SOLUTION OF THE RADIAL N-DIMENSIONAL SCHRÖDINGER
EQUATION USING HOMOTOPY PERTURBATION METHOD
SAMI AL-JABER
Department of Physics, an-Najah National University, Nablus, P.O.Box 7, Palestine.
E-mail: [email protected]
Received June 11, 2012
Homotopy Perturbation Method (HPM) is applied to formulate analytic solution of the
free particle radial dependent Schrödinger equation in N-dimensional space. This
method is based on the construction of a homotopy with an embedding parameter
δ ∈ 0, 1 . The method shows its effectiveness, usefulness, and simplicity for obtaining
approximate analytic solution. In addition, some interesting cases are analyzed and the
effect of space dimension on the solution is pointed out.
Key words: Homotopy perturbation method, higher dimensions, approximation methods.
1. INTRODUCTION
The search for analytic exact or approximate solutions of different kinds of
differential and integral equations has been of great importance over the years.
Recently, a new analytic technique based on basic ideas of homotopy, called
Homotopy Perturbation Method (HPM), was developed by He [1–4] and has been
of great potential in solving different kinds of differential and integral equations in
many areas of applied sciences and engineering. The method is a coupling between
the traditional perturbation method and homotopy, which is a highly interesting and
useful concept in topology, and deforms continuously to a simple problem which is
easily solved. The HPM requires the construction of a homotopy with an
embedding parameter p ∈ [0, 1] which is considered as a small parameter whose
range relates the initial solutions of a problem to its final solutions. If p is 0, the
problem reduces to a sufficiently simplified form, which normally admits a rather
simple solution. As p gradually increases to 1, the problem goes through a
sequence of deformation. The final state of deformation is achieved when p is 1
and the desired solution is then obtained. The method is considered as a summation
of an infinite series which usually converges rapidly to the exact solution. The HPM
Rom. Journ. Phys., Vol. 58, Nos. 3–4, P. 247–259, Bucharest, 2013
248
Sami Al-Jaber
2
has been variously developed in different areas of applied sciences: It has been
applied to different types of nonlinear problems [5-12], to integral equations [1316], and to nonlinear differential equations with fractional time derivatives [1721]. The HPM has also been used to find solutions to nonlinear Schrödinger
equation [22-24] and to some kind of boundary value problems [25, 26].
Furthermore, the HPM was employed to find solutions to heat transfer problems
and heat distribution for systems with variable thermal conductivity [27-30]. Over
the last two decades, problems in the N-dimensional space are becoming
increasingly important in different areas: For example, in quantum field theory
[31], in quantum chemistry [32], in random walks [33], in Casimir effect [34], and
in harmonic oscillators [35, 36]. Furthermore, the N-dimensional radial
Schrödinger equation has been examined for different kind of potentials [37-39]. In
the present paper, the HPM is proposed to provide approximate solutions for the
free–particle radial part of Schrödinger equation in higher N–dimensional space.
The outline of this paper is as follows: Section 2 gives details of He's homotopy
perturbation method and shows how it can be applied to the N–dimensional
Schrödinger equation. Section 3 presents details of application of HPM to our
problem and the results obtained. Section 4 is devoted for conclusions.
2. HOMOTOPY PERTURBATION METHOD
To illustrate the basic ideas of the HPM, we consider the following nonlinear
differential equation:
A(u ) − F (r ) = 0, r ∈ Ω
(1)
with boundary conditions:
B (u ,
du
) = 0, r ∈ Γ
dn
(2)
Where A is a general differential operator, F (r ) is a known analytic function, B is
a boundary operator and Γ is the boundary of the domain Ω . The operator A can
be divided into two parts: Linear operator L and nonlinear operator M so that Eq.
(1) becomes
L(u ) + M (u ) − F (r ) = 0.
(3)
Following He [1, 2], we construct a homotopy: V (r , p ) : Ω × [0, 1] → R which
satisfies
H (v, p ) = (1 − p ) [ L(v) − L(u0 )] + p [ A(v) − F ( r ) ] ,
or
p ∈ [0, 1], r ∈ Ω
(4)
3
Solution of the radial N-dimensional Schrödinger equation
249
H (v, p ) = L(v) − L(u 0 ) + p [ N (v) − F ( r ) ] = 0, p ∈ [0, 1], r ∈ Ω
(5)
Where p ∈ [0, 1] is an embedding parameter, u 0 is an initial approximation of Eq.
(1) which satisfies the boundary conditions. Eq.'s (4) and (5) give respectively;
H (v, 0) = L(v) − L(u0 ) = 0
(6)
H (v, 1) = A(v) − F (r ) = 0
(7)
The changing values of p from zero to unity is just that of V (r , p ) from u0 (r ) to
u (r ) . In topology, this is called deformation and L(v) − L(u0 ), A(v) − F (r ) are
called homotopic. Considering the embedding parameter p, as a small parameter,
and applying the traditional perturbation technique, we can assume that the solution
of Eq.(4) or (5) can be expanded as a power series in p; namely
v = v0 + pv1 + p 2 v 2 + p 3 v3 + p 4 v 4 + .......
(8)
Setting p = 0 yields the initial approximation u0 (r ) , while p = 1 gives u (r ) as
u = lim p →1 (u ) = v0 + v1 + v2 + v3 + v4 + ......
(9)
3. FREE-PARTICLE N-DIMENSIONAL RADIAL SCHRÖDINGER
EQUATION
The Laplacian operator in the N-dimensional spherical coordinates
(r , θ1 , θ 2 , ..., θ N − 2 , ϕ) has the form
∇ 2 = r 1− N
∂ N −1 ∂
1
(r
) + 2 Λ2 ,
∂r
∂r
r
(10)
where Λ 2 is a partial differential operator on the unit sphere S N −1 given by [40]
−2
 k


  N −2
∂ 
N − k −1 ∂
Λ = ∑  ∏ sin θ j  (sin θ k ) k +1− N
sin
θ

 +  ∏ sin θ j 
k
∂θ k 
∂θ k   j =1
k =1  j =1


N −2
2
−2
∂
(11)
∂ϕ2
The separation of variables method separates the free-particle N-dimensional
Schrödinger equation into two second-order differential equations [41]; namely
Λ 2 Y + βY = 0
d 2 R ( N − 1) dR  2 β
+
+ k − 2
dr 2
r
dr 
r
(12)

 R = 0,

(13)
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Sami Al-Jaber
4
where k 2 = 2mE / h 2 and β is a separation constant whose values are given by [40]
β = l ( l + N − 2) .
(14)
The solutions to Eq.(12) are the hyper-spherical harmonics, Yl{ } of degree l
on the unit sphere S N −1 . For each non-negative integer l , the number of hyperspherical harmonics is given by [41]
m
nl =
(2l + N − 2)(l + N − 3)!
,
l !( N − 2)!
(15)
and are characterized by the integers m1 , m2 , m3 , ......, m N − 2 with the restrictions
l ≥ m1 ≥ m2 ≥ m3 ≥ ...... ≥ m N −2 ≥ 0.
(16)
The hyperspherical harmonics form an orthonormal set and thus they form a
standard basis of the irreducible representations of the rotation group SO( N ) in
the space of square integrable functions defined over the surface of the
N-dimensional unit sphere with the invariant measure [40]
N −2
dΩ = ∏ ( sin θ j )
j =1
N −1− j
dθ j dϕ .
(17)
In order to find the solution to the radial part given in Eq.(13), we consider
the differential equation
d 2 Y  1 − 2 a  dY 
(a 2 − p 2 c 2 ) 
c −1 2
+
+
bcx
+
(
)


 Y = 0,
dx 2  x  dx 
x2

(18)
whose solution is given by [38]
Y ( x) = x a  AJ p (bx c ) + BN p (bx c )  ,
(19)
where J p and N p are the ordinary Bessel and Neumann functions, respectively,
and a, b, c and p are constants. Comparing Eq.(13) with Eq.(18) yields b = k ,
c = 1, a = (2 − N ) / 2, and p = n + ( N − 2) / 2 . Therefore, with the use of Eq. (19),
the solution to Eq. (13) can be immediately written down:
R (r ) = r (2 − N ) / 2  A J n + ( N − 2) / 2 (kr ) + B N n + ( N − 2) / 2 (kr )  ,
(20)
where A and B are constants. The function R (r ) must be finite at r = 0 and thus
the second term in Eq. (20) must be rejected due to the singular behavior of the
Neumann function at the origin. Therefore, the solution in Eq. (20) reduces to
5
Solution of the radial N-dimensional Schrödinger equation
251
R (r ) = A r (2 − N ) / 2 J l + ( N − 2) / 2 (kr ) ,
(21)
which can be written in terms of the spherical Bessel function as
R(r ) = A r ( 3− N ) / 2 j l + ( N −3) / 2 (kr ) .
(22)
The common representation of j m (kr ) is given by
∞
jm (kr ) = ∑ Am
p =0
(−1) p 2 m k 2 p (m + p )! m + 2 p
,
r
p ! (2m + 2 p + 1)!
(23)
and thus Eq. (22) gives the series form of the radial solution R (r ) , namely
∞
R (r ) = ∑ AlN
p =0
(−1) p 2 l + ( N −3) / 2 k 2 p (l + p + ( N − 3) / 2)! l + 2 p
.
r
p ! (2l + 2 p + N − 2)!
(24)
4. SOLUTION OF THE RADIAL PART OF THE FREE-PARTICLE'S
SCHRÖDINGER EQUATION IN N DIMENSIONS USING HPM
In this section, the HPM is implemented, in an efficient way, to find
approximate solutions for the radial part of Schrödinger equation in N-dimensional
space subject to the condition that R (r ) is finite at r = 0 . Writing the Eq.(13) in
the form
r N −1
d2 R
dR
+ ( N − 1)r N − 2
+  k 2 r N −1 − l(l + N − 2)r N −3  R = 0,
2
dr
dr 
(25)
and in view of Eq.(4) or (5), the homotopy for Eq.(25) can be constructed as
H ( R, p ) = r N −1
d 2 R ( N − 1) dR  pk 2 l(l + N − 2) 
+ 2− N
+
−
 R = 0,
dr 2
dr  r 1− N
r
r 3− N

(26)
with p ∈ [0, 1] . The basic assumption of the HPM is that the solution R (r ) can be
expressed as a power series in p, namely
R (r ) = ∑ p n Rn (r ) = R0 (r ) + pR1 (r ) + p 2 R2 (r ) + p 3 R3 (r ) + .....
(27)
n=0
Considering terms up to third power in the parameter p and substituting
Eq.(27) into Eq.(26) yields
252
Sami Al-Jaber
6
r N −1  R0″ + pR1″ + p 2 R2″ + p 3 R3″  +


( N − 1) r N − 2  R0′ + pR1′ + p 2 R2′ + p 3 R3′  +


2
l(l + N − 2) 
 pk
2
3
 r 1− N −
  R0 + pR1 + p R2 + p R3  = 0.
3− N
r


(28)
Summing up the coefficients of equal power of p and setting each sum to
zero, gives the following equations;
p 0 : r N −1 R0″ + ( N − 1)r N − 2 R0′ − l(l + N − 2)r N − 3 R0 = 0
(29)
p1 : r N −1 R1″ + ( N − 1)r N − 2 R1′ − l(l + N − 2)r N −3 R1 = − k 2 r N −1 R0
(30)
p 2 : r N −1 R2″ + ( N − 1)r N − 2 R2′ − l(l + N − 2)r N −3 R2 = − k 2 r N −1 R1
(31)
p 3 : r N −1 R3″ + ( N − 1)r N − 2 R3′ − l(l + N − 2)r N − 3 R3 = − k 2 r N −1 R2
(32)
p n : r N −1 Rn ″ + ( N − 1)r N − 2 Rn′ − l(l + N − 2)r N − 3 Rn = − k 2 r N −1 Rn −1
(33)
where Rn (0) = Rn′ (0) for n = 1, 2, 3, .....
In order to solve Equations (29-30), a series method can be used. This gives
R0 (r ) = C0 r l +
C
r
1
l + N −3
,
(34)
Where C0 and C1 are constants. Since R (0) is finite, then the constant C1 must be
rejected and therefore Eq.(34) becomes
R0 (r ) = C0 r l .
(35)
Substituting R0 (r ) into Eq.(30) and again applying the series method, we get
R1 (r ) =
− k 2 C0
r l+2 .
2(2l + N )
(36)
Similarly, the substitution of R1 (r ) into Eq. (31) and solving the resulting
equation gives
R2 (r ) =
k 4 C0
r l+4 .
8(2l + N )(2l + N + 2)
(37)
A further substitution of R2 (r ) into Eq.(32) and solving the resulting
equation by series method yields
7
Solution of the radial N-dimensional Schrödinger equation
R3 (r ) =
253
− k 6 C0
r l + 6 . (38)
2 3 3!(2l + N )(2l + N + 2)(2l + N + 4)
In general the n th term of the solution is given by
Rn (r ) =
C0 (−1) n k 2 n (2l + N − 2)! (l + n + ( N − 3) / 2)! l + 2 n
r
.
(l + ( N − 3) / 2)! n ! (2l + N − 2 + 2n)!
(39)
The above equation can be proved by mathematical induction as follows:
For n = 1 , Eq.(39) reduces to our result given in Eq.(36). Employing Eq.(33) for
n + 1 and using Eq.(39), we get
1
r
1− N
Rn′′+1 +
N −1
l(l + N − 2)
RN′ +1 −
RN +1 = −k 2 f (n)r N + l + 2 n −1 ,
2− N
r
r 3− N
(40)
where f (n) is the coefficient of Rn (r ) given by Eq.(39). Assuming a series
solution of the form
Rn +1 (r ) = ∑ C j r j ,
(41)
j
which upon its substitution into Eq.(40) gives
∑ C [ j ( j − 1) + j ( N − 1) − l(l + N − 2)] r
N + j −3
j
= −k 2 f ( n)r N + l + 2 n −1 .
(42)
j
Equating powers of r on both sides of Eq.(42) gives
j = l + 2n + 2 ,
(43)
and thus
− k 2 f (n) = C j [ (l + 2n + 2)(l + 2n + 1) + ( N − 1)(l + 2n + 2) − l(l + N − 2)]
= C j [ (l + 2n + 2)(l + 2n + N ) + 2l − l(l + N )]
= C j [ (l + 2n + 2)(l + N ) + 2n (l + 2n + 2) + 2l − l(l + N )]
(44)
= C j [ (l + N )(2n + 2) + 2n (2n + 2) + 2l (n + 1)]
= 2 C j [ (n + 1)(2l + 2n + N ]
which gives
Cj =
− k 2 f (n)
.
2(n + 1)(2l + 2n + N )
The substitution for f (n) by the coefficient of Rn (r ) of Eq. (39) yields
(45)
254
Sami Al-Jaber
Cj =
8
C0 (−1) n +1 k 2( n +1) (2l + N − 2)! (l + n + ( N − 3) / 2)!
.
2 (l + ( N − 3) / 2)!( n + 1) n !(2l + 2n + N )(2l + 2n + N − 2)!
(46)
Using the factorial property ( p + 1)! = ( p + 1) p ! , we can write
(l + n + ( N − 1) / 2)!
(l + n + ( N − 1) / 2)
(2l + 2n + N )!
(2l + 2n + N − 2)! (2l + 2n + N ) =
(2l + 2n + N − 1)
(l + n + ( N − 3) / 2)! =
(47)
The substitution of Eq.(47) into Eq.(46) and performing simple algebra, we get
Cj =
C0 (−1) n +1 k 2( n +1) (2l + N − 2)! (l + n + ( N − 1) / 2)!
,
(l + ( N − 3) / 2)! (n + 1)! (2l + 2n + N )!
(48)
and thus, with the help of Eq.(43), Eq.(41) immediately yields
Rn +1 (r ) =
C0 (−1) n +1 k 2( n +1) (2l + N − 2)! (l + n + ( N − 1) / 2)! l + 2 n + 2
r
.
(l + ( N − 3) / 2)! (n + 1)! (2l + 2n + N )!
(49)
It is clear to note that letting n → n + 1 in Eq.(39) yields exactly the result in
Eq.(49) and thus we proved our assertion in Eq.(39) by mathematical induction.
In order to compare the solution predicted by the HPM, given by Eq.(39),
with the exact solution given by Eq. (24), we let m ≡ l + ( N − 3) / 2 , so that
(2l + N − 2)! (2m + 1)!
=
= 2 m (2m + 1)!!
N −3
m!
(l +
)!
2
(50)
The last step can be proved by mathematical induction as follows: It is trivial
for m = 1 , for m → m + 1 , we get
(2m + 3)! (2m + 3)(2m + 2)(2m + 1)!
,
=
(m + 1)!
(m + 1)m !
which, upon using Eq.(50), we get
(2m + 3)!
= 2 m +1 (2m + 3)!! , QED
(m + 1)!
Therefore, upon comparing Eq.(24) with Eq.(39),
C0 2 l + ( N −3) / 2 (2l + N − 2)!! = AlN 2 l + ( N − 3) / 2 , and thus we have
AlN = C0 (2l + N − 2)!!
(51)
we
get
(52)
9
Solution of the radial N-dimensional Schrödinger equation
255
Therefore, the HPM solution yields the exact solution by choosing the
arbitrary constant C0 to satisfy Eq.(52).
It is worth to check the convergence of the solution just obtained by the
HPM. One can check this by calculating the ratio Rn +1 / Rn , which can be computed
from Eq.’s (39) and (49) with the result
Rn +1
− k 2 (l + n + 1 + ( N − 3) / 2)
=
r2 .
Rn
(n + 1)(2l + N + 2n)(2l + N + 2n − 1)
Using the identity ( p + 1)! = ( p + 1) p ! , the above ratio becomes
Rn +1
−k 2
=
r2 .
Rn
2(n + 1)(2l + N + 2n)
(53)
It is noted that, for a given r , l, N , this ratio decreases as 1/ n 2 .
5. SOME INTERESTING CASES
In this section, we analyze some interesting special cases that shed some light
on the HPM solution.
5.1. THE CASE l = 0
In this case, Eq.(39) yields
Rn =
C0 (−1) n k 2 n ( N − 2)!(n + ( N − 3) / 2))! 2 n
r ,
(( N − 3) / 2)! n !(2n + N − 2)!
(54)
which immediately gives the first few terms, namely
R0 = C0
R1 =
R2 =
R3 =
−C0 k 2 2
r
2N
C0 k 4
r4
2 2 2! N ( N + 2)
−C0 k 6
r6
23 3! N ( N + 2)( N + 4)
It is interesting to write R up to the third term, namely
(55)
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Sami Al-Jaber
10
R = R0 + R1 + R2
The extremes of R can be found by setting dR / dr = 0 to get
rmin =
1
2( N + 2) ,
k
(56)
and one can easily check that d 2 R / dr 2 > 0 so that R exhibits minimum at the
above value of r. The value of this minimum is
Rmin =
C0 ( N − 2)
2N
(57)
It is obvious that the minimum of R increases with the space dimension N. It starts
at C0 / 6 in the three-dimensional space and C0 / 2 in the infinite-dimensional
space. It is tempting to consider the large N limit where one may deduce from Eq.
(55) that
Rn =
C 0 (−1) n k 2 n 2 n
r ,
2 n n! N n
(58)
so that the solution could be written as
n
 −k 2 r 2  1
 −k 2 r 2 
R = C0 ∑ 
= C0 exp 

,
 2N  n!
 2N 
(59)
which exhibits no extreme values.
5.2. THE CASE l = 1
Using Eq.(39), we get
R0 = C0 r
R1 =
R2 =
R3 =
−C0 k 2 3
r
2( N + 2)
C0 k 4
r5
2 2 2!( N + 2)( N + 4)
−C0 k 6
r7
23 3!( N + 2)( N + 4)( N + 6)
For the large N limit we have,
(60)
11
Solution of the radial N-dimensional Schrödinger equation
Rn =
257
C0 (−1) n k 2 n 2 n +1
r ,
2n n ! N n
and thus the solution becomes
n
∞
 −k 2 r 2  1
R = C0 r ∑ 
= C0 r exp ( −k 2 r 2 / 2 N )

 n!
n=0  2 N
(61)
The extremes of the above solution can be found by setting dR / dr = 0 to get
rmax =
N
k
(62)
It is a simple matter to check that d 2 R / dr 2 is negative at r = rmax and thus R
exhibits a maximum value, given by
Rm = C0
N −1 / 2
e
k
(63)
It is clear that as the space dimension increases, the position of the maximum
and the value of the maximum shift towards higher values and in the infinite
dimensional space ( N → ∞ ), both go to infinity.
6. CONCLUSION
In this work, the solution of the radial N-dimensional Schrödinger equation
was obtained and analyzed using HPM. The obtained results show the effectiveness
and usefulness of the homotopy perturbation method. It has been demonstrated that
the solution obtained by HPM yields the exact solution by a proper choice of the
arbitrary constant as seen in Eq.(52). For computational purposes, two special cases
had been considered: For the case l = 0 , we included the first three terms in the
solution and it was found that the solution exhibits a minimum value of
C0 ( N − 2) / 2 N at rmax = 2( N + 2) / k . For the case l = 1 , our results show that,
for the large N limit, the series solution predicted by HPM has a closed form. This
closed form has a maximum value of C0 N / ke at rmax = N / k and both are
increasing with the increase of the space dimension, N. Furthermore, in the infinitedimensional space, for this second case, the maximum value becomes infinite at
infinity. Therefore, this work demonstrates the powerful of the HPM and illustrates
the effect of the space dimension on the solution.
258
Sami Al-Jaber
12
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