arXiv subjects
Publications and source records attributed to Martin Erik Horn.
The three anti-commutative two-dimensional Pauli Pascal triangles can be generalized into multi-dimensional Pauli Pascal hyperpyramids. Fibonacci and Jacobsthal numbers are then generalized into Pauli Fibonacci numbers, Pauli Jacobsthal numbers, and Pauli Fibonacci numbers of higher order. And the question is: are Pauli rabbits killer rabbits?
In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. At the same time, Hestenes geometric algebra provides a didactically promising instrument to model phenomena in physics mathematically and in a tangible manner. Quaternions particularly have a catchy interpretation in the context of geometric algebra which can be used didactically. The relation between quaternions and geometric algebra is presented with a view to analysing its didactical possibilities.
In high school physics and physics at university level anti-commutative and non-commutative quantities play an outstanding role at the theoretical description of physical relations. But while using commutative quantities, the basic formal relations of mathematical physics are introduced clearly and step-by-step, a similar clear presentation of non-commuatative relations with didactical sensible examples is lacking. On the basis of a q-analog of the Pascal triangle (sometimes called quantum Pascal triangle) this paper discusses whether a less abstract introduction into non-commutative relations is possible. The didactical relevance of this approach is analysed. In an attachment all three triangles of the Pauli Pascal plane are presented.
The bilateral binomial theorem with step width two gives a bilateral hypergeometric formula for 2H2(a, a+1/2; c, c+1/2; z).
Analyses of interviews with secondary school students about their conceptions of light at the University of Potsdam indicate that numerous students have a deterministic view of light. With regard to these results the model of energy flow lines, which has been discussed recently in the didactical literature, is of special interest. Following this model, light is presumed to move along energy flow lines as trajectories. In an analysis of the model of energy flow lines four didactical dimensions (didactical content, internal structure, present-day relevance and future significance) are investigated. It can be shown that a discussion of this model in physics at school can increase the meta-conceptional knowledge of the students about the models of light. On the other hand, this can promote deterministic conceptions and the Bohm interpretation of quantum mechanics. But the question remains: Should the nature of light really be described as deterministic?
Due to its amazing three-dimensional effects, holography is a very motivating, yet very demanding subject for physics classes at the upper level in school. For this reason an instruction unit on holography that supplement holographic experiments with computer-supported work sessions and a simulation program was developed. The effects of the lessons on holography were determined by a pre-post-test design. In addition to videotaping the lessons, knowledge and motivational tests as well as student interviews, students were asked to prepare concept maps, which were used to track processes of model construction. The way this knowledge was applied largely depends on the students' understanding of models. In particular it was shown that the participating students' demonstrated capacity for distinguishing between the different models of light is of great importance. Only students with a developed capacity for distinguishing between models are able to reason in an problem-oriented manner. They recognize the limits of models and are not only able to specifically designate changes from models and levels of models, but also to apply these in a physically correct manner.
Throughout all years of study, students of physics are confronted with the question 'what exactly is light?' - a question that is impossible to answer correctly and, therefore, continuously discussed within the framework of models. Numerous models have been introduced for this purpose and subsequently compete with one another for validity. On the one hand, this may lead to learning difficulties. Yet, on the other hand, this may be considered a didactic chance, if thinking about the model is compounded with the consideration of models in general. Research in secondary school level II physics classes and with physics students provides insight into the processes of model construction as well as that of overcoming models, which will be presented and analyzed here. The possibilities which computer simulations provide for the discussion of various models of light are demonstrated and their influence on a structured construction of models will be discussed.
In a lesson on holography a tension is to be found between the ray model, the wave model, the phasor model and the particle model. Lessons depend on the previous experiences of the students, the intentions of the teacher as well as on other parameters and therefore provide a diversity in emphasis while explaining holographic effects. As class discussion concerning selected areas of interference optics and specifically holography develops in a questioning manner and with limited intervention from the teacher, numerous student conceptions with regard to interference optics and different model conceptions of light can be identified. Student interviews in which these discovered concepts are explored in greater detail can provide in depth information. Primary student conceptions, which were gathered in the main research conducted for a dissertation project will be presented here and analysed with a view to their didactic consequences.
A new and easy way of deriving Gauss's Generalized Hypergeometric Theorem is presented by using the Bilateral Binomial Theorem.
Part I: The two-dimensional Pascal Triangle will be generalized into a three-dimensional Pascal Pyramid and four-, five- or whatsoever-dimensional hyper-pyramids. Part II: The Bilateral Binomial Theorem will be generalised into a Bilateral Trinomial Theorem resp. a Bilateral Multinomial Theorem.
Holography as a subject is neglected largely in current physics lessons and in school books. Even though this topic might be complex, it is applicable and viable in the world that we live in. Holography lends itself to further develop knowledge with regard to inference optics. However, misconceptions of students impede the learning process. Research conducted by our research group shows that problematic perceptions regarding interference optics still exist with physics students at the intermediate level and on occasion are even sustained by illustrations in school books. Based on this analysis, a concept for a series of lessons is introduced. This series can be taught in physics lessons at the secondary school level II and in basic courses. Initial results from dealing with the subject of holography in an advanced training course will be presented.
As an expansion of complex numbers, the quaternions show close relations to numerous physically fundamental concepts. In spite of that, the didactic potential provided by quaternion interrelationships in formulating physical laws are hardly regarded in the current physics curriculum. In particular, many approaches emerge that are useful in conveying the unity of seemingly distinct theories in a didactically convincing manner. This will be exemplified with the help of special relativity. The diverse examples of spatial and space-time rotations are merged into a didactic whole by introducing quaternion transformations and comparing them to the representation using rotation matrices common in physics books.