class: titlepage .title[ Advanced Scientific Programmation ] .title[ 2D Solver for the time-dependent non-relativistic Schröding equation ] <br/><br/> .subtitle[ Alexis PUDDU & Sylvain MARET ] .footnote[ [:book:](../toc/index.html) ] ??? Some slide notes. Want some `\(\LaTeX\)` ? `$$a_{0}=\frac{1}{4}b_{1}$$` Some items: * first * second * third --- layout: true class: animated fadeIn middle .footnote[ PSA Presentation -A. Puddu & S. Maret - ENSIIE - 2018 - [:book:](../toc/index.html) ] --- class: toc top #Table of content 1. Project Presentation -- 2. Project Architecture -- 3. Field Generator -- 4. Solver -- 5. Unitary Tests -- --- # Project Presentation `Problem to solve` `$$i\hbar\frac{\partial}{\partial t}\psi(x,y,t) = \hat{\mathcal{H}}_{(x,y)}\psi(x,y,t)$$` with the 2D-Hamiltonian operator defined as `$$\hat{\mathcal{H}}_{(x,y)}\equiv \frac{\hat{p}_{(x)}^2}{2m} + \frac{\hat{p}_{(y)}^2}{2m} + \hat{V}(x,y),$$` and the momentum operators defined as `$$\forall u \in \{x,y\},\hspace{5mm}\hat{p}_{(u)}\equiv -i\hbar\frac{\partial}{\partial u}.$$` `Objective of the project` :arrow_right: Generate the initial wave function :arrow_right: Generate the potential field from an image file or a formula :arrow_right: Implement solver with 3 different methods of finite difference --- #Project Architecture .hcenter.shadow.w80.animated.fadeInUp.wait1s[  ] --- # Field Generator `Initial wave $$ \psi_0 $$` .hcenter.shadow.w50.animated.fadeInUp.wait1s[  ] --- # Field Generator `Initial potential from a formula` .hcenter.shadow.w25.animated.fadeInUp.wait1s[  ] --- # Field Generator `Initial potential from an image` .hcenter.shadow.w25.animated.fadeInUp.wait1s[  ] .hcenter.shadow.w50.animated.fadeInUp.wait1s[  ] .block.hcenter[ Conversion of an image to an array ] --- # Solver The main idea of the solver is to apply ftcs, btcs or ctcs method onto the result given by the wave generator and then determine the plane wave to see the propagation of the wave. <pre id="terminal"> for(i = 0; i < nb_it-1; i++){ for( x = 0; x < st_x ; x++){ for( y = 0; y < st_y ; y++){ arma::cx_mat tmpdx=arma::shift(res.slice(i),1,0); arma::cx_mat tmpdx2=arma::shift(res.slice(i),-1,0); arma::cx_mat tmpdy=arma::shift(res.slice(i),1,1); arma::cx_mat tmpdy2=arma::shift(res.slice(i),-1,1); res_i = (potential/HB)%res.slice(i)+(HB*dt/(2*M*dx*dx))*(tmpdx-2*res.slice(i)+tmpdx2) +(HB*dt/(2*M*dy*dy))*(tmpdy-2*res.slice(i)+tmpdy2); } } } </pre> .block.hcenter[ Main part of the solver ] --- # Solver : Crank-Nicholson demonstration `$$i\hbar\frac{\partial}{\partial t}\psi(x,y,t) = \hat{\mathcal{H}}_{(x,y)}\psi(x,y,t)$$` with `$$\frac{\partial}{\partial t}\psi(x,y,t) = \frac{\psi(x,y,t+dt) - \psi(x,y,t)}{h_t}$$` `$$\frac{\partial^2}{\partial x^2}\psi(x,y,t)= \frac{1}{2h_x^2}(\psi^{(t+h_t)}_{x+h_x} - 2\psi^{(t+h_t)}_{x} + \psi^{(t+h_t)}_{x-h_x}+ \psi^{t}_{x+h_x} - 2\psi^{t}_{x} + \psi^{t}_{x-h_x}$$` `$$\frac{\partial}{\partial x}\psi(x,y,t)= \frac{1}{4h_x^2}(\psi^{(t+h_t)}_{x+h_x} - \psi^{(t+h_t)}_{x-h_x}+ \psi^{t}_{x+h_x} - \psi^{t}_{x-h_x})$$` --- #Solver : Crank-Nicholson demonstration `$$i\hbar\frac{\partial}{\partial t}\psi(x,y,t) = \hat{\mathcal{H}}_{(x,y)}\psi(x,y,t)$$` `$$\begin{eqnarray}\frac{-i\hbar}{h_t}(\psi^{t+h_t} - \psi^{t}) &=& V(x,y)(\psi^{t} + \psi^{t+h_t}) - \frac{\hbar}{2m}(\frac{1}{2h_x^2}(\psi^{(t+h_t )}_{x+h_x}- 2\psi^{t+h_t}{x} + \psi^{t+h_t}_{x-h_x}\\ &+& \psi^{t}_{x+h_x} - 2\psi^{t}_{x} + \psi^{t}_{x-h_x})\\ &+& \frac{1}{2h_y^2}(\psi^{(t+h_t )}_{y+h_y}- 2\psi^{t+h_t}_{y} + \psi^{t+h_t}_{y-h_y} + \psi^{t}_{y+h_y} - 2\psi^{t}_{y} + \psi^{t}_{y-h_y}) \end{eqnarray}$$` which after reorganisating the member give us : `$$\begin{eqnarray} \left(\frac{2i\hbar}{h_t}-V(x,y)-\frac{\hbar^2}{mh_x^2}-\frac{\hbar^2}{mh_y^2}\right)\psi^{(t+h_t)} &=& \frac{-\hbar^2}{2mh_x^2}\left(\psi_{x+h_x}+\psi_{x-h_x}\right)\\ &-& \frac{\hbar^2}{2mh_y^2}\left(\psi_{y+h_y}+\psi_{y-h_y}\right)\\ &-& \frac{\hbar^2}{2mh_x^2}\left(\psi^{(t+h_t)}_{x+h_x}+\psi^{(t+h_t)}_{x-h_x}\right)\\ &-& \frac{\hbar^2}{2mh_y^2}\left(\psi^{(t+h_t)}_{y+h_y}+\psi^{(t+h_t)}_{y-h_y}\right)\\ &+& \left(V(x,y) + \frac{2i\hbar}{h_t}+\frac{\hbar^2}{mh_x^2}+\frac{\hbar^2}{mh_y^2}\right)\psi \end{eqnarray}$$` --- # Solver `Bindings` :arrow_right: Creation of module solmod :arrow_right: Adapt numpy array to fortran array in python <pre id="terminal"> module1= Extension('_solmod', include_dirs=[numpy.get_include(),'./include/armanpy'], libraries=['m','z','armadillo'], sources=['solmod.i','solver.cpp'], swig_opts=["-c++","-Wall","-I.","-I./include/armanpy"], extra_compile_args=['--std=c++11']) </pre> <pre id="terminal"> import solmod pot=np.asfortranarray(pot,dtype=float) sol=solmod.Solver(st_x,st_y,time,nb_it,pot) </pre> --- # Unitary Tests `Test to check the proper functionning of Armadillo` `Test to check the normality of matrix after fcts` ---