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numexercises7_14 [2014/07/14 12:06] bogner |
numexercises7_14 [2014/07/14 12:29] bogner |
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* 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 $R_{nl}(r)$. | * 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. 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| }}. 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 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 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 $\hbar\omega$). |