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numexercises7_14 [2014/07/14 13:19] bogner |
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* 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. | ||
- | * 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 $V(r)=1/r$. | + | * 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 ($e=m_e=\hbar=1/4\pi\epsilon_0$). |
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+ | * Modify your code in the previous step so that it calculates $\langle nl|V|n'l\rangle$ for any user-supplied potential $V(r)$. | ||
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+ | * 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. Keep HO basis states $n,n'<N_{max}$ and diagonalize it. 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 $b=\sqrt{\hbar/(m\omega)$. | ||