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Matthieu Verstraete edited Schroedinger equation.tex
over 9 years ago
Commit id: 480c330a45e962b2473cd3caaeecbc44a91f3f9f
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diff --git a/Schroedinger equation.tex b/Schroedinger equation.tex
index bf6f1bd..da19921 100644
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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
($1 $1 /
\sqrt{(x_j-x_k)^2+1}$), \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}