In this study, students' conceptual difficulties about some basic concepts in quantum mechanics like one-dimensional potential well problems and probability density of tunneling particles were identified. For this aim, a multiple choice instrument named Quantum Mechanics Conceptual Test has been developed by one of the researchers of this study and administered to 95 upper-class undergraduates and 15 graduate level students at physics education and physics departments at three universities in Turkey. In addition, in order to be able to gather more information about how students understand these concepts, semi-structured interviews were conducted with selected five undergraduate and five graduate students. Common student difficulties which are deemed to stem from thinking based on classical mechanics were determined. These difficulties should be taken into account by instructional strategies that focus on improving student understanding of potential well problems and quantum tunneling concepts. (Contains 2 figures and 4 tables.)