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.本研究采用基于生成函数（generating function）的全新Python代码，复现了莫什诺夫斯基谐振子括号（Moshinsky harmonic oscillator brackets）。我们通过变换两组耦合粒子谐振子的波函数$Phi_{n_1l_1,n_2l_2,Lambda}^{m_1,m_2,lambda}left(vec{r}_1,vec{r}_2 ight)$与$Phi_{n_al_a,n_bl_b,Lambda}^{m_a,m_b,lambda}left(vec{r}_a,vec{r}_b ight)$，构建了上述括号项。为实现两个参考系下给定位置与动量坐标的相互转换，我们对低角动量及高角动量的原子核实施了正交变换（orthogonal transformation）。 在推导过程中，我们基于谐振子波函数，在球坐标系（spherical coordinates）下对生成函数$e^{2vec{p}cdotvec{r}-p^2}$与$e^{2cp_icdot p_j}$进行了展开。在针对双耦合谐振子态修正莫什诺夫斯基谐振子括号时，我们采用了含双变量的生成函数。相较于既往文献中的谐振子括号，本方法的指标数量已大幅缩减；程序代码迭代流程中的这一简化，已取得了极具影响力的研究成果。相较于既往莫什诺夫斯基谐振子括号代码版本，全新的Python代码具备更优异的易用性。本研究借助该代码，可针对大范围量子数（quantum number）集合计算莫什诺夫斯基谐振子括号。根据本次研究的发现，为生成函数增添更多变量，可使适用于多体相互作用的莫什诺夫斯基谐振子括号数量得到提升。