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. 


= r cos θj’1 sin θj sin θj+1 . . . . . . sin θD’1 
xj 



. 

. 
. 



xD’1 = r cos θD’1 sin θD’1 



xD = r cos θD’1


© 2001 by Chapman & Hall/CRC
where
D = 3, 4, 5 . . .
0 < r<∞
(A4)
0 ¤ θ1 < 2π
0 ¤ θj ¤ π, j = 2, 3, . . . D ’ 1
The Laplacian ∇2 can be written as
D

1 D’1 ‚ h‚
∇2 = (A5)
D
h2 ‚θi
h i=0 ‚θi i

where
D’1
θ0 = r, h = hi (A6)
i=0
and the scale factors hi are given by
D 2
‚xk
h2 = i = 0, 1, 2, . . . , D ’ 1 (A7)
i
‚θi
k=1

Explicitly
2 2 2
‚x1 ‚x2 ‚xD
h2 = + + ... + =1
0
‚θ0 ‚θ0 ‚θ0
2 2
‚x1 ‚x2
h2 = r2 sin2 θ2 sin2 θ3 . . . sin2 φD’1
= +
1
‚θ1 ‚θ1
2 2 2
‚x1 ‚x2 ‚x3
h2 = + +
2
‚θ2 ‚θ2 ‚θ2

= r2 sin2 θ3 sin2 θ4 . . . sin2 φD’1
.
.
.
h2 = r2 sin2 θj+1 sin2 θj+2 . . . sin2 θD’1
j
.
.
.
h2 2
D’1 = r
(A8)
Thus h is
h = h0 h1 . . . hD’1
(A9)
2 3 D’2
rD’1 sin θ
= 2 sin θ3 sin θ4 . . . sin θD’1


© 2001 by Chapman & Hall/CRC
From (A5), the ¬rst term of ∇2 is
D



1‚ h‚
=
h ‚θ0 h2 ‚θ0
0
1 ‚ D’1 ‚
sin θ2 . . . sinD’2 θD’1
= r
rD’1 sin θ2 sin2 θ3 . . . sinD’2 θD’1 ‚r ‚r

1‚ D’1 ‚
= r .
rD’1 ‚r ‚r
(A10)
The last term of ∇2 is
D



1‚ h ‚
=
h ‚θD’1 h2D’1 ‚θD’1
1 ‚
=
rD’1 sin θ2 sin2 θ3 . . . sinD’2 θD’1 ‚θD’1
(A11)
rD’1 sin θ2 ... sinD’2 θD’1 ‚
r2 ‚θD’1

1 ‚ ‚
sinD’2 θD’1
=
r2 sinD’2 θD’1 ‚θD’1 ‚θD’1


Other terms of ∇2 are of the forms
D



1‚ h‚
=
h ‚θj h2 ‚θj
j

1 ‚
=
rD’1 sin θ2 . . . sinj’1 θj sinj θj+1 . . . sinD’2 θD’1 ‚θj

rD’1 sin θ2 . . . sinj’1 θj . . . sinD’2 θD’1 ‚
r2 sin2 θj+1 . . . sin2 θD’1 ‚θj

1 1 ‚ ‚
sinj’1 θj
=
r2 sin2 θj+1 . . . sin2 θD’1 sinj’1 θj ‚θj ‚θj
(A12)


© 2001 by Chapman & Hall/CRC
Using (A10)-(A12), we get from (A5) the representation
D’2
1 ‚ D’1 ‚ 1 1
∇2 = r +2
D
sin2 θj+1 . . . sin2 θD’1
rD’1 ‚r ‚r r j=1


1 ‚ ‚ (A13)
sinj’1 θj
sinj’1 θj ‚θj ‚θj

1 1 ‚ ‚
sinD’2 θD’1
+
sinD’2 θD’1 ‚θD’1
r2 ‚θD’1

We note also that the Laplacian ∇2 obeys the relation
D

L2
1
‚ D’1 ‚
’ D’1
∇2 = r (A14)
D
r2
rD’1 ‚r ‚r
with
L2 = Lij Lij , i = 1, 2, . . . j ’ 1
n
(A15)
i,j
j = 2, . . . D
and the angular momentum components Lij are de¬ned as the skew
symmetric tensors
Lij = ’Lji
= xi pj ’ xj pi , i = 1, 2, . . . j ’ 1 (A16)
j = 2, . . . D
To prove (A14), we ¬rst note that we can express pk as
D’1
‚ ‚θr ‚
pk = ’i¯
h = ’i¯
h
‚xk ‚xk ‚θr
r=0 (A17)
D’1
1 ‚xk ‚
= ’i¯
h
h2 ‚θr ‚θr
r
r=0

where we have used the relations
D’1
‚xl ‚xl
= δir h2 ,
i
‚xi ‚θr
l=0
(A18)
D’1
‚θi ‚xl
= δ kl
‚xk ‚θi
l=0



© 2001 by Chapman & Hall/CRC
We next note that the following commutation relation holds (see
Appendix B)

[Lij , Lkl ] = i¯ δjl Lik + i¯ δik Ljl ’ i¯ δjk Lil ’ i¯ δil Ljk
h h h h (A19)

Further if we set
·
L2 = Lij Lij , i = 1, 2, . . . j ’ 1
k
(A20)
i,j
j = 2, 3, . . . k + 1

we can obtain [see Appendix B]

‚2 

L2 
=’ 2 
1 
‚θ1 








L2
1 ‚ ‚ 

1
L2 =’ sin θ2 ’ 

2 
2
sin θ2 ‚θ2 ‚θ2 sin θ2 


.
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