Quantum mechanics is a fundamental core course for undergraduate students in physics major, and its content is closely related to many research frontiers in modern physics. In this paper, we use the abstract formulas of representation transformation theory in quantum mechanics to study a specific case in the context of neutrino oscillations. In the teaching, we first introduce the flavor and mass representations of neutrinos, and then we can build up the representation transformation matrix between the two representations intuitively, where the matrix elements have a clear physical meaning: to represent the mixing angle between different neutrino mass eigenstates. Using this representation transformation theory, we can analyze the time evolution behavior of neutrinos and compare with the experimental neutrino oscillation phenomena. We found that the theoretical analysis can explain the experimental phenomena very well. The introduction of this cutting-edge example in teaching not only helps students to understand the representation transformation theory more intuitively, but also expands their horizons and stimulates their interest and enthusiasm for further research.