deletions | additions       diff --git a/Schroedinger equation.tex b/Schroedinger equation.tex  index 7ce6cb1..d8dd6b0 100644  --- a/Schroedinger equation.tex  +++ b/Schroedinger equation.tex 
...
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_{\stackrel{\scriptstyle j,k=1}{j\neq k}}^N\frac{1}{\sqrt{(x_j-x_k)^2+1}}, k}}^N v_{int}(x_j, x_k),  \end{equation}  where theusual Coulomb  interaction has been 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, interaction ($ \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 in one dimension. non-integrable.  Mathematically, the Hamiltonian (\ref{eq:1dham}) is equivalent to the one that  of a single electron in $N$ dimensions 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\frac{1}{\sqrt{(x_j-x_k)^2+1}}. k}}^N v_{int}(x_j, x_k)}.  \end{equation}  For small $N$ one can numerically solve the $N$-dimensional Schr\"odinger equation  \begin{equation}