deletions | additions       diff --git a/Schroedinger equation.tex b/Schroedinger equation.tex  index 9278072..5a594f6 100644  --- a/Schroedinger equation.tex  +++ b/Schroedinger equation.tex 
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The time dependent propagation of the Schr\"odinger equation can be carried out in the same spirit, since the Hamiltonian is given explicitly and each ``single particle orbital'' represents a full state of the system. A laser or electric field perturbation can also be applied, depending on the charge of each particle (given in the input), and taking care to apply the same effective field to each particle along the polarization direction of the field (in 1D the diagonal of the hyper-cube).   Solving Eq.\ (\ref{eq:SENd}) leaves the problem of constructing a wave function which satisfies the antisymmetry properties of $N$ electrons in one dimension. In particular, for fermions, one needs to ensure that those spatial wave functions $\Psi_j$ which are not the spatial 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 is given by  theclassical  Young diagrams (or tableaux)  for permutation symmetries, where each electron is asigned a box, and those the  boxes are then arranged stacked  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. %% NB: this is basic quantum mechanics, possibly not the place to add it here. I recommend we chop it out and add a reference to some QM textbook, even if we feel that it is not simple or well explained anywhere. A summary of the operation and the output of octopus would be sufficient.  All possible decorated  Young diagrams for three and four electrons are shown in Fig.\ \ref{fig:young}. Since there are two different spin states for electrons, our Young diagrams for the  allowed spatial  wave functions contain at most two columns. The diagram d) is not allowed for thespatial  wave function of 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). \left[ \Psi(x_1,x_2,x_3)+\Psi(x_2,x_1,x_3)\right]- \left[\Psi(x_3,x_2,x_1)+\Psi(x_3,x_1,x_2)\right].  \end{equation}  Hence, one symmetrizes with respect to an interchange of the first two variables and antisymmetrizes with respect to the first and third variable. We note that we are referring to the position of the variable in the list list,  not their index the index,  and that symmetrization always comes before antisymmetrization. At the end of thesesymmetrization  operations one calculates the norm of the resulting wave function and if function. If  it passes a certain threshold, by default set to  $10^{-5}$, one keeps the obtained function as a proper fermionic spatial part. If the norm is below the threshold, one continues with the next allowed diagram until either a norm larger than the threshold is found or all diagrams are used up. If a solution $\Psi_j$ does not yield a norm above the threshold for any diagram it is removed since it corresponds to a wave funtion with only bosonic or other non-fermionic character. Generally, as the number of forbidden diagrams increases with $N$, the number of wave functions that need to be removed also increases quickly  with $N$. $N$, in particular in the lowest part of the spectrum.  The case of two electrons is specific specific,  as all solutions of Eq.\ (\ref{eq:SENd}) correspond to allowed fermionic wave functions: the symmetric ones to the singlet states and the antisymmetric ones to the triplet states.