Schrödinger equation - Revision history

APirnat at 14:19, 19 July 2019

2019-07-19T14:19:31Z

← Older revision		Revision as of 14:19, 19 July 2019
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	[[File:infinite_well_2D_error.png\|400px]]		[[File:infinite_well_2D_error.png\|400px]]
	[[File:infinite_well_2D_error_fit.png\|400px]]		[[File:infinite_well_2D_error_fit.png\|400px]]

			The complete exampe is avalible at [https://gitlab.com/e62Lab/medusa/blob/dev/examples/quantum_mechanics/infinite_well_2D.cpp infinite_well_2D.cpp].

	== 3D infinite cubic well ==		== 3D infinite cubic well ==
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	[[File:infinite_well_3D_rsol.png\|400px]]		[[File:infinite_well_3D_rsol.png\|400px]]
	[[File:infinite_well_3D_csol.png\|400px]]		[[File:infinite_well_3D_csol.png\|400px]]

			The complete exampe is avalible at [https://gitlab.com/e62Lab/medusa/blob/dev/examples/quantum_mechanics/infinite_well_3D.cpp infinite_well_3D.cpp].

	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

APirnat at 14:35, 15 July 2019

2019-07-15T14:35:30Z

← Older revision		Revision as of 14:35, 15 July 2019
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	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

			__FORCETOC__

	== 2D infinite square well ==		== 2D infinite square well ==
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	\]		\]

	remain unchanged, ~~only~~ the domain changes to unit cube, $\Omega=[0, 1]\times[0, 1]\times [0, 1]$.		remain unchanged, merely the domain changes to unit cube, $\Omega=[0, 1]\times[0, 1]\times [0, 1]$. The only step left is to select an initial state of the wave function, we choose
	~~For~~ initial state ~~$\Psi(x~~, ~~y, z,t= 0)$~~ we choose
	\[		\[
	\Psi(x,y,t=0) = \sin{(\pi x)} \sin{(\pi y) \sin(\pi z)}.		\Psi(x,y,t=0) = \sin{(\pi x)} \sin{(\pi y) \sin(\pi z)}.
	\]		\]
	The plot of the real and imaginary part of the solution $\Psi(x, y, z)$ at time $T=0.1$ is ~~presented~~ bellow.		The plot of the real and imaginary part of the solution $\Psi(x, y, z)$ at time $T=0.1$ is shown bellow.

	[[File:infinite_well_3D_rsol.png\|400px]]		[[File:infinite_well_3D_rsol.png\|400px]]

APirnat at 14:21, 15 July 2019

2019-07-15T14:21:31Z

← Older revision		Revision as of 14:21, 15 July 2019
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	== 3D infinite cubic well ==		== 3D infinite cubic well ==
	Now we are dealing with a problem very similar to the ~~previus~~ one, the only ~~diffrenece~~ is, that in this case we have a ~~quantom~~ particle in a 3-dimensional box. ~~Equation~~ and the boundary condition		Now we are dealing with a problem very similar to the previous one, the only difference is, that in this case we have a quantum particle in a 3-dimensional box. The equation and the boundary condition

	\[		\[
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	For initial state $\Psi(x, y, z,t= 0)$ we choose		For initial state $\Psi(x, y, z,t= 0)$ we choose
	\[		\[
	\Psi(x,y,t=0) = \sin{(\pi x)} \sin{(\pi y) \sin(\~~piz~~)}.		\Psi(x,y,t=0) = \sin{(\pi x)} \sin{(\pi y) \sin(\pi z)}.
	\]		\]
	The plot of the real and imaginary part of the solution $\Psi(x, y, z)$ at time $T=0.1$ is ~~plotted~~ bellow.		The plot of the real and imaginary part of the solution $\Psi(x, y, z)$ at time $T=0.1$ is presented bellow.

	[[File:infinite_well_3D_rsol.png\|400px]]		[[File:infinite_well_3D_rsol.png\|400px]]

APirnat at 14:19, 15 July 2019

2019-07-15T14:19:00Z

@@ Line 73: / Line 73: @@
 [[File:infinite_well_2D_error.png|400px]]
 [[File:infinite_well_2D_error_fit.png|400px]]
 Go back to [[Medusa#Examples|Examples]].

APirnat at 13:17, 15 July 2019

2019-07-15T13:17:39Z

← Older revision		Revision as of 13:17, 15 July 2019
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	\]		\]

	With this potential ~~Schrodinger~~ equation simplifies to		With this potential Schrödinger equation simplifies to
	\[		\[
	-{\hbar^2 \over 2m} \nabla^2 \Psi(x,y,t) = i\hbar {\partial \over \partial t}\Psi(x,y,t),		-{\hbar^2 \over 2m} \nabla^2 \Psi(x,y,t) = i\hbar {\partial \over \partial t}\Psi(x,y,t),
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	\Psi = 0 \quad \text{on} \quad \partial \Omega,		\Psi = 0 \quad \text{on} \quad \partial \Omega,
	\]		\]
	where $\Omega=[0, 1]\times[0, 1]$ denotes the square box domain and $\partial \Omega$ is the boundary of the domain. In order to get the unique solution for the time propagation of the wave function the initial state $\Psi(x, y,t= 0)$ has to be selected. We choose		where $\Omega=[0, 1]\times[0, 1]$ denotes the square box domain and $\partial \Omega$ is the boundary of the domain. In order to get the unique solution for the time propagation of the wave function, the initial state $\Psi(x, y,t= 0)$ has to be selected. We choose
	\[		\[
	\Psi(x,y,t=0) = \sin{(\pi x)} \sin{(\pi y)}.		\Psi(x,y,t=0) = \sin{(\pi x)} \sin{(\pi y)}.
	\]		\]

	Once we prepare the domain and operators, we can turn the ~~Schrodinger~~ equation into code. As we are solving the time propagation implicitly, we will have to solve a matrix equation for every step. For this reason a space matrix $M$ is constructed. Next the equations are implemented, $psi$ denotes the matrix solution for each step and $rhs$ the right-hand side of the matrix equation which is in our case equivalent to the previous state $psi$ .		Once we prepare the domain and operators, we can turn the Schrödinger equation into code. As we are solving the time propagation implicitly, we will have to solve a matrix equation for every step. For this reason a space matrix $M$ is constructed. Next the equations are implemented, $psi$ denotes the matrix solution for each step and $rhs$ the right-hand side of the matrix equation, which is in our case equivalent to the previous state $psi$.

	<syntaxhighlight lang="~~c++~~" ~~line~~>		<syntaxhighlight lang="cpp">
	~~Eigen::~~SparseMatrix<~~std::~~complex<double>, ~~Eigen::~~RowMajor> M(N, N);		SparseMatrix<complex<double>, RowMajor> M(N, N);

	// ~~set~~ initial state		// Set initial state.
	for(int i : domain.interior()) {		for(int i : domain.interior()) {
	double x = domain.pos(i, 0);		double x = domain.pos(i, 0);
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	}		}

	// ~~set~~ equation on interior		// Set equation on interior.
	for(int i : domain.interior()) {		for(int i : domain.interior()) {
	op.value(i) + (-1.0i) * dt * op.lap(i) = rhs(i);		op.value(i) + (-1.0_i) * dt * op.lap(i) = rhs(i);
	}		}

	// ~~set~~ equation on boundary		// Set equation on boundary.
	for (int i: domain.boundary()){		for (int i: domain.boundary()){
	op.value(i) = 0;		op.value(i) = 0;
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	</syntaxhighlight>		</syntaxhighlight>

	At each time step the matrix equation is solved using an Eigen linear equation solver, in this case, the BICGStab algorithm, and $rhs$ is updated.		At each time step the matrix equation is solved using an Eigen linear equation solver, in this case the BICGStab algorithm, and $rhs$ is updated.

	<syntaxhighlight lang="~~c++~~" ~~line~~>		<syntaxhighlight lang="cpp">
	// ~~solve~~ matrix system		// Solve matrix system.
	psi = solver.solve(rhs);		psi = solver.solve(rhs);

	// ~~update~~ rhs		// Update rhs.
	rhs = psi;		rhs = psi;
	</syntaxhighlight>		</syntaxhighlight>

	Time propagation of the real part of the solution $\Psi(x,y; t)$ from times 0 to $T=0.63$ is presented below. Time step used was $10^{-5}$.		Time propagation of the real part of the solution $\Psi(x,y,t)$ from times 0 to $T=0.63$ is presented below. Time step used was $10^{-5}$.

	[[File:video (1).mp4\|400px]]		[[File:video (1).mp4\|400px]]

	As analytical solutions for infinite potential well are known this enables comparison between analytical and numerical solution computed with Medusa. On the graph bellow the error of the numerical solution is presented as maximal absolute error, meaning the largest absolute difference between numeric and its analytical solution in node at specific time. Expected linear convergence of error on a log - log scale can be observed.		As analytical solutions for infinite potential well are known this enables comparison between analytical and numerical solution computed with Medusa. On the graph bellow the error of the numerical solution is presented as maximal absolute error $e_{\infty}$, meaning the largest absolute difference between numeric and its analytical solution in node at specific time. Expected linear convergence of error on a log - log scale can be observed.

	[[File:~~infinite_well_2D_convergence~~.png\|400px]]		[[File:infinite_well_2D_error.png\|400px]]
	[[File:~~infinite_well_2D_convergence_fit~~.png\|400px]]		[[File:infinite_well_2D_error_fit.png\|400px]]

	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

APirnat at 20:33, 17 June 2019

2019-06-17T20:33:35Z

← Older revision		Revision as of 20:33, 17 June 2019
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	</syntaxhighlight>		</syntaxhighlight>

	Time propagation of the real part of the solution \Psi(x,y; t)$ from times 0 to $T=0.63$ is presented below. Time step used was $10^{-5}$.		Time propagation of the real part of the solution $\Psi(x,y; t)$ from times 0 to $T=0.63$ is presented below. Time step used was $10^{-5}$.

	[[File:video (1).mp4\|400px]]		[[File:video (1).mp4\|400px]]

APirnat at 20:32, 17 June 2019

2019-06-17T20:32:14Z

← Older revision		Revision as of 20:32, 17 June 2019
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	As analytical solutions for infinite potential well are known this enables comparison between analytical and numerical solution computed with Medusa. On the graph bellow the error of the numerical solution is presented as maximal absolute error, meaning the largest absolute difference between numeric and its analytical solution in node at specific time. Expected linear convergence of error on a log - log scale can be observed.		As analytical solutions for infinite potential well are known this enables comparison between analytical and numerical solution computed with Medusa. On the graph bellow the error of the numerical solution is presented as maximal absolute error, meaning the largest absolute difference between numeric and its analytical solution in node at specific time. Expected linear convergence of error on a log - log scale can be observed.

	[[File:~~infinite_well_2D_convergence_fit~~.~~pdf~~\|400px]]		[[File:infinite_well_2D_convergence.png\|400px]]
	[[File:~~infinite_well_2D_xerr_convergence~~.~~pdf~~\|400px]]		[[File:infinite_well_2D_convergence_fit.png\|400px]]

	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

APirnat at 20:22, 17 June 2019

2019-06-17T20:22:53Z

← Older revision		Revision as of 20:22, 17 June 2019
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	<syntaxhighlight lang="c++" line>		<syntaxhighlight lang="c++" line>
	Eigen::SparseMatrix<std::complex<double>, Eigen::RowMajor> M(N, N);		Eigen::SparseMatrix<std::complex<double>, Eigen::RowMajor> M(N, N);

	// set initial state		// set initial state
	for(int i : domain.interior()) {		for(int i : domain.interior()) {
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	</syntaxhighlight>		</syntaxhighlight>

	~~The animation~~ of the real part of solution ~~$\Re~~ \Psi(x,y)$ is presented below.		Time propagation of the real part of the solution \Psi(x,y; t)$ from times 0 to $T=0.63$ is presented below. Time step used was $10^{-5}$.

	[[File:video (1).mp4\|400px]]		[[File:video (1).mp4\|400px]]

			As analytical solutions for infinite potential well are known this enables comparison between analytical and numerical solution computed with Medusa. On the graph bellow the error of the numerical solution is presented as maximal absolute error, meaning the largest absolute difference between numeric and its analytical solution in node at specific time. Expected linear convergence of error on a log - log scale can be observed.

			[[File:infinite_well_2D_convergence_fit.pdf\|400px]]
			[[File:infinite_well_2D_xerr_convergence.pdf\|400px]]

	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

APirnat at 09:44, 17 June 2019

2019-06-17T09:44:43Z

← Older revision		Revision as of 09:44, 17 June 2019
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	With this potential Schrodinger equation simplifies to		With this potential Schrodinger equation simplifies to
	\[		\[
	-{\hbar^2 \over 2m} \nabla^2 \Psi(x,y;t) = i\hbar {\partial \over \partial t}\Psi(x,y;t),		-{\hbar^2 \over 2m} \nabla^2 \Psi(x,y,t) = i\hbar {\partial \over \partial t}\Psi(x,y,t),
	\]		\]
	where $m$ is the mass of the particle, $\hbar$ is reduced Planck constant and $\Psi(x,y;t)$ is the wave function. We set $\hbar / 2m = 1$ and $L=1$.		where $m$ is the mass of the particle, $\hbar$ is reduced Planck constant and $\Psi(x,y,t)$ is the wave function. We set $\hbar / 2m = 1$ and $L=1$.

	The solution $\Psi(x,y;t)$ must therefore satisfy		The solution $\Psi(x,y,t)$ must therefore satisfy
	\[		\[
	\nabla^2 \Psi = -i{\partial \over \partial t}\Psi \quad \text{in} \quad \Omega,		\nabla^2 \Psi = -i{\partial \over \partial t}\Psi \quad \text{in} \quad \Omega,
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	\Psi = 0 \quad \text{on} \quad \partial \Omega,		\Psi = 0 \quad \text{on} \quad \partial \Omega,
	\]		\]
	where $\Omega=[0, 1]\times[0, 1]$ denotes the square box domain and $\partial \Omega$ is the boundary of the domain. In order to get the unique solution for the time propagation of the wave function the initial state $\Psi(x, y,\, 0)$ has to be selected. We choose		where $\Omega=[0, 1]\times[0, 1]$ denotes the square box domain and $\partial \Omega$ is the boundary of the domain. In order to get the unique solution for the time propagation of the wave function the initial state $\Psi(x, y,t= 0)$ has to be selected. We choose
	\[		\[
	\Psi(x,y;t=0) = \sin{(\pi x)} \sin{(\pi y)}.		\Psi(x,y,t=0) = \sin{(\pi x)} \sin{(\pi y)}.
	\]		\]

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	</syntaxhighlight>		</syntaxhighlight>

	The animation of the real part of solution $\Re \Psi(x,y~~; t~~)$ is presented below.		The animation of the real part of solution $\Re \Psi(x,y)$ is presented below.

	[[File:video (1).mp4\|400px]]		[[File:video (1).mp4\|400px]]

	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

APirnat at 09:39, 17 June 2019

2019-06-17T09:39:21Z

← Older revision		Revision as of 09:39, 17 June 2019
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	</syntaxhighlight>		</syntaxhighlight>

	At each time step the matrix equation is solved using an Eigen linear equation solver, in this case, the BICGStab algorithm.		At each time step the matrix equation is solved using an Eigen linear equation solver, in this case, the BICGStab algorithm, and $rhs$ is updated.

	<syntaxhighlight lang="c++" line>		<syntaxhighlight lang="c++" line>