Diploma thesis

3 Complex square well potential
The most important information these plots give us is the fact that the imaginary part of all
states is always negative for 0 < c < ∞. This consolidates the assumption that the model of an
imaginary potential describes dissipation since for the time evolution of the density (2.8) it yields
a damping effect.
Furthermore, the results of the real limit c → 0 are confirmed in Figs. 3.2(a) – 3.2(l) since the
real part of the energy starts at some squared natural number ε R (c = 0) = m 2 and the imaginary
part exactly at zero. Thus these solutions include the states of the real square well potential.
It immediately catches the eye that we can generally differ between two kinds of solutions which
are characterized by their imaginary part. For the states of the one type it is lim c→∞ ε I = 0 and
for the other type a deeper consideration yields a linear decay lim c→∞ ε I = lim c→∞ (−c) = −∞.
These limits correspond exactly to the imaginary part of the energies we obtained by evaluating
ω → 0 and ω → π, which is ε 2 I,0 = 0 and ε I,∞ = −c in dimensionless variables. So while we
counted all states for c = 0 by m, now for c → ∞ states with lim c→∞ ε I = 0 are counted by k so
that we call them k-states ε k 0, and states with lim c→∞ ε I = −∞ are counted by n so that we call
them n-states ε n ∞.
Observing all 12 pictures one can see that for small waists there are only k-states among the lowest
six states. For some ω < 0.1 the first n-state enters these lowest states namely the state starting
at m = 5. In the course of increasing ω more and more states become n-states, so for some big
waist ω > 1.1 even the k = 1-state is dropped out. Then we are left with 6 n-states among the
lowest 6 energy levels so that for ω = π we have reached the constellation that we calculated in
2
Section 2 for w = L. Thus both limits are connected by a continuous rearrangement of k- and
n-states among the lowest states.
Now let us take a look at the situations where the waist is very close to 0 or π which yields that
2
area 2 or area 1 and 3 approximately vanish, respectively.
3.2.1 Regime of vanishing areas
We start with comparing Fig. 3.2(a) and Fig. 3.2(l) which represent a very small waist ω 0 and
a very large waist ω π , respectively. From our former discussion we expect that approximately
2
the same constellation is represented as we calculated for the limits ω = 0 and ω = π, that 2
means ε k 0 ≈ k 2 = m 2 and ε n ∞ ≈ n 2 − ic = m 2 − ic. This means that in both cases the real part
is approximately unaffected of the dissipation c so for every strength of dissipation it should be
nearly equal to m 2 . The imaginary part of the k-states for ω 0 is nearly equal to zero for all
values of c and the imaginary part of the n-states for ω π approximately decreases with −c to
2
−∞, so ε n I,∞ + c is nearly equal to zero for all values of c.
Let us compare this with the respective images and start with the n-states. Fig. 3.2(l), where
ω = 1.57 ≈ π, shows that the real part is nearly constant and equal to 2 n2 = m 2 so the real part
is correct. To prove for the imaginary part ε n I,∞ ≈ −c it seems to be more comfortable to have a
look at ε n I,∞ + c and to show that it tends to be equal to zero for all c and ω → π:
2
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