| Quantum mechanics has revolutionized the way we view the physical world. This theory has required a dramatic revision in the structure of the laws of mechanics governing the behavior of the particles and led to the discovery of macroscopic quantum effects ranging from lasers and superconductivity to neutron stars and radiation from black holes. Though its validity is well confirmed by the experimental evidence available, quantum mechanics remains somewhat of a mystery.; The purpose of this study is to identify students' conceptual and mathematical difficulties in learning the core concepts of introductory quantum mechanics, with the eventual goal of developing instructional material to help students with these difficulties. We have investigated student understanding of several core topics in the introductory courses, including quantum measurement, probability, Uncertainty Principle, wave functions, energy eigenstates, recognizing symmetry in physical systems, and mathematical formalism. Student specific difficulties with these topics are discussed throughout this dissertation.; In addition, we have studied student difficulties in learning, applying, and making sense out of complex mathematical processes in the physics classroom. We found students' achievement in quantum courses is not independent of their math backgrounds (correlation coefficient 0.547 for P631 and 0.347 for P263). In addition, there is a large jump in the level of mathematics at which one needs to succeed in physics courses after the sophomore level in The Ohio State University's physics curriculum.; Many students do not have a functional understanding of probability and its related terminologies. For example, many students confuse the "expectation value" with "probability density" in measurement and some students confuse "probability density" with "probability amplitude" or describe the probability amplitude as a "place" or "area."; Our data also suggested that students tend to use classical models when interpreting quantum systems; for example, some students associate a higher energy to a larger amplitude in a wave function. Others, have difficulty differentiating wave functions from energy eigenstates. Furthermore, some students do not use the relationship between the wave function and the wavenumber as a primary resource in for qualitative analysis of wave functions in regions of different potential. Many students have difficulty recognizing mathematical symbols for a given graph and lack the ability to associate the correct functions with their respective graphs. I addition, students do not distinguish an oscillatory function such as e-ix from an exponential decay function such as e-x.; The results reported suggest recommendations for further study of student understanding of quantum mechanics and for the development of materials to aid understanding. These recommendations have potentially important implications for the teaching of introductory quantum mechanics and for the development of teaching aids, texts, and technology resources. |
