Page 81 - IJOCTA-15-1

Conformable Schr¨odinger equation in D-dimensional space

h (D s − (N − 1))
2β β And J −βq (ρ) it is called the second solution of the
∂ + ∂ (17)
θ N−2 θ β θ N−2 conformable Bessel equation of order βq
tan N−2
β
 2s−q
)
J −βq (ρ) = ρ β(1− D s P ∞ (−1) s ρ β .
2
 Θ
+ λ N−3 − λ N−2  β(N−2) = 0. s=0 s!Γ(s−q+1) 2
β
θ
sin 2 N−2 (25)
β ρ β
β
after substituting r = K , we have
. . .
(D s − N) ∞ s β 2s+q
2β β X (−1) r K
∂ Θ β(N−1) + ∂ Θ β(N−1) J βq (rK) = . (26)
θ N−1 θ β θ N−1 s!Γ(s + q + 1) 2
tan N−1 s=0
β
+ λ N−2 Θ β(N−1) = 0. (18) ∞ s β 2s−q
X (−1) r K
J −βq (rK) = .(27)
s!Γ(s − q + 1) 2
s=0
We will now solve these equations separately for
β
β
R β (r ) andΘ iβ (θ ) The solution of the radial part is represented as:
2β
β
2
∂ r R β + β(D s−1) β 2 K − λ r R β = 0 R β = (r K) (1− D s ) [AJ βq (rK) + BJ −βq (rK)] .(28)
∂ r R β + β
2
r β r 2β
(19)
The constant B, must equal zero since the
β
β
where K 2 = 2m β E . Let r β = ρ β → ∂ r = Schr¨odinger equation’s solution must be finite and
2β
ℏ β 2 K
β regular
2β
β
2 2β
K∂ ρ → ∂ r = K ∂ ρ . Thus, we have
β
R β = (r K) (1− D s ) (29)
2 AJ βq (rK).
h i
2β β(D s−1) β 2
∂ ρ R β + ∂ ρ R β + β 1 − λ r R β = 0.
ρ β ρ 2β The change of variable is employed to find the
(20) solution of the angular part eq.(18). Let
Then,
Θ β(N−1) = X β(N−1) ,
h i
2β 2β
β β
ρ ∂ R β + βaρ ∂ R β + β 2 ρ 2β − λ r R β = 0(21) θ β
ρ
ρ
cos N−1 = x β ,
N−1
β
where a = (D s −1) and λ r is called separation con- β
stant λ r = ℓ(ℓ+D s −3). 49 This equation is called ∂ β = − 1 sin θ N−1 ∂ ,
β
Conformable Bessel’s differential equation. 50 ac- θ N−1 β β x 1
cording to the solution for this equation in, 50 we θ β
2β 1 2 N−1 2β
have ∂ = sin ∂ ,
θ N−1 β 2 β x N−1
(1 − x 2β )
2β
N−1

1−a
R β = ρ β 2 AJ βq(ρ) + BJ −βq (ρ) , (22) ∂ θ N−1 = β 2 ∂ 2β .
x N−1
√ Thus, the eq.(18) becomes
2
where q = a +1−2a+4λ r . After substituting
2
a = (D s − 1) and λ r = ℓ(ℓ + D s − 3), we obtain 2β 2β
(1 − x )∂ X
N−1 x N−1 β(N−1)
β ∂ β X
− β(D s − N)x
D s
R β = ρ β(1− 2 ) [AJ βq (ρ) + BJ −βq (ρ)] . (23) N−1 θ N−1 β(N−1)
2
+ β λ N−2 X β(N−1) = 0. (30)
q
2
where q = D s − D s + 1 + ℓ(ℓ + D s − 3).
4 When comparing this equation with the Con-
J βq (ρ) it is called the first solution of the con- 51
formable Bessel equation of order βq formable Gegenbauer equation provided in Ref
β
2β
β
β 2s+q (1 − x )D D T γ N−1 (x) (31)
x
s
)
J βq (ρ) = ρ β(1− D s P ∞ (−1) ρ . x βm N−3
2
s=0 s!Γ(s+q+1) 2 β β γ N−1
− 2β(γ N−1 + 1)x D T (x)
(24) x βm N−3
2
+ β m N−3 (m N−3 + 2γ N−1 + 1)T γ N−1 (x) = 0.
βm N−3
75