- #1
ognik
- 626
- 2
Hi - I have been through quite a few articles on the Lagrange Mesh method and mostly follow it, but still find it confusing to understand which are the practical equations I should be using.
I want to find the expression(s) for the Kinetic energy matrix elements for the Laguerre mesh to be used in solving the 1-D Schrödinger eqtn. - so that I can write a fortran program to calculate the elements.
I will assume the potential V everywhere = 0, then the Schrödinger eqtn I want to solve is (from a couple of the articles, mostly with Baye an author) $ \hat{T}\phi = E\phi $, where $ \hat{T} = - \d{^2{}}{{x}^2} + \frac{\alpha(\alpha -2)}{4x^2} $
Not stated in the article, but I will assume that the usual $\frac{{\hbar}^{2}}{2m}$ term has been set to 1 for convenience.
The articles actually recommend the Hermite mesh for 1-D, but my task is to use the Laguerre mesh. What I couldn't be certain of from the articles therefore, is that the $\frac{\alpha(\alpha -2)}{4x^2}$ term seems to be for the Laguerre mesh in a radial situation, therefore for the 1-D case I think I should use $\hat{T} = - \d{^2{}}{{x}^2}$ but would appreciate confirmation?Either way, the articles state the following eqtns for the matrix elements:
$ T_{ij} =\frac{{\alpha + 1}^{2}}{(4x_i)^2} + S_{ij}, i=j $
$ T_{ij} = {(-1)}^{(i-j)} \left[ \frac{1}{2}\left(\alpha+1{\left(x_i x_j\right)}^{-\frac{1}{2}}\right)
\left( {x_i}^{-1}+{x_j}^{-1} \right) + S_{ij} \right] , i \ne j$
where $ S_{ij}=\left(x_ix_j\right)^{\frac{1}{2}} \sum_{k \ne i,j} {x^{-1}_k }(x_k - x_i)^{-1}(x_k-x_j)^{-1} $
But how big do I make the matrix? I think that I should sum i,j from 1 to N and look at the accuracy with different values of N
The Laguerre mesh is over $[0, \infty)$ and some of the error in the method are the mesh points between N and $\infty$ that we ignore. An example in an article used N=4, but with no justification.
However the column vectors of T will be an orthonormal basis for the space, which should be 1-D so I can't see how T can be an NxN matrix, with N large?
The N I choose also determines the Laguerre eqtn from which the mesh points $x_i$ can be determined, the roots from $L^{\alpha}_N(x_i) = 0 $ This all makes me think there is more than 1 N in play?
I have read through these articles until the letters dribbled off the page in protest, the answer to N eludes me.
Even if you don't have all the answers, I'd appreciate all assistance, thanks.
I want to find the expression(s) for the Kinetic energy matrix elements for the Laguerre mesh to be used in solving the 1-D Schrödinger eqtn. - so that I can write a fortran program to calculate the elements.
I will assume the potential V everywhere = 0, then the Schrödinger eqtn I want to solve is (from a couple of the articles, mostly with Baye an author) $ \hat{T}\phi = E\phi $, where $ \hat{T} = - \d{^2{}}{{x}^2} + \frac{\alpha(\alpha -2)}{4x^2} $
Not stated in the article, but I will assume that the usual $\frac{{\hbar}^{2}}{2m}$ term has been set to 1 for convenience.
The articles actually recommend the Hermite mesh for 1-D, but my task is to use the Laguerre mesh. What I couldn't be certain of from the articles therefore, is that the $\frac{\alpha(\alpha -2)}{4x^2}$ term seems to be for the Laguerre mesh in a radial situation, therefore for the 1-D case I think I should use $\hat{T} = - \d{^2{}}{{x}^2}$ but would appreciate confirmation?Either way, the articles state the following eqtns for the matrix elements:
$ T_{ij} =\frac{{\alpha + 1}^{2}}{(4x_i)^2} + S_{ij}, i=j $
$ T_{ij} = {(-1)}^{(i-j)} \left[ \frac{1}{2}\left(\alpha+1{\left(x_i x_j\right)}^{-\frac{1}{2}}\right)
\left( {x_i}^{-1}+{x_j}^{-1} \right) + S_{ij} \right] , i \ne j$
where $ S_{ij}=\left(x_ix_j\right)^{\frac{1}{2}} \sum_{k \ne i,j} {x^{-1}_k }(x_k - x_i)^{-1}(x_k-x_j)^{-1} $
But how big do I make the matrix? I think that I should sum i,j from 1 to N and look at the accuracy with different values of N
The Laguerre mesh is over $[0, \infty)$ and some of the error in the method are the mesh points between N and $\infty$ that we ignore. An example in an article used N=4, but with no justification.
However the column vectors of T will be an orthonormal basis for the space, which should be 1-D so I can't see how T can be an NxN matrix, with N large?
The N I choose also determines the Laguerre eqtn from which the mesh points $x_i$ can be determined, the roots from $L^{\alpha}_N(x_i) = 0 $ This all makes me think there is more than 1 N in play?
I have read through these articles until the letters dribbled off the page in protest, the answer to N eludes me.
Even if you don't have all the answers, I'd appreciate all assistance, thanks.