We discuss an investigation of student difficulties with the corrections to the energy spectrum of the hydrogen atom for the intermediate field Zeeman effect using degenerate perturbation theory (DPT). The investigation was carried out in advanced quantum mechanics courses by administering free-response and multiple-choice questions and conducting individual interviews with students. We find that students share many common difficulties related to relevant physics concepts. They had difficulty with mathematical sense making in this context of quantum mechanics, which requires the ability to interpret the implications of the degeneracy in the unperturbed energy spectrum and how the Zeeman perturbation will impact the splitting of the energy levels. Many of the common student difficulties arise from challenges in mathematical sense making and applying linear algebra concepts incorrectly in this novel context of quantum mechanics. We describe how the research on student difficulties was used as a guide to develop and evaluate a Quantum Interactive Learning Tutorial (QuILT), which strives to help students develop a functional understanding of the concepts necessary for finding the corrections to the energy spectrum of the hydrogen atom for the intermediate field Zeeman effect using the DPT. We also discuss the development and validation of the DPT QuILT focusing on these issues and its in-class evaluation in the undergraduate and graduate courses.