Reasoning with mathematics plays an important role in university students' learning throughout their courses in the scientific disciplines, such as physics. In addition to understanding mathematical concepts and procedures, physics students often must mathematize physical constructs in terms of their associated mathematical structures and interpret mathematical entities in terms of the physical context. In this study, we investigate physics students' reasoning about mathematics in relation to physics content addressed in two quantum mechanics problems. Through qualitative analysis of interview data from twelve students, results show that 1) students use intricate, nonuniform problem-solving methods with reasoning that moves fluidly between structural (mathematizing and interpreting) and technical (conceptual and procedural) skills in quick succession, and 2) student reasoning about orthonormal bases, change of basis, inner products, and probability informed their flexibility in choosing problem-solving approaches. We illustrate the results with examples of student reasoning and discuss the inextricability of mathematics and physics in students' reasoning.