Stefan Heusler, Institute of Didactics of Physics,
University of Münster, Wilhelm-Klemm-Str. 10
D-48149 Münster, Germany
In this article, we motivate our interest in theatre, film and show techniques for science education and explain our methods with one specific example taken from the DVD-project “QED – Matter, Light and the Void”
Mathematical formulas are like pieces of music. They need to be performed to come alive.
Mathematics is a language which is needed to communicate observations and findings in physical research. But are mathematical formulas the best medium to reveal the fascination for the laws of nature to students? The answer to this question is obvious using the analogy to a piece of music: Are music scores the best medium to reveal the fascination for music to students?
Obviously, the answer is no. A fascinating and emotional experience of music can only be achieved if music comes alive through performance. It is this experience which motivates students to learn the technicalities, that is, performing the music scores.
How can we motivate students to learn abstract, mathematical models in physics?
I claim that theatre, film and show techniques are important to create a fascinating and emotional experience of science. It is this experience which motivates students to learn the language of mathematics.
The DVD and internet project ”QED – Matter, Light and the Void“
The interaction of electromagnetic radiation with electrons and positrons is successfully described by quantum electrodynamics (QED). A fascinating history of research has led to the success of this theory. J. Schwinger, one of the fathers of QED, describes the development of the theory as follows:
“Only when the theory is finally frozen in the textbooks can one speak of the ‘physicist‘s conception‘. At any interesting moment of the development of the theory, there are discordant viewpoints of individual physicists. “
These discordant viewpoints stem from our limited understanding of nature. Our mathematical models are not the reality but only an attempt to describe some parts of reality. Only if the predictions of a model agree with observations in nature, does the model survive and develop further.
Mathematical models reflect concepts and ideas the scientist has for his view on nature. Even if the mathematical implementation of the model becomes very complicated, the underlying concepts and ideas can be simple and beautiful.
How can we teach pupils at high school the simple and beautiful ideas which stand behind the mathematical model without teaching the complicated technicalities of the mathematical model itself?
We approach the problem how the simple and beautiful concepts underlying quantum electrodynamics (QED) can be explained on three levels:
Level I: A puppet animation movie about QED. In five chapters, the concepts behind the theory are introduced without mathematical equations. Here, we use the following methods:
- Two puppet characters discuss the question what light is and debate different models to explain their experimental observations.
- Visualization of the models and underlying physical concepts using modern computer graphics.
- Performance of experiments, comparison with the models, and once more: discussion of the results.
Level II: 30 short clips (3-4 minutes each) in which the intuitive concepts introduced in the puppet animation movie are related to mathematical equations.
Level III: Further explanations of the models introduced on the DVD are provided through the internet on the webpage”Cinema and Science“, (www.cisci.net). This material enables teachers to use parts of the movie in classroom.
The EU-funded project “Cinema and Science“ (CISCI, www.cisci.net) combines two media, the internet and the DVD to raise the interest of young people for science.
In classroom, short sequences (3-4 minutes) of the movies can be presented to introduce a topic and to motivate the scientific analysis. From the DVD “QED – Matter, Light and the Void“, more than a dozen short clips can be used in classroom to introduce the physical properties of light.
The so-called commutator is of uppermost importance in physics. In quantum mechanics, the position x and the momentum p of a particle do not commutate, meaning that “x*p” is not equal to “p*x”. Rather, the commutator x*p – p*x is proportional to Planck’s constant h. Likewise, creation and annihilation operators of photons in an electromagnetic field do not commutate.
In level I of the QED-project, the commutator does not occur explicitly. However, the consequence of the non-commutativity of position and momentum is shown in the staircase model of the atom (Chapter IV). Furthermore, we introduce a simple model of the absorption and emission of light quanta (Chapter II).
In level II of the QED-project (Chapter IV b), a simple mathematical model of the commutator is introduced, which will be discussed below.
In level III of the QED-project, additional educational material about the commutator is provided on the webpage www.cisci.net.
How can we perform the concept of the commutator? The model which we propose is the following: We use an empty glass, a bottle of water and two operations: The first operation turns the glass upside down (described by the operator U) and the second operation pours water from the bottle (described by the operator W).
The operators U and W are applied to a glass. It is a simple (and funny!) exercise to show that the two operations do not commutate. Before any equations are introduced, pupils can experience the effect of these operations by using a real glass and a bottle of water. Doing so, the fact that ordering is important if operators are applied to a state emerges naturally. It is also possible to assign the role of the operators U and W to two pupils and to show the non-commutativity as a little on-stage demonstration.
After this demonstration, pupils feel motivated to learn how this experiment can be translated into mathematical equations. It is fascinating that there exists a fundamental relation between mathematical equations and the experiment. Once the mathematical model is introduced, we can compare predictions which follow from the equations with experiments which are performed on the glass with the operators U and W.
We introduce three possible states of the glass:
- Glass upright, empty
- Glass turned upside down, empty
- Glass upright, filled
Operating with U and W changes the state of the glass from an initial state to a final state. We can introduce for U a 3 times 3 matrix which shows all possible initial states (line of the matrix) and all possible final states (column of the matrix). The final state which emerges when applying the operator is defined by the entry “1“ in the matrix, all other entries are “0“. With this definition, the operations U and W are represented as two different 3 times 3 matrices. Using this mathematical representation, we can compare predictions of the theory with experiments, and vice versa.
The experiment tells us that the operations U and W do not commutate. Translated into mathematical language, this signifies that the matrix product U*W – W*U does not vanish. Indeed, the calculation shows that this is the case.
In level III of the QED-project, we introduce a basis transformation and diagonalize the matrices U and W. It is interesting to discuss the experimental realization of these mathematical calculations, which is rather unexpected.
More examples for pupils’ exercises are:
1.) Calculate U2 and W2 using matrix multiplication. Compare the resulting matrices with the experimental realization of the U2 and W2 operators.
2.) Discuss the inverse of the operations U and W both experimentally and mathematically.
3.) Find non-commuting operations which are isomorphic to U and W
The DVD and internet project “QED – Matter, Light and the Void“ is briefly introduced in this article. The project is part of the EU-funded initiative “Cinema and Science“ (www.cisci.net). Our aim is to find possibilities to perform mathematical equations, that is, to a give direct, intuitive and emotional access to the underlying ideas behind the abstract formalism. The backbone of the project consists of computer graphics and character animation techniques.
List of references
Schwinger, “A Report on Quantum Electrodynamics”, in J. Mehra (ed.), The Physicist’s Conception of Nature (Dordrecht: Reidel, 1973).