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numexercises7_14 [2014/07/14 13:13] bogner |
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* In the code {{:coulomboscrelme.f90.zip|}}, you will find a subroutine (laguerre_general) that 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 $R_{nl}(r)$. Note that for large $n,l$ values, the factorial and double factorial functions that appear in $R_{nl}$ lead to overflow if you code them according to their naive expressions. How might you avoid this problem? [Hint: Recall that $Log{(AB\cdots)}=Log{A} + Log{B} + \cdots$]. | * In the code {{:coulomboscrelme.f90.zip|}}, you will find a subroutine (laguerre_general) that 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 $R_{nl}(r)$. Note that for large $n,l$ values, the factorial and double factorial functions that appear in $R_{nl}$ lead to overflow if you code them according to their naive expressions. How might you avoid this problem? [Hint: Recall that $Log{(AB\cdots)}=Log{A} + Log{B} + \cdots$]. | ||
- | * Check numerically that the constructed HO wf's are orthonormal. I.e., evaluate $\int r^2dr R_{nl}(r)R_{n'l'}(r)$. You will want to use Gaussian quadrature to discretize the integrals. Gaussian quadrature is discussed some in {{:ho_spherical.pdf| }}. While Gauss-Laguerre quadrature is ideal for this problem, plain Gauss-Legendre quadrature, which is widely available in canned routines you can easily find via Google, is sufficient. | + | * Check numerically that the constructed HO wf's are orthonormal. I.e., evaluate $\int r^2dr R_{nl}(r)R_{n'l'}(r)$. You will want to use Gaussian quadrature to discretize the integrals. Gaussian quadrature is discussed some in {{:ho_spherical.pdf| }}. While Gauss-Laguerre quadrature is ideal for this problem, plain Gauss-Legendre quadrature, which is widely available in canned routines you can easily find via Google, is sufficient. |
- | * Construct the matrix elements of the Coulomb potential given by. | + | * Next, write a function/subroutine that calculates the matrix elements of the Coulomb potential, $\langle nl|V|n'l\rangle$. Use atomic units ($e=m_e=1$) where $V(r)=1/r$. |
- | \begin{equation} | ||
- | \langle nl|V|n'l\rangle = \int_0^{\infty}r^2dr R_{nl}(r}\frac{1}{r}R_{n'l}(r) | ||
- | \end{equation} | ||