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--- numexercises7_14 [2014/07/14 12:50]
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+++ numexercises7_14 [2014/07/14 13:36]
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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.
+  * Write a function/subroutine that calculates the matrix elements of the Coulomb potential between an electron and a proton, $\langle nl|V|n'l\rangle$. Use atomic units ($e=m_e=\hbar=1/4\pi\epsilon_0$).
+  * Modify your code in the previous step so that it calculates $\langle nl|V|n'l\rangle$ for any user-supplied potential $V(r)$.
+  * Using the analytical expressions for the kinetic energy matrix elements $\langle nl|T|n'l\rangle$,construct the hamilton matrix $\langle nl|H|n'l\rangle$ for the hydrogen atom for $l=0$. Keep HO basis states $n,n'<N_{max}$ and diagonalize $H$. For a given $N_{max}$ value, repeat the calculation at different HO frequencies and plot the ground state energy versus $\omega$ (or the oscillator length scale, defined as $b=\sqrt{\hbar/(m\omega)}$.
-  * 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 matrix elements of the Coulomb potential given by
-\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}
   * 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 $\hbar\omega$).

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