Abstract
The renormalized Numerov algorithm is applied to solving time-independent Schrödinger equation relating to atom-atom collisions at ultralow temperature. The proprieties of Feshbach resonance in <sup>39</sup>K-<sup>133</sup>Cs collisions are investigated as an example. The results show that the renormalized Numerov method can give excellent results for ultracold colliding process. In contrast to improved log derivative method, the renormalized Numerov method displays a certain weakness in computational efficiency under the same condition. However, it is much stable in a wide range of grid step size. Hence a new propagating method is proposed by combining renormalized Numerov and logarithmic derivative method which can save computational time with a better accuracy. This algorithm can be used to solve close-coupling Schrödinger equation at arbitrary temperature for two-body collisions.
Highlights
Total errors of resonant positions by using the method combining RN method in short range with LOGD with variable step size in long range, where the connected point is located at 20a0
The second and third columns, NRN and NLOGD, denote the steps propagated in RN and LOGD methods, respectively
[27] Tiecke T G, Goosen M R, Ludewig A, Gensemer S D, Kraft S, Kokkelmans SJJMF, Walraven J T M 2010 Phys
Summary
式中, μ0 表示玻尔磁子, B 是磁场强度, μa(b) 表示 原子 a(b) 的核磁矩, Ia(b) 和 MIα(β)分别表示原子 a (b) 的核自旋及其在磁场方向的投影量子数. 式中, ga(b) 表示原子 a(b) 的超精细相互作用常数, Sa(b) 是原子 a(b) 的电子自旋量子数. SαmSα Sβ mSβ IαmIα Iβ mIβ |lml⟩ , 其中 mSα(β) 表示原子 a(b) 电子自旋在磁场方向的 投影量子数, ml 是 l 的磁量子数, 可得一套耦合二 阶微分方程组 ψ′′(R) = (2μV (R) − E) ψ(R),. 在数值计算中, 通常把原子核 间距 R 离散为等间距的格点, 然后求解不同格点 位置下的波函数. 在数值计算中, 一般设置初始格点 Qn = 1 = 1020 × I, 将 (10) 式由短程向长程传播, 可以得到 y 矩阵. Ground electronic potential curves of singlet and triplet states in 39K-133Cs. 由于 39K 和 133Cs 原子最外层各有 1 个电子, 所以它们的电子自旋量子数都是 1/2, 核自旋量子 数分别为 3/2 和 5/2, 在磁场中能量最低的超精细 态 通 道 为 |fK = 1, mfK = 1⟩ + |fCs = 3, mfCs = 3⟩ , 这里 f 和 mf 分别表示单个原子的总自旋量子数及. 当入射 通道为 s 波|fK = 1, mfK = 1⟩ + |fCs = 3, mfCs = 3⟩ 时, 它是唯一开通道, 并和另外 7 个 Mf = 4 的闭通 道通过塞曼超精细相互作用耦合在一起.
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