Diploma thesis

3 Complex square well potential
that this changes for intermediate waists c ≈ c crit from m to k or n the states for large dissipation
c ≫ c crit , the states are characterized only by the integer number k or n, which is not clearly
ascribable to a particular m as we just stated.
Here always two ε k I,0-states are fusing for c → ∞ so we denoted them with the same k. These
pairs consist on a k-state starting at some odd m and one starting at one adjoining even m ± 1 so
a symmetric and an antisymmetric state. Moreover we know from Fig. 3.2(a) that it is the state
starting at some odd m which approaches to the state started at some even m. The indeterminacy
of the assignment m ↔ k if ω ≈ ω k,n
crit does not matter since anyhow, if the considered ω is
approximately equal to some critical waist, both possible k-states starting at m or m + 2 yield the
same parity which is opposite to this of the m + 1-state.
It is interesting that all pairs of such fusing k-states have this property that they start at adjoining
values of m for all ω. This is related to the fact that the n-states are ordered by the real part of
the energy and never intersect each other, that means that the real part of states with higher n
run at higher energies than states with smaller n for all c. We already know that the energy, these
real parts run at, decreases for increasing ω so that interchanges with the upcoming real parts of
the k-states occur but the order of the n-states does not change. So after a k-state starting at
some odd m pairing with the k-state starting at m + 1, which is even, has interchanged with some
symmetric n-state, it then starts at m + 2, so the property of pairs m and m ± 1 is still fulfilled.
Because of the order of the n-states the next downcoming n-state is an antisymmetric one, which
interchanges with the other k-state which starts at m + 3 after that and the property is once more
fulfilled. This procedure repeats until all k-states populating the potential well for ω = 0 have
been redistributed to much more higher energies for ω → π until exclusively n-states are present
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when this limit is reached.
Next we still consider the limit c → ∞. For the imaginary part the limits lim c→∞ ε k I,0 = 0 and
lim c→∞ ε n I,∞ = lim c→∞ (−c) = −∞ seem to be accurate. For the real parts we assume that they
are converging to some finite real number which seems to be plausible since on the one hand it is
implied by the numerical evaluation and on the other hand we know that they are converging for
ω = 0 and ω = π since they are constant in these cases. These should be continuously reached
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limit cases so the real part should converge for all waists.
So let us make the ansatz lim c→∞ ε k I,0 = 0 and lim c→∞ ε n I,∞ = −c and insert this into the quantization
conditions (3.26) and (3.27) to derive expressions for symmetric and antisymmetric k- and
n-states from these assumptions. Performing this and summing up the results of both (3.26) and
(3.27) provides indeed analytical results:
( ) 2
ε k,sat π
R,0 = k
π
− ω , ε n,sat
2
R,∞ =
(
n π ) 2
. (3.28)
2ω
The results of these equations coincide perfectly with the numerical solutions for large c. They
confirm that for ω = 0 there are only k-states present, since then ε n R,∞ diverges, and for ω = π 2
the k-states are vanishing since then the energy they run at goes to infinity.
They also include the fact that the symmetric k-states tend to the antisymmetric ones as we can
see in Fig 3.2(a), because
lim
ω→0 εk,sat R,0 = (2k) 2 , (3.29)
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