摘要：提出了超细颗粒的三维量子振动系统的实湮灭算符和实产生算符,结合SU(1,1)Lie代数及SO(3)Lie代数,得到系统的能级和波函数。

关键词：超细颗粒;能级;波函数

Abstract：Based on the theory of real unitary transformation and SU（1，1）Lie algebra and SO（3）Lie algebra the new theory of three dimensional quantum oscillation for ultrafine particle is discussed．T he energy level and the wave function of the system are obtained． 

Keywords：ultrafine particle；energy level；wave function