Moshinsky brackets for a wide range of quantum numbers using generating functions

We used a new Python code to reproduce the brackets for the Moshinsky harmonic oscillator, which was based on the generating function. We made these brackets by transforming the wave functions of two groups of coupled particle harmonic oscillators, $\Phi_{n_1l_1,n_2l_2,\Lambda}^{m_1,m_2,\lambda}\left(\vec{r}_1,\vec{r}_2\right)$ and $\Phi_{n_al_a,n_bl_b,\Lambda}^{m_a,m_b,\lambda}\left(\vec{r}_a,\vec{r}_b\right)$. To convert between the supplied position and momentum coordinates in both frames, we performed orthogonal transformations on nuclei with both low and high angular momentum.
In our derivation, we have used the expansion of the generating functions $e^{2\vec{p}.\vec{r}-p^2}$ and $e^{2cp_i.p_j}$ in spherical coordinates in terms of harmonic oscillator wave functions. When we modified the Moshinsky brackets for two-coupled oscillator states, we used generating functions with two variables. The number of indices has significantly decreased compared to the oscillator brackets in previous references; this reduction in the program code's iterative process has yielded influential results. Compared to the previous version of the Moshinsky brackets code, the new Python code is easier to use. Our approach utilizes this code to assess Moshinsky brackets across a broad spectrum of quantum numbers. According to the revelation, adding more variables to the generating function makes the number of Moshinsky brackets that work for the higher body interactions increase.

本研究采用一种新颖的Python编程语言，复现了Moshinsky谐振子的括号结构，该结构基于生成函数理论。我们通过对两组耦合粒子谐振子波函数(Phi_{n_1l_1,n_2l_2,Lambda}^{m_1,m_2,lambda}(vec{r}_1,vec{r}_2))和(Phi_{n_al_a,n_bl_b,Lambda}^{m_a,m_b,lambda}(vec{r}_a,vec{r}_b))进行变换，构建了这些括号。为了在两个参考系中实现所提供的位置和动量坐标之间的转换，我们对具有低角动量和高角动量的核子进行了正交变换。在推导过程中，我们采用了在球坐标系中将生成函数(e^{2vec{p}.vec{r}-p^2})和(e^{2cp_i.p_j})展开成谐振子波函数的形式。在修改Moshinsky括号以适应两个耦合谐振子状态时，我们使用了具有两个变量的生成函数。与先前参考文献中的谐振子括号相比，指数显著减少；这种在程序代码迭代过程中的缩减产生了显著的结果。与之前的Moshinsky括号代码版本相比，新的Python代码更加易于使用。我们的方法利用此代码评估了广泛量子数范围内的Moshinsky括号。据揭示，向生成函数中添加更多变量将增加适用于更高阶相互作用的Moshinsky括号数量。