Number of functions in a linear superposition

I was interested how many wave functions are essential in the sum that gives the final wave function, formula (22) of the paper of B. S. and E. J. H. It is interesting that numerical calculations show that only two terms in the sum are essential (give more 99% of contribution) for examples shown on figures 2, 3, and 4, and 4 terms are essentials for figures 1, 5, 6, and it is valid even for n=100. So, that examples are not "wave packets", but sums of a few pure states. Results of that numerical calculations are on figures in Internet file.

I plot on figures weights c_j=<j,n-j|Psi_i>^2 vs r_j=(j+1)/n*100%. Part of energy going to x-mode (R_x) is simply a sum of c_j*r_j over j. I found that weights c_j are essentially non-zero only for a few j, between 2 and 4, for examples from the paper. However, for my own examples, "worst" and "the best", see the last two pages, all weights are of the same order, and all terms contribute to the sum. On my figures, I marked by a vertical bar the percentage of energy going to x-mode, R_x. It is typically close to the maximum of weights c_j, except for the example of "worst" accuracy.

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