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Talmi-Moshinsky coefficients for momentum space wavefunctions

  1. Jul 15, 2014 #1
    I've found a number of papers about how to calculate Talmi-Moshinsky coefficients. For example W. Tobocman Nucl. Phys A357 (1981) 293-318 and FORTRAN code base on it Y.-P. Gan et al. Comput. Phys. Commun. 34 (1985) 387.
    This works well if I want to calculate matrix elements that only depend on the position or relative position, such as ##\sum_{i<j} \frac{1}{2}\mu_{ij}\omega r_{ij}^2##. I also want to use this to calculate momentum dependent terms, like ##\sum_i \frac{p_i^2}{2m_i}##. The end goal is to eventually calculate terms that depend on both momentum and position, such as
    $$\left( \frac{1}{\sqrt{p_i^2+m_i^2}} \frac{1}{\sqrt{p_j^2+m_j^2}} \right)^{1/2+\epsilon} V(r_{ij}) \left( \frac{1}{\sqrt{p_i^2+m_i^2}} \frac{1}{\sqrt{p_j^2+m_j^2}} \right)^{1/2+\epsilon}.$$
    I could go through Tobocman and recreate what he did step by step in momentum space, and possibly rewrite the Gan et al. code, but I was hoping that someone had already done something similar.

    Some background:
    The 3D harmonic oscillator hamiltonian has the form ##H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2##, which has the solutions ##\psi_{nlm}(\vec{r})## in position space and ##\phi_{nlm}(\vec{p})## in momentum space.
    The positions of the particles in an N-body system can be written in terms of the center of mass coordinate ##\vec{R}## and N-1 Jacobi coordinates ##\vec{\rho}_i##. For N=3, the choice of coordinates are ##\vec{\rho_1^\alpha}=\vec{r_\beta}-\vec{r_\alpha}, \vec{\rho_2^\alpha}=\vec{r_\gamma}-\frac{m_\alpha \vec{r_\alpha}+m_\beta \vec{r_\beta}}{m_\alpha+m_\beta}## where ##\alpha\beta\gamma## could be any of 123,231,312.
    The wavefunction of an N-body system can be written as the product of harmonic oscillator basis wave functions in each coordinate $$\Psi_{\{q\}}^\alpha=\left[\prod_{i=1}^{N-1} \psi_{n_i l_i m_i}(\vec{\rho_i^\alpha})\right]_{\{L\}}$$ where ##\{q\}## is the set of quantum numbers, including all ##n_i,l_i## as well as total (and intermediate total) angular momentum while ##\{L\}## is the set of total and intermediate total angular momentum quantum numbers. I can write these wavefunctions in one basis in terms of the wavefunctions in another basis:
    $$\Psi_{\{q\}}^\alpha=\sum_{\{q'\}} a_{\{q'\}}^{\{q\}}(\alpha,\alpha') \Psi_{\{q'\}}^{\alpha'}$$
    where ##a_{\{q'\}}^{\{q\}}(\alpha,\alpha')## is the Talmi-Moshinsky coefficient.
    If I want to calculate the matrix elements in this basis, I simply have to calculate
    $$\left<\{q'\}^\alpha\right| V \left|\{q\}^\alpha\right> = \int\left(\Psi_{\{q'\}}^\alpha\right)^* V \Psi_{\{q\}}^\alpha \prod d^3\rho$$.
    This is understandably easier for some terms than others. For example ##\vec{r_{21}}=\vec{r_2}-\vec{r_1} = \vec{\rho_1^1}## while ##\vec{r_{32}}=\vec{r_3}-\vec{r_2}= -\vec{\rho_2^1} - \frac{m_2}{m_1+m_2}\vec{\rho_1^1}##. In the ##\alpha\beta\gamma=123## basis, it is nearly trivial to calculate terms that only depend on ##\vec{r_{21}}## only, but you have to account for cross terms when dealing with ##\vec{r_{13}}## and ##\vec{r_{32}}##, making it difficult, if not impossible, to calculate the more complicated terms. That's where Talmi-Moshinsky transformations come in. I can calculate all the terms involving ##\vec{r_{21}}## in the 123 basis, all ##\vec{r_{32}}## terms in the 231 basis, and all ##\vec{r_{13}}## terms in the 312 basis then transform all the matrices into the 123 basis.
    The harmonic oscillator basis is ideal because ##\psi_{nlm}= N_{nl} \alpha^{3/2}(\alpha r)^l L_n^{l+1/2}((\alpha r)^2) e^{-(\alpha r)^2/2} Y_l^m(\hat{r})## while ##\phi_{nlm}= (-i)^{2n+l} N_{nl} \alpha^{-3/2}(p/\alpha)^l L_n^{l+1/2}((p/\alpha)^2) e^{-(p/\alpha)^2/2} Y_l^m(\hat{p})## where ##\alpha^2=m\omega/\hbar##. Because the wavefunctions have the same form in both position and momentum space, it makes calculations easier. The only hiccup is that ##\alpha r## transforms differently (much more nicely, getting rid of cross terms) than ##p/\alpha##, which also means that the Talmi-Moshinsky coefficients are different in momentum space and position space.
     
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