deletions | additions       diff --git a/Schroedinger equation.tex b/Schroedinger equation.tex  index 1594fbc..c9821ea 100644  --- a/Schroedinger equation.tex  +++ b/Schroedinger equation.tex 
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\label{eq:SENd}  \hat{H}\Psi_j(x_1...x_N)=E_j\Psi_j(x_1...x_N)  \end{equation}  which provides a spatial wave function for a single electron in $N$ dimensions. This leaves the problem of constructing a wave function which satisfies the antisymmetry properties of equivalence is not restricted to one-dimensional problems. One can generally map  a wave function problem  of $N$ electrons in one dimension. Especially, one needs to ensure that those spatial wave functions $\Psi_j$ which cannot be part $d$ dimensions onto the problem  of a properly antisymmetric wave function are removed as allowed solutions for single electron in $Nd$ dimensions. In order to keep our notation relatively simple, we will continue to discuss  the $N$-electron problem. case of an originally one-dimensional problem with $N$ electrons.  Solving Eq.\ (\ref{eq:SENd}) leaves the problem of constructing a wave function which satisfies the antisymmetry properties of a wave function of $N$ electrons in one dimension. Especially, one needs to ensure that those spatial wave functions $\Psi_j$ which cannot be part of a properly antisymmetric wave function are removed as allowed solutions for the $N$-electron problem.  A graphical representation of which wave functions are allowed are the so-called Young diagrams, where each electron is asigned a box, and those boxes are then arranged in columns and rows. Each box is labeled with a number from 1 to $N$ such that the numbers increase from top to bottom and left to right. All possible Young diagrams for three and four electrons are shown in Fig.\ \ref{fig:young}. Since there are two different spin directions for the electrons our Young diagrams for allowed wave functions contain maximally two columns. Therefore, the diagram d) is not allowed for three particles with spin $1/2$ and diagrams k) to n) are not allowed for four particles. To connect a given wave function $\Psi_j$ with a diagram one has to symmetrize the wave function according to the diagram. For example, for diagram b) one would perform the following operations \begin{equation}  \Psi(x_1,x_2,x_3)+\Psi(x_2,x_1,x_3)-\Psi(x_3,x_2,x_1)-\Psi(x_3,x_1,x_2).  \end{equation}