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Conformable Schr¨odinger equation in D-dimensional space

q  
β
r = x 2β + x 2β + · · · + x 2β + x 2β 2
1 2 N−1 N 2β 2β β(D s − 1) β β 2β (D s − 2) β
∇ = ∂ + ∂ +  ∂ + ∂ 
D r r β r r 2β θ 1 θ β θ 1
tan 1
β β
θ x 1  
1 , 2
β = arccos q 2β 2β 2β 2β β 2β (D s − 3) β
x + x + · · · + x + x +  ∂ + ∂ 
1 2 N−1 N 2 θ β θ 2 θ β θ 2
r 2β sin 1 tan 2
β β β
θ x 2  
2 , 2
β = arccos q 2β 2β 2β β 1 2β (D s − 4) β
x + · · · + x + x +  ∂ + ∂  + . . .
2 N−1 N r 2β 2 θ β 2 θ β θ 3 θ β θ 3
sin 1 sin 2 tan 3
β β β β
θ x 3
3 , β 2 2β
β = arccos q 2β 2β 2β + β β β ∂ θ N−1
x + · · · + x + x 2 θ 2 θ 2 θ
3 N−1 N r 2β sin 1 sin 2 . . . sin N−2
β β β
· · · 
β D s − N β 
θ + ∂
N−2 x N−2 θ β θ N−1 
= arccos q , tan N−1
β 2β 2β 2β β
x + x + x
N−2 N−1 N (6)
β
θ
N−1 x N−1
= arccos q .
β 2β 2β
x + x
N−1 N
(3)
3. Conformable Schr¨odinger equation
Considering the scalar Laplacian operator in N- in N-dimensional
Dimensional as suggested in Ref, 48 and utilizing
the conformable gradient and conformable scalar In this section, our objective is to derive
Laplacian operator in 3-D as employed in. 31,34 We the Conformable Schr¨odinger Equation in N-
can express the Conformable scalar Laplacian op- Dimensional space. The time-dependent Con-
erator in N-Dimensional as formable Schr¨odinger equation in 3-D is given
by 32
N α i − 1
X
β
∇ 2β = ∂ 2β + ∂ . (4)
D x i β x i
x !
i=1 i 2β
β ℏ β β β
− ∇ 2β + V β (ˆx β ) Ψ β (x i , t) = iℏ D Ψ β (x i , t).
Let us analyze the conformable Schr¨odinger equa- 2m β β t
tion for any N, denoted by eq(4).
(7)
In a stationary state 32 when
α 1 − 1
2β 2β β 2β β
∇ ψ x i ) = ∂ + β ∂ + ∂ − i E β t
D ( x 1 β x 1 x 2 β β
β
x Ψ β (x i , t) = ψ β (x i )T β (t ) = ψ β (x i )e ℏ β .
1
α 2 − 1 β 2β So,
+ β β ∂ x 2 + · · · + ∂ x N
x
2
!
!
α N − 1 β 2m β
β
+ β β ∂ x N ψ x i ) ∇ 2β − (V β (ˆx β ) − E ) ψ β (x i ) = 0. (8)
(
x ℏ 2β
N β
Choosing α N as the single parameter for the non-
This equation represents the time-independent
integer dimensions with α 1 = α 2 = · · · = α N−1 = 32
1, D s = α N + (N − 1). In this case, equation (10) Conformable Schr¨odinger equation in 3-D. To
can be expressed as form generalize of conformable Schr¨odinger equation to
N-Dimensions, we obtain

2β 2β 2β 2β
∇ ψ x i ) = ∂ + ∂ + · · · + ∂ (5)
D ( x 1 x 2 x N !
! 2β 2m β β
α N − 1 β ∇ D − (V β (ˆx β ) − E ) ψ β (x i ) = 0. (9)
+ β β ∂ x N ψ x i ) ℏ 2β
(
x β
N
After performing the calculations, we obtain After substituting eq.(6), we have
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