# VISUALIZING QUANTUM PHYSICS USING MATHEMATICA

Mechanics studies how a physical system, given its initial set of conditions (e.g. initial position and initial momentum), may evolve as time passes under the influence of certain forces or constrains. Classically speaking, this evolution is governed by Newton laws, commonly studied by students as early as secondary levels. At the classical level, one is expected to deal with physical system one experiences daily; pulley, wheeled-cart, ball, etc. Quantum Mechanics tries to address the same issue, except that it deals with subatomic particles. For instance, while one can easily visualize how a tennis ball, placed inside a solid square container, would behave when given an initial kick, how could one imagine an electron, bounded by the so-called potential square-well and given the same initial kick, would behave? Simply solving analytically the Schrödinger’s equation (the quantum-equivalent to Newton laws in classical mechanics) for this particular system may not help much in term of students’ understanding, although those solutions are well-studied and understood by experts. In fact, from experience, students (even the best ones in class) tend to treat this problem merely as a mathematical exercise. They are, however, unsure to answer a “simple” physically-relevant question, such as: will the electron simply bounce back as it hits the potential well, as it is expected in the case of tennis ball?

Students’ frustration is understandable, given the fact that the relevant mathematics to solve such problems can be pretty complex, and that the solution can be counterintuitive as well. We address this problem by incorporating MathematicaTM into our Introductory Quantum Mechanics course, meant for sophomore physics students. We will start with basic premises in Quantum Mechanics, which leads to the mathematical description of an quantum evolution. This evolution, embedded in the wave-function solution to the time-dependent Schrödinger’s equation, is then simulated and visualized using MathematicaTM. By making this as a graded term-project, students' feedbacks have been encouraging. First and foremost, students are able to visualize wave function evolving in the system. It can be seen that while this wave function does behave as classically predicted (e.g. traversing and bouncing off the barrier), it also exhibits its quantum properties (e.g. tunneling though the barrier, collapsing and reviving again over time). Students then start to explore more complicated quantum system, involving simple-harmonic and linear potential well. Furthermore, students realize the importance of learning basic programming skill (to write the MathematicaTM code) and computational tool in problem solving. They do not simply stop at this course; they are exploring the potential use of MathematicaTM to solve tutorial problems and analyze experimental data in other courses as well. In my talk, I will further address the detailed execution, challenges faced, and assessment mode of this MathematicaTM project.

This work follows up our earlier work presented in: Dewanto, A, Yeo Ye and Ariando. Visually engaging pedagogical strategies for Gen-Y Physics Students. Teaching and Learning in Higher Institute (TLHE) 2011, ed. Center of Development for Teaching and Learning (CDTL) (2011). Singapore: Center of Development for Teaching and Learning (CDTL). (Teaching and Learning in Higher Institute (TLHE) 2011, 6 - 9 Dec 2011, National University of Singapore, Singapore)