2.4 Pöschl-Teller-Ginocchio Potential
39
depend on the value of the integer N = 2n + + 1, where n is its radial quantum
number (see Exercise IX. Note that this exercise is rather difficult and can be omitted
during first reading).
Exercise IX
Demonstrate Eqs. (2.114), (2.115), and (2.116).
The nature of the state depends on the following boundary values:
B Λ = Λ
1 − a
Λ 2 (1 − a) − 2
ν +
1
2
−
1
2
, Λ >
2
1 − a
(2.117)
= +∞ , Λ ≤
2
1 − a
(2.118)
B ν = [ν] , ν ∈ R, ν ∈ N
(2.119)
B ν = ν − 1 , ν ∈ N
(2.120)
Below, we will show that bound states occur when N ≤ B Λ and N ≤ B ν . Antibound
states and complex states of negative real energy occur for N ≤ B Λ and N > B ν
(N > ν if ν ∈ N). Resonances of positive energy and width are found for N > B Λ .
The ± sign in Eqs. (2.114) and (2.115) is “+” for bound states and antibound states
of physical importance, that is, whose linear momentum k is close to the real k-axis,
and states of negative real energy and width. It is “−” for resonance, other antibound
states, and virtual states.
One will first consider the case Δ < 0 in Eq. (2.116). Using Eq. (2.117) and
assuming that Λ 2 (1 − a) > 2, the real and imaginary parts of Eq. (2.115) read:
(e) =
¯
h 2 s 2
2m 0 (1 − a) 2
Λ
2 (1 − a) − 2
(B Λ + N + 1)(N − B Λ )
(2.121)
(e) = ±
¯
h 2 s 2
2m 0 (1 − a) 2
(2N + 1)
√
−Δ .
(2.122)
Hence, (e) > 0 occurs only if Λ 2 (1 − a) > 2 and N > B Λ , and (e) < 0
can happen only in the class of states denoted by the “−” sign in Eqs. (2.114)
and (2.115). Thus one obtains resonances of positive energy and width if N > B Λ
and Λ 2 (1 − a) > 2. The class of states denoted by “+” sign contains capturing
resonances, which are a complex conjugate of the physical decaying resonances.
All cases different from physical resonances clearly bear N ≤ B Λ .
The only possibility for k to lie on the positive imaginary axis is to have Δ >
(N + 1/2)
2 while using the “+” sign, as can be seen from Eq. (2.114). This directly
translates in the N < ν equation, where N = ν is suppressed for ν ∈ N as it
provides the value k = 0 with which the Pöschl-Teller-Ginocchio wave function
39
depend on the value of the integer N = 2n + + 1, where n is its radial quantum
number (see Exercise IX. Note that this exercise is rather difficult and can be omitted
during first reading).
Exercise IX
Demonstrate Eqs. (2.114), (2.115), and (2.116).
The nature of the state depends on the following boundary values:
B Λ = Λ
1 − a
Λ 2 (1 − a) − 2
ν +
1
2
−
1
2
, Λ >
2
1 − a
(2.117)
= +∞ , Λ ≤
2
1 − a
(2.118)
B ν = [ν] , ν ∈ R, ν ∈ N
(2.119)
B ν = ν − 1 , ν ∈ N
(2.120)
Below, we will show that bound states occur when N ≤ B Λ and N ≤ B ν . Antibound
states and complex states of negative real energy occur for N ≤ B Λ and N > B ν
(N > ν if ν ∈ N). Resonances of positive energy and width are found for N > B Λ .
The ± sign in Eqs. (2.114) and (2.115) is “+” for bound states and antibound states
of physical importance, that is, whose linear momentum k is close to the real k-axis,
and states of negative real energy and width. It is “−” for resonance, other antibound
states, and virtual states.
One will first consider the case Δ < 0 in Eq. (2.116). Using Eq. (2.117) and
assuming that Λ 2 (1 − a) > 2, the real and imaginary parts of Eq. (2.115) read:
(e) =
¯
h 2 s 2
2m 0 (1 − a) 2
Λ
2 (1 − a) − 2
(B Λ + N + 1)(N − B Λ )
(2.121)
(e) = ±
¯
h 2 s 2
2m 0 (1 − a) 2
(2N + 1)
√
−Δ .
(2.122)
Hence, (e) > 0 occurs only if Λ 2 (1 − a) > 2 and N > B Λ , and (e) < 0
can happen only in the class of states denoted by the “−” sign in Eqs. (2.114)
and (2.115). Thus one obtains resonances of positive energy and width if N > B Λ
and Λ 2 (1 − a) > 2. The class of states denoted by “+” sign contains capturing
resonances, which are a complex conjugate of the physical decaying resonances.
All cases different from physical resonances clearly bear N ≤ B Λ .
The only possibility for k to lie on the positive imaginary axis is to have Δ >
(N + 1/2)
2 while using the “+” sign, as can be seen from Eq. (2.114). This directly
translates in the N < ν equation, where N = ν is suppressed for ν ∈ N as it
provides the value k = 0 with which the Pöschl-Teller-Ginocchio wave function