Thank,s for the hint with double naming. That was the problem
It is irrelevant to use ctc1[t] or ConjugateTranspose[c1][t].
When I solve and Plot the System with this code :
eoMkitaevchaindiracnormal = {
-I c1'[t] == -\[Omega] (c2[t] + ConjugateTranspose[c2][t]) +
Exp[-((x1 - v*t)^2/(2 \[Sigma]^2))]/
2 c1[t], -I ConjugateTranspose[c1]'[
t] == \[Omega] (c2[t] + ConjugateTranspose[c2][t]) +
Exp[-((x1 - v*t)^2/(2 \[Sigma]^2))]/
2 ConjugateTranspose[c1][t], -I c2'[
t] == -\[Omega] (c1[t] - ConjugateTranspose[c1][t]) +
Exp[-((x2 - v*t)^2/(2 \[Sigma]^2))]/
2 c2[t], -I ConjugateTranspose[c2]'[
t] == \[Omega] (-c1[t] + ConjugateTranspose[c1][t]) +
Exp[-((x2 - v*t)^2/(2 \[Sigma]^2))]/2 ConjugateTranspose[c2][t],
c1[-100] == 1, c2[-100] == 1, ConjugateTranspose[c1][-100] == 1,
ConjugateTranspose[c2][-100] == 1};
solutionkitaevchaindiracnormal =
NDSolve[eoMkitaevchaindiracnormal /. {v0 -> 1,
v -> 1, \[Sigma] -> 0.75, x1 -> 1, x2 -> 2, \[Omega] -> 1}, {c1[
t], c2[t], ConjugateTranspose[c1][t],
ConjugateTranspose[c2][t]}, {t, -100, 100}]
ParametricPlot[{Re[#], Im[#]} &@
Evaluate[{ConjugateTranspose[c1][t]*c1[t]} /.
solutionkitaevchaindiracnormal], {t, -10, 10}, PlotRange -> All,
PlotStyle -> {Blue,Thick}]
I become this solution for
$c_1^\dagger c_1$
But
$c_1^\dagger c_1$ is a density Operator.
I think that the imaginary part of
$c_1^\dagger c_1$ should be 0 and
ConjugateTranspose[c1] and ctc1
is not
$c_1^\dagger$. But what is the correct description of
$c_1^\dagger$?
