Diploma thesis

4 Complex harmonic potential
oscillator length l = λ −1/2
1 and the energy scale by the ground-state energy 1 Ω of a real harmonic
2
potential to make the calculations and the results more clearly:
ε := E Ω
2
, c := C Ω
, χ := 1 l x, ω := 1 w and κ := l k. (4.19)
l
2
Furthermore we introduce the following abbreviation:
θ = θ(c, ω) =
From this we can directly calculate λ and γ and thus finally obtain
a 1 = 1 4 (1 − ε) and a 2 = 1 4
Using these new variables the general solutions now read
√
1 + i c
ω 2 . (4.20)
(
1 − ε + ic
θ
)
. (4.21)
⎧
⎪⎨
ψ s (χ) =
A s 1e − 1 2 χ2 1F 1
(
a1 ; 1 2 ; χ2) + A a 1χe − 1 2 χ2 1F 1
(
a1 + 1 2 ; 3 2 ; χ2)
A s 2e − 1 2 θχ2 1F 1
(
a2 ; 1 2 ; θχ2)
, χ < −ω
, −ω < χ < ω
, (4.22)
⎪⎩
A s 1e − 1 2 χ2 1F 1
(
a1 ; 1 2 ; χ2) − A a 1χe − 1 2 χ2 1F 1
(
a1 + 1 2 ; 3 2 ; χ2)
, ω < χ
⎧
⎪⎨
ψ a (χ) =
A s 1e − 1 2 χ2 1F 1
(
a1 ; 1 2 ; χ2) + A a 1χe − 1 2 χ2 1F 1
(
a1 + 1 2 ; 3 2 ; χ2)
A a 2θχe − 1 2 θχ2 1F 1
(
a2 + 1 2 ; 3 2 ; θχ2)
, χ < −ω
, −ω < χ < ω
. (4.23)
⎪⎩
−A s 1e − 1 2 χ2 1F 1
(
a1 ; 1 2 ; χ2) + A a 1χe − 1 2 χ2 1F 1
(
a1 + 1 2 ; 3 2 ; χ2)
, ω < χ
Now we demand continuity of the wave function and its first derivative at χ = ±ω. With the
symmetry conditions (4.14) it is sufficient to evaluate this at χ = −ω and to ensure that ψ 1 (χ)
vanishes for χ → −∞, because then the same is automatically fulfilled for ψ 3 at χ = +ω and
χ → ∞. We start with the normalizability:
lim ψ 1(χ) =
χ→−∞
[ (
lim
1
χ→−∞ e− 2 χ2 A s 1 1F 1 a 1 ; 1 )
(
2 ; χ2 + A a 1χ 1 F 1 a 1 + 1 2 ; 3 )]
2 ; χ2
!
= 0. (4.24)
If the confluent hypergeometric functions (4.10) reduce to polynomials, this would be automatically
fulfilled because of the exponential function. In the real case this allows us immediately
to formulate the quantization condition a 1 = −n for some n ∈ N. Unfortunately this does not
work here since a is, in general, a complex number so that the Pochammer symbol (a 1 ) ν does not
vanish even for any n ∈ R and 1 F 1
(
a1 ; 1 2 ; χ2) does not become a polynomial. Therefore we have
to demand that the sum of two confluent hypergeometric functions vanishes for χ → −∞:
lim
[A s
χ→−∞
1 1F 1
(a 1 ; 1 )
2 ; χ2 + A a 1χ 1 F 1
(a 1 + 1 2 ; 3 )]
2 ; χ2 !
= 0. (4.25)
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