The multidimensional Bessel and Airy uniform approximations developed earlier in this series for the semiclassical S matrix are applied to the atom rigid−rotor system. The need is shown for (a) using a geoemetrical criterion for determining whether a stationary phase point (s.p.pt) is a maximum, minimum, or saddle point; (b) choosing a proper quadrilateral configuration of the s.p.pts. with the phases as nearly equal as possible; and (c) choosing a unit cell to favor near−separation of variables. (a) and (b) apply both to the Airy and to the Bessel uniform approximations, and (c) to the Bessel. The use of a contour plot both to understand and to facilitate the search in new cases is noted. The case of real and complex−valued stationary phase points is also considered, and the Bessel uniform−in−pairs approximation is applied. Comparison is made with exact quantum results. As in the one−dimensional case, the Bessel is an improvement over the Airy for ’’k = 0’’ transitions, while for other transitions they give similar results. Comparison in accuracy with the results of the integral method is also given. As a whole, the agreement can be considered to be reasonable. The improvement of the present over various more approximate results is shown.
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机译:在本系列中较早时针对半经典S矩阵开发的多维Bessel和Airy均匀逼近应用于原子刚性转子系统。显示了对以下方面的需求:(a)使用几何标准确定固定相位点(s.p.pt)是最大,最小还是鞍点; (b)选择合适的四边形构形。相位尽可能接近相等; (c)选择一个单元格以利于变量的近距离分离。 (a)和(b)都适用于Airy和贝塞尔统一逼近,并且(c)适用于贝塞尔。注意了在新情况下使用等高线图来理解和促进搜索。还考虑了实值和复值固定相点的情况,并应用了Bessel成对一致近似。比较精确的量子结果。与一维情况一样,对于“ k = 0”过渡,Bessel是对Airy的改进,而对于其他过渡,它们提供了相似的结果。还给出了精度与积分法结果的比较。总体而言,该协议可以被认为是合理的。显示了本发明相对于各种更近似结果的改进。
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