deletions | additions       diff --git a/Schroedinger equation.tex b/Schroedinger equation.tex  index ed99b8d..34fc20c 100644  --- a/Schroedinger equation.tex  +++ b/Schroedinger equation.tex 
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In one dimensional systems the Hamiltonian for $N$ electrons has the following form  \begin{equation}  \label{eq:1dham}  \hat{H}=\sum_{j=1}^N \left(-\frac{d^2}{dx_j^2}+v_{ext}(x_j)\right)+\sum_{j\lt \left(-\frac{d^2}{dx_j^2}+v_{\rm ext}(x_j)\right)+\sum_{j\lt  k}^N v_{int}(x_j, v_{\rm int}(x_j,  x_k), \end{equation}  where the interaction potential $v_{int}(x_j, $v_{\rm int}(x_j,  x_k)$ is usually Coulombic, though the following discussion also applies for other types of interaction, including more than two-body ones. In 1D one often uses the soft-Coulomb interaction $1 / \sqrt{(x_j-x_k)^2+1}$, where a softening parameter (usually set to one) is introduced in order to avoid the divergence at $x_j=x_k$, which is non-integrable in 1D. Mathematically, the Hamiltonian (\ref{eq:1dham}) is equivalent to that of a single (and hence truly independent) electron in $N$ dimensions, with the $N$-dimensional external potential   \begin{equation}  v_{ext}^{Nd}(x_1...x_N)=\sum_{j=1}^N v_{\rm ext}^{Nd}(x_1...x_N)=\sum_{j=1}^N  v_{ext}(x_j)+\sum_{j\lt k}^N v_{int}(x_j, v_{\rm int}(x_j,  x_k). \end{equation}  For small $N$ one can numerically solve the $N$-dimensional Schr\"odinger equation  \begin{equation}