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