The effective beam quality factor and the effective parameter are introduced to obtain the effective abcd formulation of propagator by means of heisenberg picture . the atom - atom interaction in the atom laser beam makes the beam quality factor nonconservative , and the effective beam quality factor describes the beam quality factor and the atom - atom interaction together 无论是否考虑原子激光内部原子间的相互作用，传播子表达式和abcd变换矩阵都仅与所通过的势场有关，可以用以研究各种原子束的传输，而且计算极为简单。
The equation for the quasi - continuous atom laser beam wave - function is given by schrodinger equation , the beam width , divergence curvature radius and quality factor were introduced to represent the atom laser beam whose transverse potential could be separated . for the conservative quality factor system in some direction , we can ge t the propagator of abcd formulation 本文在含时量子系统传播子的abcd形式理论的基础上，引入某一方向的束宽、发散角、曲率半径和品质因子等光束传输参数来表征横向势能满足x和y分量相互独立的原子激光的传输。
