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--- numexercises7_14 [2014/07/14 10:26]
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+++ numexercises7_14 [2014/07/14 12:29]
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   * 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.
-  * The code {{:coulomboscrelme.f90.zip|}} calculates the relative matrix elements $\langle nl|V|n'l\rangle$ in HO states. From this, construct a subroutine that returns the properly normalized $r$-space HO wf's. For some basic background on HO wf's, see  {{:ho_spherical.pdf| here}}.
+  * The code {{:laguerre_general.f90.zip|}} calculates the generalized Laguerre polynomials that appear in the definition of the HO wf's, see {{:ho_spherical.pdf| here}}. Use this subroutine to create a function or subroutine that calculates the HO wf's $R_{nl}(r)$.
-  * Check numerically that the constructed HO wf's are orthonormal. To do this, you will want to use Gaussian quadrature to discretize the integrals. Gaussian quadrature is discussed a bit in {{:ho_spherical.pdf| }}. If you don't have a routine to calculate quadrature points/weights, take advantage of Google to find a canned routine to do this for you.
+  * Check numerically that the constructed HO wf's are orthonormal. I.e., evaluate $\int r^2dr R_{nl}(r)R_{n'l'}(r)$. To do this, you will want to use Gaussian quadrature to discretize the integrals. Gaussian quadrature is discussed a bit in {{:ho_spherical.pdf| }}. (Gauss-Laguerre quadrature is ideal for this problem, but plain old Gauss-Legendre quadrature, which is widely available in canned routines you can easily find via Google, is sufficient.)
-  * Construct the hamilton matrix $\langle nl|H|n'l\rangle$ keeping all HO basis states $n,n'<N_{max}$. Diagonalize the matrix for increasing $N_{max}$ values for different values of the oscillator length parameter $b$ ("oscl" in the code.) $b$ and $\hbar\omega$ are related by $b = \sqrt{(\hbar/(m\omega)}$.
+  * Construct the matrix elements of the Coulomb potential given by
+\begin{equation}
+\langle nl|V|n'l> = \int_0^{\infty}r^2dr R_{nl}(r}\frac{1}{r}R_{n'l}(r)
+\end{equation}
+  * Construct the hamilton matrix $\langle nl|H|n'l\rangle$ keeping all HO basis states $n,n'<N_{max}$. Diagonalize the matrix for increasing $N_{max}$ values for different values of the oscillator length parameter $b$ ("oscl" in the code.) $b$ and $\hbar\omega$ are related by $b = \sqrt{(\hbar/(m\omega)}$.  Plot your lowest eigenvalue for each $N_{max}$ as a function of $b$ (or $\hbar\omega$).
   * Construct a subroutine that returns relative HO matrix elements of the Minnesota NN potential. The definition of the Minnesota potential and a sketch of how to proceed will be given on the black board.

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