TalentDFT

In today's exercises, we complete the tasks from yesterday, which unfortunately had ambiguous and confusing instructions!

• Install Git, and try out some of the commands covered in Morten or Nicolas's lecture slides (computing.pdf talentdftguides.pdf) for your codes in the following problems.

• In your favorite programming language, make a program to construct a real symmetric $NxN$ matrix. Diagonalize it using the appropriate LAPACK or GSL routine, and write out some number of the lowest eigenvalues. (Suggestion: You might find it useful to use Mathematica (available on the ECT* computers) to diagonalize a small matrix that you can benchmark against.) This will help you test that you've linked to the GSL or LAPACK library and things are properly installed.

• In the code here, find the subroutine that computes the generalized laguerre polynomials $L_n^{l+1/2}(x)$, “laguerre_general”.

Recall that the HO wavefunctions are given by \begin{equation} R_{nl}(r) = \frac{A_{nl}}{b^{3/2}}x^le^{-x^2/2}L_{n}^{l+1/2}(x^2)\,, \end{equation}

where the oscillator length $b\equiv \sqrt{\hbar/(m\omega)}$, and the dimensionless variable $x\equiv r/b$. The normalization constant is given by

\begin{equation} A_{nl}=\sqrt{\frac{2^{n+l+2}n!}{\pi^{1/2}(2n+2l+1)!!}} \end{equation}

• Using the uploaded code for guidance, make your own subroutine to calculate $L_{n}^{\alpha}(x)$. Then, using the definition of the HO wf's, make a subroutine/function to calculate $R_{nl}(r)$.

• As a check on your code, check the orthogonality of the $R_{nl}$,

\begin{equation} \int_{0}^{\infty}R_{nl}(r)R_{n'l}(r)r^2dr= \delta_{nn'} \end{equation}

• As discussed in the HO notes, the “best” type of Gaussian quadrature over HO wf's is Gauss-Laguerre. However, as you see in the code here, you can use Gauss-Legendre as well. (Note: there are plenty of freely downloadable Gauss-Legendre subroutines.) In this way, the orthogonality check takes the form

\begin{equation} \int_{0}^{\infty}R_{nl}(r)R_{n'l}(r)r^2dr\approx \sum_{i=1}^{N}w_i r_i^2 R_{nl}(r_i)R_{n'l}(r_i)\,, \end{equation}

where the sum is over the N mesh points and $w_{i}$ are the quadrature weights.