Research on difficulties of upper division students with mathematics in physics generally shows that the major issues concern the relation between physical meaning and formal structures. This is particularly true for quantum mechanics (QM), where the interaction between concepts and formal aspects presents multiple layers of complexity. There is a need to explore this interaction from student perspective, starting from the most fundamental elements, in order to understand challenges and potential of the interplay between quantum physics and mathematics for educational purposes. On the basis of a content and literature analysis, we designed a research on physics student understanding of QM, primarily focused on the central concept of quantum state and its formal representations. Here we discuss first the multifaceted role of mathematics in learning QM, highlighting the importance of the structural aspects. Then, we report results of the second calibration stage of the study. The research was conducted on six 3rd years physics students by means of a multi-perspective questionnaire, examining the same aspects from a global perspective and in the application context of specific problems, as well as by means of different representations. Two students were interviewed on each item, a third one on a selection of items. Results and consequences concerning the use of math in the building of conceptual knowledge in QM are discussed.