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. In addition, we have studied student difficulties in learning, applying, and making sense out of complex mathematical processes in the physics classroom. Most students show difficulties formalizing their conceptual understandings in terms of mathematical symbols. We found students' achievement in quantum courses correlates with their math scores (correlation coefficient 0.547 for P631 and 0.347 for P263). Furthermore, students have difficulty recognizing mathematical symbols for a given graph and lack the ability to associate the correct functions with their respective graphs. In addition, students do not distinguish an oscillatory function such as e-ix from an exponential decay function such as e-x. 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." Furthermore, students' difficulties with the concepts of probability often interfere with their ability to understand and apply the Uncertainty Principle. Some students have difficulty with the concept of the wave function as a probability amplitude. Most students have difficulties calculating a probability density from a given wave function. For example, as a common mistake, students do not square or normalize the wave function before finding the probabilities. Some students have difficulty differentiating wave functions from energy eigenstates and do not write the wave function in terms of its energy eigenfunctions in order to determine the wave function in a later time. 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. Furthermore, 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.