Starting from Dirac's δ-function, we derive the sum rule of the spherical hamonics. We obtain a generalized y-transform which covers several traditional integral transformations. We also construct a new type of integral transformation and its inverse transformation. Furthermore, we introduce the Schrdingers equation by making use of the Dirac function, which effectively avoid the plane wave hypothesis for a quantum particle moving in an external potential. The de Broglie matter wave relations are naturally derived via dimension analysis. Our method provides a new point of view to understanding quantum mechanics for students.