deletions | additions       diff --git a/Schroedinger equation.tex b/Schroedinger equation.tex  index d8dd6b0..12897d8 100644  --- a/Schroedinger equation.tex  +++ b/Schroedinger equation.tex 
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\label{eq:1dham}  \hat{H}=\sum_{j=1}^N \left(-\frac{d^2}{dx_j^2}+v_{ext}(x_j)\right)+\sum_{\stackrel{\scriptstyle j,k=1}{j\neq k}}^N v_{int}(x_j, x_k),  \end{equation}  where the interaction potential is usually Coulombic, though for the following any type (including more than 2 body) will do. In 1D $v_{int}$ is often replaced by the soft-Coulomb interaction ($ \frac{1}{\sqrt{(x_j-x_k)^2+1}$), ($\frac{1}{\sqrt{(x_j-x_k)^2+1}$),  the last term in Eq.\ (\ref{eq:1dham}), in order to avoid the divergence at $x_j=x_k$ which is non-integrable. Mathematically, the Hamiltonian (\ref{eq:1dham}) is equivalent to that of a single electron in $N$ dimensions, with the $N$-dimensional external potential \begin{equation}  v_{ext}^{Nd}(x_1...x_N)=\sum_{j=1}^N v_{ext}(x_j)+\sum_{\stackrel{\scriptstyle j,k=1}{j\neq k}}^N v_{int}(x_j, x_k)}.  \end{equation}