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

Abstract

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, ${\Phi }_{n_1{l}_1,n_2{l}_2,\Lambda }^{m_1,m_2,\lambda }({\overrightarrow{r}}_1,{\overrightarrow{r}}_2)$ and ${\Phi }_{n_a{l}_a,n_b{l}_b,\Lambda }^{m_a,m_b,\lambda }({\overrightarrow{r}}_a,{\overrightarrow{r}}_b)$. 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 $e^{2\overrightarrow{p}.\overrightarrow{r}-p^2}$ and $e^{2c{p}_i.p_j}$ 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.

Program summary

Program Title: GFBRACKETS

CPC Library link to program files: https://doi.org/10.17632/jvbnwp35rm.1

Licensing provisions: MIT

Programming language: Python

Supplementary material: Appendix

Nature of problem: The generating functions were used to obtain a concise representation of the oscillator states for single particles. This was done to formulate and compute the generalized version of the Moshinsky brackets and matrix elements of two-body operators. By enlarging sets of basis oscillator states, it becomes possible to cover interactions involving more than two bodies. This can be easily achieved using the expanded generating function to compute the Moshinsky brackets for high-quantum numbers.

Solution method: The Python code requires fewer iterations than similar codes generated by the method of the generating functions.

Additional comments including restrictions and unusual features: When calculating Moshinsky brackets for large quantum numbers, restrictions arise. There is no data for quantum numbers greater than 10 in the formation of reaction potential. Additionally, a transformation matrix can be used to switch from single-particle to centre-of-mass coordinates on a two-particle harmonic oscillator basis. This transformation is applicable even when the masses of the two particles are unequal.