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Minimize the Gross-Pitaevskii functional and demonstrate the effect of choice of function space of the Gradient on convergence. More...
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Typedefs | |
typedef double | RealT |
Functions | |
int | main (int argc, char **argv) |
Minimize the Gross-Pitaevskii functional and demonstrate the effect of choice of function space of the Gradient on convergence.
Minimize the one-dimensional Gross-Pitaevskii (GP) energy functional
\[ J[\psi] = \int \frac{1}{2} |\nabla\psi|^2 + V(x)|\psi|^2 +g|\psi|^4 \,\mathrm{d}x \]
Subject to the equality constraint that the particle density be normalized.
\[ e(\psi) = \int |\psi|^2\,\mathrm{d}x - 1 = 0 \]
For simplicity, we will assume the wavefunction \(\psi\) to be real-valued, the potential function \( V(x)\geq 0\), the computational domain is the interval \([0,1]\), and that \(\psi(0)=\psi(1)=0\). We also discretize the problem using second-order centered finite differences on a uniform grid.
\[ \psi''(x_i) \approx = \frac{\psi(x_{i-1})-2\psi(x_i)+\psi(x_{i+1})}{\Delta x^2} \]
Definition in file gross-pitaevskii/example_01.cpp.
typedef double RealT |
Definition at line 74 of file gross-pitaevskii/example_01.cpp.
int main | ( | int | argc, |
char ** | argv ) |
Definition at line 76 of file gross-pitaevskii/example_01.cpp.