deploy: 7f24d69

.buildinfo

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+# Sphinx build info version 1
+# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
+config: c729e624b1d095b2c5341f61d75097ce
+tags: 645f666f9bcd5a90fca523b33c5a78b7

_sources/doc_pages/examples.rst.txt

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+Examples
+########

_sources/doc_pages/spherical_coordinates.rst.txt

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+Spherical coordinates
+#####################
+
+Spherical coordinate system
+===========================
+
+.. math::
+
+    x &= r \sin \theta \cos \phi \\
+    y &= r \sin \theta \sin \phi \\
+    z &= r \cos \theta
+
+where :math:`r \in [0,\infty)`, :math:`\theta \in [0,\pi]` and :math:`\phi \in [0,2\pi)`. 
+
+Furthermore, the volume element is given by 
+
+.. math:: 
+    
+    dV = r^2 \sin \theta  dr d\theta d\phi 
+
+and the Laplacian is given by 
+
+.. math::
+
+    \nabla^2 &= \frac{1}{r^2} \frac{\partial}{\partial r}\left( r^2 \frac{\partial}{\partial r} \right) + \frac{1}{r^2} \left[\frac{1}{\sin(\theta)}\frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial}{\partial \theta}\right) +\frac{1}{\sin^2(\theta)}\frac{\partial^2}{\partial \phi^2}\right] \\
+    &= \frac{1}{r} \frac{\partial^2}{\partial r^2} r - \frac{\hat{L}^2}{r^2}
+
+
+Wavefunction paramtetrization
+=============================
+
+In spherical coordinates, we parametrize the wavefunction as
+
+.. math::
+
+    \Psi(\mathbf{r}) = \sum_{l=0}^{l_{max}} \sum_{m=-l}^{l} r^{-1} u_{l,m}(r) Y_{l,m}(\theta, \phi),
+
+where :math:`Y_{l,m}(\theta, \phi)` are the spherical harmonics and :math:`l_{max}` the maximum angular momentum.
+
+.. math::
+
+    \nabla^2 \Psi(\mathbf{r}) = \sum_{l,m} \frac{1}{r} \left(\frac{d^2 u_{l,m}(r)}{d r^2} - \frac{l(l+1)}{r^2} u_{l,m}(r) \right) Y_{l,m}(\theta, \phi)
+
+The time-independent Schrödinger equation
+=========================================
+
+The time-independent Schrödinger equation
+
+.. math::
+    \left(-\frac{1}{2}\nabla^2 + V(\mathbf{r}) \right) \Psi_k(\mathbf{r}) = E_k \Psi_k(\mathbf{r}), \ \ k=1,2,3,\cdots
+
+For a spherical symmetric potential :math:`V(r)`, the eigenfunctions can be taken as 
+
+.. math::
+    \Psi_{n,l,m}(\mathbf{r}) = r^{-1} u_{n,l}(r) Y_{l,m}(\theta, \phi),
+
+and the (radial) TISE becomes 
+
+.. math::
+
+    -\frac{1}{2}\frac{d^2 u_{n,l}(r)}{d r^2}+\frac{l(l+1)}{2 r^2} u_{n,l}(r) + V(r)u_{n,l} = \epsilon_{n,l} u_{n,l}(r).
+
+Gauss-Legendre-Lobatto quadrature is defined on :math:`x \in [-1,1]`. 
+We need to map the grid/Lobatto points :math:`x_i \in [-1,1]` into radial points :math:`r(x): [-1,1] \rightarrow [0, r_{\text{max}}]`. 
+In that case, :math:`\psi(r) = \psi(r(x))`.
+
+The (rational) mapping 
+
+.. math::
+    
+    r(x) = L \frac{1+x}{1-x+\alpha}, \ \ \alpha = \frac{2L}{r_{\text{max}}} \label{x_to_r},
+
+has often been used in earlier work. 
+The parameters :math:`L` and :math:`\alpha` control the length of the grid and the density of points near :math:`r=0`. 
+
+Another option is to use the linear mapping given by 
+
+.. math::
+
+    r(x) = \frac{r_{\text{max}}(x+1)}{2}.
+
+In order to use the Gauss-Legendre-Lobatto pseudospectral method, we must first formulate 
+the radial TISE with respecto to :math:`x`.
+
+By the chain rule we find 
+
+.. math::
+    
+    \frac{d \psi}{dr} &= \frac{1}{\dot{r}(x)} \frac{d \psi}{dx}, \\
+    \frac{d^2 \psi}{dr^2} &= \frac{1}{\dot{r}(x)^2} \frac{d^2 \psi}{dx^2} - \frac{\ddot{r}(x)}{\dot{r}(x)^3} \frac{d \psi}{dx},
+
+and the radial TISE becomes
+
+.. math::
+    :name: eq:unsymmetric_radial_TISE
+    
+    -\frac{1}{2} \left( \frac{1}{\dot{r}(x)^2} \frac{d^2 \psi}{dx^2} - \frac{\ddot{r}(x)}{\dot{r}(x)^3} \frac{d \psi}{dx} \right) + V_l(r(x)) \psi(r(x)) = \epsilon \psi(r(x)),
+    
+where we have defined :math:`V_l(r(x)) \equiv V(r(x)) + \frac{l(l+1)}{2 r(x)^2}`. Note that this is, in general, an unsymmetric eigenvalue problem 
+due to the presence of the term :math:`\frac{\ddot{r}(x)}{\dot{r}(x)^3}`.
+
+In order to reformulate the above as a symmetric eigenvalue problem, we define the new wavefunction :math:`f(x) = \dot{r}(x)^{1/2} \psi(r(x))`.
+Insertion of this into Eq. :ref:`Link title <eq:unsymmetric_radial_TISE>` yields 
+
+.. math::
+    
+    \left(-\frac{1}{2} \frac{1}{\dot{r}(x)} \frac{d^2}{dx^2} \frac{1}{\dot{r}(x)} + V_l(r(x))+\tilde{V}(r(x)) \right) f(x) = \epsilon f(x),
+    :label: symmetric_radial_TISE
+
+where we have introduced the new potential :math:`\tilde{V}(x) \equiv \frac{2\dddot{r}(x)\dot{r}(x)-3\ddot{r}(x)^2}{4\dot{r}(x)^4}`.
+
+We discretize this equation with the pseudospectral method by expanding :math:`f(x)` and :math:`f(x)/\dot{r}(x)` as 
+
+.. math::
+
+    f(x) &= \sum_{j=0}^N f(x_j) g_j(x), \\
+    \frac{f(x)}{\dot{r}(x)} &= \sum_{j=0}^N \frac{f(x_j)}{\dot{r}(x_j)} g_j(x).
+
+Inserting these expansions into Eq. :math:ref:`symmetric_radial_TISE` we have that 
+
+.. math::
+
+     \sum_{j=0}^N \left(-\frac{1}{2} \frac{1}{\dot{r}(x)} \frac{f(x_j)}{\dot{r}(x_j)} g^{\prime \prime}_j(x) + V(r(x)) f(x_j) g_j(x) \right) = \epsilon \sum_{j=0}^N f(x_j) g_j(x)
+
+Next, we multiply through with :math:`g_i(x)` and integrate over :math:`x`, 
+
+.. math::
+
+     \sum_{j=0}^N \left(-\frac{1}{2}  \frac{f(x_j)}{\dot{r}(x_j)} \int \frac{g_i(x)}{\dot{r}(x)} g^{\prime \prime}_j(x) dx +  f(x_j) \int g_i(x) V(r(x)) g_j(x) dx \right) = \epsilon \sum_{j=0}^N f(x_j) \int g_i(x) g_j(x) dx
+

_sources/doc_pages/theory.rst.txt

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+Theory
+######
+
+The pseudospectral method
+=========================
+In the pseudospectral method a function :math:`f(x)` is approximated as a linear combination
+
+.. math::
+    
+    f(x) \simeq f_N(x) = \sum_{j=0}^N f(x_j) g_j(x)
+
+
+of cardinal functions :math:`g_j(x)` where the expansion coefficients are values of 
+:math:`f(x)` at the grid points :math:`x_j`. For Lobatto-type grids, the interior grid points 
+:math:`\{ x_j \}_{j=1}^{N-1}` are defined as the zeros of :math:`P_N^\prime(x)`, where :math:`P_N(x)` is a 
+:math:`N`-th order polynomial [#f1]_,
+
+.. math::
+    
+    P_N^\prime(x_j) = 0, \ \ j=1,\cdots,N-1,
+
+
+while the end (boundary) points :math:`x_0` and :math:`x_N` are given.
+
+The cardinal functions :math:`g_j(x)` have the property that
+
+.. math::
+    
+    g_j(x_i) = \delta_{ij}, \label{delta_property_g}
+
+
+and can be written as 
+
+.. math::
+    
+    g_j(x) = -\frac{1}{N(N+1)P_N(x_j)}\frac{(1-x^2)P_N^\prime(x)}{x-x_j}.
+
+
+The integral of a function is approximated by quadrature
+
+.. math::
+    
+    \int_{x_0}^{x_N} f(x) dx \simeq \sum_{j=0}^N w_j f(x_j), \label{quadrature_rule}
+
+
+where :math:`w_j` are the quadrature weights.
+
+.. rubric:: Footnotes
+
+.. [#f1] Which type of polynomial to choose depends on the problem. Examples include Chebyshev, Legendre, Laguerre or Hermite polynomials.
+
+
+Gauss-Legendre-Lobatto
+=======================
+
+In the Gauss-Legendre-Lobatto pseudospectral method, :math:`P_N(x)` is taken as the :math:`N`-th 
+order Legendre polynomial. The end points are :math:`x_0 = -1` and :math:`x_N=1`, while the interior 
+grid points must be found as the roots 
+
+.. math::
+    
+    P_N^\prime(x_i) = 0,
+
+and the quadrature weights are given by
+
+.. math::
+    
+    w_i = \frac{2}{N(N+1)P_N(x_i)^2}.
+
+The grid points and weights can be determined using numpy functions (REF).
+Furthermore, 
+
+.. math::
+    
+    
+    \frac{d g_j}{dx} \Bigr|_{x=x_i} &= g_j^\prime(x_i) = \tilde{g}_j^\prime(x_i) \frac{P_N(x_i)}{P_N(x_j)}, \\
+    \tilde{g}_j(x_i) &= 
+    \begin{cases}
+    \frac{1}{4}N(N+1), \ \ i=j=0, \\
+    -\frac{1}{4}N(N+1), \ \ i=j=N, \\
+    0, \ \ i=j \text{ and } 1 \leq j \leq N-1, \\
+    \frac{1}{x_i-x_j}, \ \ i \neq j,
+    \end{cases}
+    
+
+and for :math:`i,j = 1,\cdots,N-1`
+
+.. math::
+    
+    \frac{d^2 g_j}{dx^2} \Bigr|_{x=x_i} &= g_j^{\prime \prime}(x_i) = \tilde{g}_j^{\prime \prime}(x_i) \frac{P_N(x_i)}{P_N(x_j)}, \\
+    \tilde{g}_j^{\prime \prime}(x_i) &= 
+    \begin{cases}
+    -\frac{1}{3} \frac{N(N+1)}{(1-x_i^2)}, & i = j,\\
+    -\frac{2}{(x_i-x_j)^2}, & i \neq j.
+    \end{cases}
+

_sources/index.rst.txt

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+.. Test documentation master file, created by
+   sphinx-quickstart on Thu Feb 29 14:38:30 2024.
+   You can adapt this file completely to your liking, but it should at least
+   contain the root `toctree` directive.
+
+Pseudospectral TDSE documentation
+=================================
+
+Indices and tables
+==================
+
+* :ref:`genindex`
+* :ref:`modindex`
+* :ref:`search`
+
+.. Contents
+.. ========
+
+.. toctree::
+   :maxdepth: 2
+   :numbered:
+   :hidden:
+   :caption: Examples
+
+   ./doc_pages/examples
+
+.. toctree::
+   :maxdepth: 2
+   :numbered:
+   :hidden:
+   :caption: Theory
+
+   ./doc_pages/theory
+   ./doc_pages/spherical_coordinates

_static/_sphinx_javascript_frameworks_compat.js

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+/* Compatability shim for jQuery and underscores.js.
+ *
+ * Copyright Sphinx contributors
+ * Released under the two clause BSD licence
+ */
+
+/**
+ * small helper function to urldecode strings
+ *
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+ */
+jQuery.urldecode = function(x) {
+    if (!x) {
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+    }
+    return decodeURIComponent(x.replace(/\+/g, ' '));
+};
+
+/**
+ * small helper function to urlencode strings
+ */
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+
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