本文从狄拉克δ函数出发,证明了球谐函数的加法公式,将数学物理中常用的积分变换统一为一般的Y-变换,并由此推导出一组原创的积分变换及其逆变换.我们还利用狄拉克函数重新对量子力学薛定谔方程进行演绎,避免了外势场中粒子平面波的假设,并由量纲分析自然地得到了德布罗意关系,为初学者理解量子力学提供了新的视角.

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.