Similar to the wavepackets program, showing the superposition of waves in
angular-space. This program is used to teach the concept of the z-axis rotation
operator, quantization of angular momentum, and the angular function of s-,p-
and d-orbitals. 
The superposition of the three p-type spherical harmonics can be
displayed in this program. Color is used to encode phase, while intensity
(opaqueness) is proportional to the absolute value of the complex function (i.e.
the nodal plane at the equator is completely transparent, while the maxima at
the poles are solid). This program is used to teach the concept of
arbitrary-axis rotation operators and angular momentum, and basis transformation
between different axis representations. 
This is the
conventional 2D representation of the angular part of the pz orbital
(the same p-type spherical harmonic shown in the picture above).
The fact that this function is defined on a sphere is far from obvious in
the 2D representation.
This is the
mathematical formula of the above. Many students had difficulties accepting that
this is a function of two variables, as it only depends on one.
The
superposition of the first 17 eigenstates of an anharmonic oscillator is
displayed, and propagated in time. The student can control the amplitude and the
phase of each eigenstate, or create a coherent wavepacket - shift the ground
state in coordinate. At the basic level this program is used to teach the
properties of bound states, and the tradeoff between kinetic and potential
energy. At a higher level, phenomena like squeezed states and wavepacket revival
can be addressed. 
The
superposition of the hydrogen orbitals with n=1 and n=2 is displayed, and
propagated in time. The display shows the orbital iso-probability-surface, and
also a representation of the electron density. The student can control the
amplitude and the phase of each orbital. This program is used to teach about the
shapes of the orbitals, hybridization, oscillating electronic dipoles and basic
electronic spectroscopy. 
The lowest
bonding and anti-bonding orbitals of the hydrogen molecule ion are displayed,
for different inter-nuclear distances. The orbitals are determined in a
variational method, with the effective nuclear charge as a variation parameter.
The student can change the variation parameter at each inter-nuclear distance to
find the minimum energy value, and thus construct an ab-initio potential curve
for this system. This program is used to teach such concepts as the
Born-Openheimer approximation, adiabatic potential curves and the variation
principle. 
The elements of
a density operator matrix are displayed as colored bars, where the color stands
for the phase of the complex number. The data for the display comes from a
quantum mechanical simulation, in which the student can determine the potential,
the initial state, and dissipation forces acting on the system. The simulation
produces a series of frames that can be animated on the display. This program
can be used in teaching of advanced topics such as statistical quantum mechanics
and condensed phase mechanics. 
The Wigner
distribution in phase space of a density operator is displayed. This is a
different display for the same simulations described in the density operator representation. 
This program
simulates a real-life experimental technique, without requiring the students to
go to the laboratory. The student can change the accelerating potential to see
its impact on the wavelength, and decrease the beam intensity until the random
particle nature is evident. It is used for teaching the wave-particle duality.
This is a picture of
a real LEED experiment.