Diploma thesis

2 Non-Hermitian dynamics
The Hamilton operator does not depend on time so we can directly integrate (2.1) and obtain the
separation
Ψ(x, t) = exp
(− i )
tH ψ(x). (2.6)
Let ψ(x) be an eigenstate of H to the eigenvalue E so we can write
Now we can derive the time evolution of the density ρ(x, t):
Ψ(x, t) = exp
(− i )
Et ψ(x). (2.7)
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ρ(x, t) = exp
E I(t − t 0 ) ρ(x, t 0 ), (2.8)
where ρ(x, t 0 ) = ψ ∗ (x)ψ(x) is the density at a fixed time t = t 0 . One can see immediately that
this non-stationarity is a direct consequence of the complexity of the potential since H is not Hermitian,
which is followed by complex energy eigenvalues E = E R + iE I and so the time evolution
operator exp ( − i tH) is not unitary any more.
Inserting the separation (2.7) into (2.1) provides that ψ has to fulfill the time independent
Schrödinger
H(x)ψ(x) = Eψ(x), (2.9)
which we will solve in the next two chapters for two different approximations of V (x).
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