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Solution of Coulomb system in momentum space
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Annals of Physics 323 (2008) 317–336 www.elsevier.com/locate/aop
Solution of Coulomb system in momentum space De-Hone Lin
*
Department of Physics, National Sun Yat-sen University, Kaohsiung, Taiwan Received 5 February 2007; accepted 5 April 2007 Available online 12 April 2007
Abstract The solution of D-dimensional Coulomb system is solved in momentum space by path integral. From which the topological effect of a magnetic flux in the system is given. It is revealed that the flux effect represented by the two-dimensional field of Aharonov–Bohm covers any space-dimensions. 2007 Elsevier Inc. All rights reserved. PACS: 03.65.Ge; 03.65Db; 03.65Vf Keywords: Solution of H-atom in momentum representation; Topological effect in the Coulomb system; Coulomb Green’s function
1. Introduction The momentum space representation of Coulomb system is most convenient to reveal its symmetry and degeneracy [1]. It was pointed out that by Hamilton [2] the dynamics of the three-dimensional Coulomb system in momentum representation is equivalent to that of a free particle on the hyperspherical surface in four-dimensional space. The result directly reveals the SO(4) symmetry of the Coulomb system and can be generalized to the system with 1/r potential in D dimensions that exhibits the SO(D + 1) symmetry. Modern discussions about the Coulomb system in momentum representation can be dated back to 1929 in [3] by Podolsky and Pauling who performed the Fourier transformation for the eigenstate of Coulomb system in position space to obtain its momentum representation. Nevertheless, the solution by Schro¨dinger equation in momentum space was still lacking. *
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0003-4916/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2007.04.006
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
It was Fock in 1935 [4] who solved the problem in momentum space. Fock wrote the Schro¨dinger equation in three-dimensional momentum space, and then projected the momentum space onto the surface of a unit sphere in four space. A further transformation of eigenstate leaded to the Hamiltonian for the hydrogen atom was invariant under rotations, in four space, of this S3-sphere around its center and obtained the momentum eigenstate which had been obtained by Podolsky and Pauling. As the global approach of quantum mechanics, path integral approach for the Coulomb system in momentum space is formulated by Kleinert until fairly recently in Ref. [5]. From which he obtains the Coulomb Green’s function in momentum space [6] and its energy spectrum. The corresponding eigenstate and its generalization to include the topological effect are extracted and discussed in [7]. In this paper, we shall generalize Kleinert’s results in two directions: first to the Coulomb system in momentum space in arbitrary dimensions and second to include the topological effect of a magnetic flux. In Section 2 we generalize Kleinert’s results to the D-dimensional Coulomb system in momentum space. The energy spectrum and eigenstate in D dimensions are presented. In Section 3 the topological effect of a magnetic flux in the charged particle of Coulomb system in D dimensions is given in momentum representation. Conclusions are summarized in Section 4. 2. Solution of the D-dimensional Coulomb system in momentum space The momentum Green’s function for a point particle moving in a potential barrier in D space can be represented as the following path integral [8,9]: Z 1 Z Z D Z D p Gðpb ; pa ; EÞ ¼ DD x expfA½p; x; f g; dS Df U½f ð1Þ 2p 0 where the action A½p; x; f ¼
Z
S
dkfip_ x þ f ½Hðp; xÞ Eg
ð2Þ
0
in which H is the classical Hamiltonian, f(k) is an arbitrary dimensionless fluctuating scale variable, and U[f(k)] is some convenient gauge-fixing functional. The only condition on U[f(k)] is that Z Df ðkÞU½f ðkÞ ¼ 1: ð3Þ Eq. (1) reduces to that of Feynman’s original formulation [10] if U[f(k)] is taken as the delta functional d[f 1]. For the Coulomb system under consideration, H ¼ p2 =2 a=r, where natural units with h = c = M = 1 are used. The path integral can be performed if the gauge fixing functional U[f] is chosen as [5] ( 2 2 ) N N Y X 1 n 2 pn E U½f fn xn : ð4Þ exp rn 2r2n 2 n¼0 n¼0 With which the gauge condition measure is given by
R
Df U½f ¼ 1 is automatically satisfied when the sliced
D.-H. Lin / Annals of Physics 323 (2008) 317–336
Z
Df
"Z N Y n¼0
1
1
# dfn pffiffiffiffiffiffiffiffiffiffiffiffi ; 2p=n
319
ð5Þ
where the approximation ‘‘’’ becomes equality ‘‘=’’ while limNfi1. Inserting the representation of U[f] into (1), we have the action # 2 Z S " 1 2 p2 f2 fa A½p; x; f ¼ E þ 2 dk ip_ x þ x : ð6Þ 2 r 2 2r 0 The path integrations over f and x now are Gaussian and can be performed. The former gives us the factor ( ) N X n a2 exp ; ð7Þ 2 n¼0 and the latter associated with x variable has the result ( ) D=2 2 N N Y 2D ð2p=n Þ 1X ð2Dpn Þ exp : 2 2 D 2 n¼0 n ðp2n þ j2 Þ2 n¼0 ðpn þ j Þ
ð8Þ
Here we have defined (2E) = j2. The representation of Green’s function in Eq. (1) now reduces to that of the path integral in D-dimensional momentum space as follows: Z 1 D 2D ð2pÞ dS Gðpb ; pa ; EÞ D=2 D ð2pa Þ ðp2a þ j2 Þ 0 "Z # N 1 Y 2D d D p n ð9Þ exp ANE : D=2 2 D ðpn þ j2 Þ 1 ð2pn Þ n¼1 The sliced action reads " # N X 1 ð2Dpn Þ2 n a2 N AE ¼ : 2 n ðp2n þ j2 Þ2 2 n¼0
ð10Þ
The sliced integrations can be simplified by defining a stereographic projection onto a unit sphere in (D + 1) space as follows: z¼
2jp ; 2 p þ j2
z¼
p 2 j2 ; p 2 þ j2
z2 þ z2 ¼ 1:
Then the action in Eq. (10) becomes " # N X 1 ðD^zn Þ2 n a2 N AE ½^z ¼ 2 n j2 2 n¼0 and the measure in Eq. (9) turns into "Z # Z 1 D N 1 Y ð2pÞ dDþ1^zn dS ; ð2pj2 a ÞD=2 n¼1 1 ð2pj2 n ÞD=2 0
ð11Þ
ð12Þ
ð13Þ
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
where ^z ¼ (z, z) is the (D + 1)-dimensional unit vector. With the help of the definition 1/j2 = l, the momentum path integral becomes the standard representation on D sphere in (D + 1) space [11, Ch. 8] Z 1 Gðpb ; pa ; EÞ ¼ ð2pÞD dS Kðpb ; pa ; SÞ; ð14Þ 0
where the pseudo-propagator Kðpb ; pa ; SÞ
"Z N Y
1 ð2pa =lÞ
D=2 n¼1
1
1
dDþ1^zn ð2pn =lÞ
#
D=2
exp ANE ^zð1=j2 ! lÞ :
ð15Þ
The propagator is solvable and corresponds to a particle moving on a sphere in (D + 1)dimensional space. Since the stereographic projection is from the flat to the curved space, a measure correction described by the Riemannian scalar curvature R must be included that is given by (e.g. [11, p. 742]) Z S R Ameasure ¼ : ð16Þ dk 6l 0 Here R = D(D 1)/r2 for a D sphere of radius r. Thus the measure correction gives a contribution expfA measure g ¼ expfj2 DðD 1ÞS=6g to the propagator that leads to the accurate energy spectrum. With this correction, the pseudo-propagator in momentum space is found to be [9,11,12] 1 X 2J D1 þ ðD þ 1Þ 2 CððD þ 1Þ=2Þ ðDþ1 2 1Þ Kðpb ; pa ; SÞ ¼ C J D1 ðcos #Þ ðDþ1Þ=2 ðD þ 1Þ 2 2p ¼0 J D1
e f j
2 ½ðJ
2 2 D1 þ1ÞþðD3Þ=2 a
gS=2 ;
ð17Þ
where C an ðzÞ is the Gegenbauer polynomials [13, p. 218] and # is the angular between the (D + 1) vectors ^zb and ^za : cos # ¼ ^zb ^za ¼
ðp2b j2 Þðp2a j2 Þ þ 4j2 pb pa : ðp2b þ j2 Þðp2a þ j2 Þ
ð18Þ
By performing the integration on S, the momentum Green’s function of the Coulomb system in D space is found to be Gðpb ; pa ; EÞ ¼ ð2pÞD
1 X
2
2 2 J D1 ¼0 2E½ðJ D1 þ 1Þ þ ðD 3Þ=2 þ a 2J D1 þ D 1 CððD þ 1Þ=2Þ ðDþ1 1Þ 2 C J D1 ðcos #Þ: D1 2pðDþ1Þ=2
ð19Þ
The energy spectrum are determined by the poles and given by EJ D1 ¼
a2 2½ðJ D1 þ 1Þ þ ðD 3Þ=22
;
that agrees with the result given in Ref. [14]. Particularly, when D = 3, we have
ð20Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336
321
1 X n2 2 P ðcos #Þ ; 2 2 n 2p 2En þ a2 n¼1
Gðpb ; pa ; EÞ ¼ ð2pÞ3
ð21Þ
where we have defined n = (JD1 + 1) and used the equalities C 1J D1 ðcos #Þ ¼ sinðJ D1 þ 1Þ#= sin #
ð22Þ
P n ðcos #Þ ¼ sin n#=n sin #:
ð23Þ
and
The momentum representation (21) was first given by Kleinert in [5]. Their poles display the correct energy levels of hydrogen atom En ¼
a2 ; 2n2
n ¼ 1; 2; 3; . . . :
ð24Þ ðDþ11Þ
2 To extracting the momentum eigenstate of Coulomb system, we note that C J D1 has the expansion [13, p. 223]
ðDþ11Þ
2 C J D1
ðDþ11Þ
2 ðcos #Þ ¼ C J D1
¼
ðuv þ ð1 u2 Þ
CðD 2Þ C2 ðD=2 1=2Þ
J D1 X
1=2
ð1 v2 Þ
1=2
ðcos #Þ
cos D#Þ
4J D2 ðD þ 2J D2 2Þ
J D2 ¼0
CðJ D1 J D2 þ 1ÞC2 ðJ D2 þ ðD 1Þ=2Þ ð1 u2 ÞJ D2 =2 CðD þ J D1 þ J D2 1Þ
ð1 v2 Þ
J D2 =2
J
þðD1Þ=2
D2 C J D1 J D2
J
þðD1Þ=2
D2 ðuÞC J D1 J D2
D=21
ðvÞC J D2 ðcos D#Þ; ð25Þ
where we have defined u¼
ðp2b j2 Þ ; ðp2b þ j2 Þ
v¼
ðp2a j2 Þ ; ðp2a þ j2 Þ
ð26Þ
^b , p^a in D space. Since the polynomial and D# is the angular between the unit vectors p D=21 C J D2 ðcos D#Þ has the expansion into D-dimensional ultra-spherical harmonics [11, p. 667] D=21
C J D2 ðcos D#Þ ¼
D2 2pD=2 X Y pb ÞY J D2 J^ ð^pa Þ; ^ ð^ 2J D2 þ D 2 CðD=2Þ ^ J D2 J
ð27Þ
J
^ ¼ ðJ D3 ; J D4 ; . . . ; J 0 Þ with the values Jk = 0, 1, 2, . . . (Jk P Jk1; k P 1) and for where J k = 1, J0 < 0 applicable, we obtain
322
D.-H. Lin / Annals of Physics 323 (2008) 317–336 J D1 X 1 X X
Gðpb ; pa ; EÞ ¼ ð2pÞD
2
2E½ðJ D1 þ 1Þ þ ðD 3Þ=22 þ a2 CðD 1Þ 2J D1 þ D 1 C Dþ1 2 D pffiffiffi 2 D1 ðD 1Þ p C 2 C 2 J D2 J =2 4 CðJ D1 J D2 þ 1ÞC2 ðJ D2 þ ðD 1Þ=2Þ 1 u2 D2 ð1 v2 ÞJ D2 =2 CðD þ J D1 þ J D2 1Þ J D1 ¼0 J D2 ¼0
þðD1Þ=2
J
D2 C J D1 J D2
^ J
J
þðD1Þ=2
D2 ðuÞC J D1 J D2
ðvÞY J D2 J^ ð^ pb ÞY J D2 J^ ð^ pa Þ:
ð28Þ
From which the spectrum representation of momentum Green’s function can be expressed as D
Gðpb ; pa ; EÞ ¼ ð2pÞ
1 X
J D1 X X
2
J D1 ¼0 J D2 ¼0
C E½ðJ D1 þ 1Þ þ ðD 3Þ=2 þ a2 =2
^ J WJ D1 ;J D2 ;J^ ðpb ÞWJ D1 ;J D2 ;J^ ðpa Þ
þ ;
ð29Þ
where C is an independent constant. The momentum eigenstate of Coulomb system in D dimensions is found to be " #1=2 ð2J D1 þ D 1Þð2jÞ3D4 WJ D1 ;J D2 ;J^ ðpÞ ¼ 2p J D2 1=2 4 CðJ D1 J D2 þ 1ÞC2 ðJ D2 þ ðD 1Þ=2Þ 1 2 CðD þ J D1 þ J D2 1Þ ðp þ j2 ÞðD1Þ J D2 2 2 2jp J D2 þðD1Þ=2 p j 2 C pÞ: ð30Þ Y ^ ð^ J J D1 D2 p þ j2 p2 þ j2 J D2 J The normalization condition Z dD pWJ D1 ;J D2 ;J^ ðpÞ WJ D1 ;J D2 ;J^ ðpÞ ¼ 1
ð31Þ
can be checked by using the recursion formula zC mn ðzÞ ¼
1 ½ðn þ 1ÞC mnþ1 ðzÞ þ ðn þ 2m 1ÞC mn1 ðzÞ; 2ðn þ mÞ
and the orthogonality relations of Gegenbauer polynomials [13, p. 378] Z 1 p212k Cðn þ 2kÞ k1=2 dx C kn ðxÞC km ðxÞð1 x2 Þ ¼ dm;n ; n!ðk þ nÞC2 ðkÞ 1
ð32Þ
ð33Þ
where the volume element in momentum space is given by ^ ðDÞ dD p ¼ pD1 dp dX ¼ pD1 dp sinD2 HD2 sink Hk sin H1 dHD2 dHk dH1 dU
ð34Þ
with the ranges of coordinates 0 6 p 6 1, 0 6 U 6 2p, and 0 6 Hk 6 p (k P 1). Particularly, when D = 3, the result in (30) reduces to that of
D.-H. Lin / Annals of Physics 323 (2008) 317–336
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 2j5 l 4pnCðn lÞ 2 Cðl þ 1Þ Wn;l;m ðpÞ ¼ Cðn þ l þ 1Þ p l 2 1 2jp p j2 lþ1 C nl1 2 Y lm ðH; UÞ: p þ j2 ðp2 þ j2 Þ2 p2 þ j2
323
ð35Þ
Here we have identified n = (J2 + 1) and l = J1 with n = 1, 2, 3, . . . and l = 0, 1, 2, . . . The eigenstate was first obtained by Podolsky and Pauling by carrying out the Fourier transformation on the eigenstate in position space in [3]. For D = 2, the Coulomb eigenstate is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2J 0 þ 1Þð2jÞ 4J 0 CðJ 1 J 0 þ 1ÞC2 ðJ 0 þ 1=2Þ WJ 1 ;J 0 ðpÞ ¼ CðJ 1 þ J 0 þ 1Þ 2p J 0 2 2 1 2jp J 0 þ1=2 p j 2 C J 1 J 0 ð36Þ Y J 0 ð^pÞ; ðp þ j2 Þ p2 þ j2 p 2 þ j2 pffiffiffiffiffiffi where Y J 0 ð^ pÞ ¼ expfiJ 0 Ug= 2p. It is of interest to evaluate the average values of the square of the momentum in the various quantum states: Z 1 Z p Z 2p 2 ^ ðDÞ W hp2 i ¼ pD1 dp dX ð37Þ ^ ðpÞp WJ ^ ðpÞ: J D1 ;J D2 ;J D1 ;J D2 ;J 0
0
0
With the help of (33), one find 2 a 2 2 hp i ¼ j ¼ ; n þ ðD 3Þ=2
n ¼ 1; 2; 3; . . .
ð38Þ
This quantity characterizes the circular Bohr orbit of Coulomb system in D space. The average of square momentum of the electron in 3 space becomes
a2 : ð39Þ hp2 i ¼ j2 ¼ n 3. Magnetic flux effect in the eigenstate of Coulomb system in D dimensions In this section we discuss the topological effect of an Aharonov–Bohm (AB) magnetic flux [15] in the momentum eigenstate of a charged particle moving in the Coulomb field in D dimensions. Introducing the polar coordinates (r, h0, h1, . . . , hD2) in D space, the Cartesian coordinates are related by 8 x1 ¼ r cos hD2 ; > > > > > > < x2 ¼ r sin hD2 cos hD3 ; ð40Þ > > > xD1 ¼ r sin hD2 sin hD3 sin h1 cos h0 ; > > > : xD ¼ r sin hD2 sin hD3 sin h1 sin h0 ; where 0 6 r < 1, 0 6 h0 6 2p, 0 6 hk 6 p, k P 1. The magnetic flux in D space located at the force center is defined by
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
AðxÞ ¼ 2g
xD^eD1 þ xD1^eD ; x2D þ x2D1
ð41Þ
where eˆD1,D stand for the unit vector along the D 1, D axis. The field generalizes the effect of a magnetic flux in 2 space to arbitrary D space. It defines an interaction of field with a charged particle in D space that just relates to the topological winding number around the polar axis in D space (See Appendix A). The topological effect is conveniently considered by a global nonintegrable phase factor (NPF) [16–18]. The NPF represents the interaction of a charged particle with the magnetic field by the phase modulation as follows: ( Z ) pb
Gflux ðpb ; pa ; EÞ ¼ Gðpb ; pa ; EÞ exp ie
AðpÞ dp ;
ð42Þ
pa
where Gflux is the new Green’s function that has included the effect of the magnetic flux defined by the vector potential A(p). For an AB magnetic flux, the A(p) is the same as A(x) and given by (see Appendix A) AðpÞ ¼ 2g
pD^eD1 þ pD1^eD : p2D þ p2D1
ð43Þ
Using the polar coordinates (p, U, H1, . . . , HD2) in momentum pace pD1 ¼ p sin HD2 sin HD3 sin H1 cos U; pD ¼ p sin HD2 sin HD3 sin H1 sin U
ð44Þ
1=2
with p ¼ ðp21 þ p22 þ þ p2D Þ and the ranges of coordinates 0 6 p < 1, 0 6 U 6 2p, 0 6 Hk 6 p, k P 1, one find Z S Z pb Z kb _ _ ie AðpÞ dp ¼ ie dk AðpÞ pðkÞ ¼ il0 d kUðkÞ; ð45Þ ka
pa
0
where U(k) = U(p(k)), U_ ¼ dU=dk, and l0 is the dimensionless number l0 = 2eg. Given two paths C1 and C2 in momentum space connecting pa and pb, the integral only differs by an integer multiple of 2p. The winding number k is thus given by Z S 1 _ k¼ dk UðkÞ: ð46Þ 2p 0 The flux effect is purely topological, its value is then given by ( Z ) pb exp ie AðpÞ dp ¼ expfil0 ð2kp þ Ub Ua Þg:
ð47Þ
pa
To consider the topological effect, it is very convenient to use another representation of the spherical harmonics in D space in (27). Let us first consider the D = 3 case. The wellknown representation of spherical harmonics gives l X k¼l
Y lk ð^ pb ÞY lk ð^pa Þ ¼
l X 2l þ 1 Cðl k þ 1Þ k P ðcos Hb ÞP kl ðcos Ha ÞeikðUb Ua Þ : 4p Cðl þ k þ 1Þ l k¼l
ð48Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336
325
Here (H, U) are the angular of the polar coordinates of the total momentum vector in D = 3 momentum space. With the help of the relation between the associated Legendre polynomial P kl ðxÞ and the Jacobi function P ðk;kÞ ðxÞ [19] n P kl ðcos HÞ ¼ ð1Þk
Cðk þ l þ 1Þ ðk;kÞ ðcos H=2 sin H=2Þk P lk ðcos HÞ; Cðl þ 1Þ
ð49Þ
the summation of product in (48) becomes l l X X 2l þ 1 Cðl k þ 1ÞCðl þ k þ 1Þ Y lk ð^ pb ÞY lk ð^ pa Þ ¼ 4p C2 ðl þ 1Þ k¼l k¼l k Hb Ha Hb Ha cos sin sin cos 2 2 2 2 ðk;kÞ
ðk;kÞ
P lk ðcos Hb ÞP lk ðcos Ha ÞeikðUb Ua Þ :
ð50Þ
Generalizing the procedure to higher dimensions we find that (27) in D space has the representation X Y J D2 J^ ð^pb ÞY J D2 J^ ð^pa Þ ^ J
¼
J D2 X J D3 ¼0
J1 X
Y J D2 ;...;J 1 ;J 0 ððHD2 Þb ; . . . ; ðH1 Þb ; Ub Þ
J 0 ¼J 1
Y J D2 ;...;J 1 ;J 0 ððHD2 Þa ; . . . ; ðH1 Þa ; Ua Þ ( ) J D3 J D2 J2 J1 D 2 Y X X X X 1 ð2J k þ kÞCðJ k J k1 þ 1ÞCðJ k þ J k1 þ kÞ ¼ ðD1Þ 2k1 C2 ðJ k þ ðk 1Þ=2 þ 1Þ p k¼1 J D3 ¼0 J D4 ¼0 J 1 ¼0 J 0 ¼J 1 2 ( J D 2 Y ðHk Þb ðHk Þa ðHk Þb ðHk Þa k1 iJ 0 ðUb Ua Þ cos sin sin cos e 2 2 2 2 k¼1 ðJ þðk1Þ=2;J k1 þðk1Þ=2Þ ðJ k1 þðk1Þ=2;J k1 þðk1Þ=2Þ : ðcosðH Þ ÞP cosðH Þ ð51Þ P J kk1 k b k a J k1 J k J k1
The detail derivation of the result is given in Appendix B. The correctness of the representation can be proved by the following orthogonality relations of the Jacobi function [13, p. 383] Z 1 2aþbþ1 a b dxð1 xÞ ð1 þ xÞ P nða;bÞ ðxÞP ða;bÞ m ðxÞ ¼ a þ b þ 2n þ 1 1 Cða þ n þ 1ÞCðb þ n þ 1Þ dm;n ð52Þ n!Cða þ b þ n þ 1Þ which leads to Eq. (111) in Appendix B has the orthonormal relation Z ^ ðDÞ Y J 0 ;...;J 0 ;J 0 ðHD2 ; . . . ; H1 ; UÞY dX J D2 ;...;J 1 ;J 0 ðHD2 ; . . . ; H1 ; UÞ D2 1 0 ¼ dJ 0D2 J D2 dJ 01 J 1 dJ 00 J 0 ;
ð53Þ
where the solid angle in D space is given by ^ ðDÞ ¼ sinD2 HD2 sink Hk sin H1 dHD2 dHk dH1 dU: dX
ð54Þ
326
D.-H. Lin / Annals of Physics 323 (2008) 317–336
To include the flux effect by the NPF, we change summation indices by defining Jk Jk1 = nk for the k = 2, . . . , (D 1), and replaces J1 with n1 by defining J1 J0 = n1. Owing to the true equality, j 1 X X j¼0
f ðj; j lÞ ¼
l¼0
1 X 1 X n¼0
f ðn þ l; nÞ;
ð55Þ
l¼0
we obtain the new representation of (28) "
D 1 Y
1 X
#
1 X
2 i2 D1 k¼1 nk ¼0 J 0 ¼1 2E n þ J þ 1 þ ðD 3Þ=2 þ a2 k 0 k¼1 2 PD1 3 Dþ1 2 k¼1 nk þ J 0 þ D 1 C CðD 1Þ 2 5 D pffiffiffi 4 2 D1 ðD 1Þ p C 2 C 2 PD2
P . D2 n þJ 0 k¼1 k 4 n þ J þ ðD 1Þ 2 CðnD1 þ 1ÞC2 k 0 k¼1
P D2 CðnD1 þ 2 k¼1 nk þ J 0 þ ðD 1ÞÞ PD2 PD2 n þJ 0 =2 n þJ 0 =2 k¼1 k k¼1 k ð1 u2 Þ ð1 v2 Þ PD2 PD2 D
Gðpb ; pa ; EÞ ¼ ð2pÞ
h P
nk þJ 0 þðD1Þ=2
nk þJ 0 þðD1Þ=2
C nD1 k¼1 ðuÞC nD1 k¼1 ðvÞ h P i
P 8 k k1 2 2
P k : k¼1 2k1 C2 ~k¼1 n~k þ J 0 þ ðk 1Þ=2 þ 1 Pk1 n~ þJ 0 ~ k¼1 k ðHk Þb ðHk Þa ðHk Þb ðHk Þa cos cos sin sin 2 2 2 2 Pk1 Pk1 P nk ~k¼1 P k 1
n~k þJ 0 þðk1Þ=2;
n~k þJ 0 þðk1Þ=2;
P nk
~k¼1 n~k þJ 0 þðk1Þ=2
Pk1
~k¼1
~k¼1 n~k þJ 0 þðk1Þ=2
ðcosðHk Þb Þ 9 > = 1 ðcosðHk Þa Þ eiJ 0 ðUb Ua Þ : ðD1Þ > p ;2
ð56Þ With the help of Poisson’s summation formula [11, p. 143] 1 X k¼1
f ðkÞ ¼
Z
1
dy 1
1 X
e2pnyi f ðyÞ;
n¼1
the topological effect in the Coulomb Green’s function is given by
ð57Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336
" Gflux ðpb ;pa ; EÞ ¼ ð2pÞ
D
D1 1 YX
#
h P
1 X J 0 ¼1
k¼1 nk ¼0
327
2
. i2 2E þ jJ 0 þ l0 j þ 1 þ ðD 3Þ 2 þ a2 2 PD1 3 Dþ1 2 k¼1 nk þ jJ 0 þ l0 j þ D 1 C CðD 1Þ 2 5 D pffiffiffi 4 ðD 1Þ p C 2 C2 D1 2 PD2
P . D2 4 k¼1 nk þjJ 0 þl0 j CðnD1 þ 1ÞC2 n þ jJ þ l j þ ðD 1Þ 2 k 0 0 k¼1
P D2 CðnD1 þ 2 k¼1 nk þ jJ 0 þ l0 j þ ðD 1ÞÞ PD2 PD2 ð1 u2 Þ k¼1 nk þjJ 0 þl0 j =2 ð1 v2 Þ k¼1 nk þjJ 0 þl0 j =2 PD2 PD2 D1 k¼1 nk
nk þjJ 0 þl0 j þðD1Þ=2
nk þjJ 0 þl0 j þðD1Þ=2
C nD1 k¼1 ðuÞ C nD1 k¼1 ðvÞ i
P 8 h P k k1
P k : k¼1 2k1 C2 ~ n~ þ jJ þ l j þ ðk 1Þ=2 þ 1 k¼1 k
ðHk Þb ðHk Þa ðHk Þb ðHk Þa cos cos sin sin 2 2 2 2 Pk1 Pk1 n~ þjJ 0 þl0 jþðk1Þ=2;
0
0
Pk1
n~ þjJ 0 þl0 jþðk1Þ=2
~k¼1 n~k þjJ 0 þl0 j
~k¼1 k P nk ~k¼1 k ðcosðHk Þb Þ ) Pk1 Pk1 k þjJ 0 þl0 jþðk1Þ=2; k þjJ 0 þl0 jþðk1Þ=2 ~k¼1 n~ ~k¼1 n~ ðcosðHk Þa Þ P nk
1 ðD1Þ
2
p
eiJ 0 ðUb Ua Þ :
ð58Þ
From which we find the spectrum representation of Green’s function " Gflux ðpb ; pa ; EÞ ¼ ð2pÞ
D
D 1 Y
1 X
#
k¼1 nk ¼0
1 X J 0 ¼1
C h P i2 D1 E þ a2 =2 k¼1 nk þ jJ 0 þ l0 j þ 1 þ ðD 3Þ=2 WnD1 ;...;n1 ;J 0 ðpb ÞWnD1 ;...;n1 ;J 0 ðpa Þ þ :
ð59Þ
The energy spectrum is determined by the poles and reads a2 EnD1 ;...;n1 ;J 0 ¼ h P i2 : D1 2 k¼1 nk þ jJ 0 þ l0 j þ 1 þ ðD 3Þ=2 The momentum eigenstate in D space is given by
ð60Þ
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
"
D1 X
WnD1 ;;n1 ;J 0 ðpÞ ¼ 2
!
#1=2
nk þ jJ 0 þ l0 j þ D 1
k¼1
PD2
P 31=2 D2 n þjJ 0 þl0 j 2 k¼1 k 4 þ 1ÞC n þ jJ þ l j þ ðD 1Þ=2 Cðn D1 k 0 0 k¼1 j 62 7
P 4 5 D2 p2 n þ jJ þ l j þ ðD 1ÞÞ CðnD1 þ 2 0 0 k¼1 k i
P 2 h Pk 31=2 k1 D2 Y 2 ~k¼1 n~k þ jJ 0 þ l0 j þ k Cðnk þ 1ÞCðnk þ 2 ~k¼1 n~k þ jJ 0 þ l0 j þ kÞ 4 5
P k 2k1 C2 k¼1 ~k¼1 n~k þ jJ 0 þ l0 j þ ðk 1Þ=2 þ 1 PD2 nk þjJ 0 þl0 j =2 PD2 2 2 k¼1 n þjJ 0 þl0 j þðD1Þ=2 p j 1 2jp k¼1 k 2 C n D1 ðD1Þ p þ j2 p2 þ j2 ðp2 þ j2 Þ 8 k 1 P >DY n þjJ þl j 2 < Hk Hk ~k¼1 ~k 0 0 cos sin > 2 2 : k¼1 2
2D4 3D4
P k 1
n~k þjJ 0 þl0 jþðk1Þ=2;
~
k 1 P
n~k þjJ 0 þl0 jþðk1Þ=2
~k¼1
P nk k¼1
9 > = ðcosHk Þ eiJ 0 U : > ;
The normalization condition Z dD pWnD1 ;...;n1 ;J 0 ðpÞWnD1 ;...;n1 ;J 0 ðpÞ ¼ 1
ð61Þ
ð62Þ
can be proved by Eqs. (33) and (52). When D = 3, we obtain the energy spectrum En2 ;n1 ;J 0 ¼
a2 2ðn2 þ n1 þ jJ 0 þ l0 j þ 1Þ
ð63Þ
2
and the momentum eigenstate pffiffiffiffiffiffiffi 2 2j5 ðn1 þjJ 0 þl0 jÞ 1=2 2 ðn2 þ n1 þ jJ 0 þ l0 j þ 1Þ Wn2 ;n1 ;J 0 ðpÞ ¼ p 1=2 ½2ðn1 þ jJ 0 þ l0 jÞ þ 1Cðn2 þ 1ÞCðn1 þ 1ÞCðn1 þ 2jJ 0 þ l0 j þ 1Þ Cðn2 þ 2ðn1 þ jJ 0 þ l0 jÞ þ 2Þ n1 þjJ 0 þl0 j 2 2 1 2jp n1 þjJ 0 þl0 jþ1 p j C n2 p 2 þ j2 ðp2 þ j2 Þ2 p2 þ j2 P nðjJ1 0 þl0 j;jJ 0 þl0 jÞ ðcos HÞ ðcos H=2 sin H=2Þ which has been obtained in [7]. For D = 2, one get 2 1 a : En1 ;J 0 ¼ 2 n1 þ jJ 0 þ l0 j þ 1=2
jJ 0 þl0 j
eikU
ð64Þ
ð65Þ
The result agrees with that in Ref. [20]. The normalized momentum eigenstate in D = 2 reads
D.-H. Lin / Annals of Physics 323 (2008) 317–336
"
j2 4jJ 0 þl0 j ½2ðn1 þ jJ 0 þ l0 jÞ þ 1Cðn1 þ 1ÞC2 ðjJ 0 þ l0 j þ 1=2Þ Wn1 ;J 0 ðpÞ ¼ p2 Cðn1 þ 2jJ 0 þ l0 j þ 1Þ jJ 0 þl0 j 2 2 1 2jp jJ 0 þl0 jþ1=2 p j 2 C eiJ 0 U : n1 ðp þ j2 Þ p2 þ j2 p 2 þ j2
329
#1=2
ð66Þ
Due to the important development of anionic theory in the past 3 decade, the effect of a magnetic flux in the Coulomb eigenstate in D = 2 is of interest for many researcher and maybe have important applications in the cosmic string, D = 3 gravity theories, quantum Hall effect, superconductivity, and repulsive Bose gases. Before finalizing the paper, let us evaluate the average values of the square of the momentum in the various quantum states: Z 2 ^ ðDÞ Wn ;...;n ;J ðpÞp2 W hp i ¼ pD1 dpdX ð67Þ D1 1 0 nD1 ;...;n1 ;J 0 ðpÞ: With the help of Eqs. (37) and (52), one find 2 32 a . 5: hp2 i ¼ j2 ¼ 4 P D1 k¼1 nk þ jJ 0 þ l0 j þ 1 þ ðD 3Þ 2
ð68Þ
This quantity characterizes the modified circular Bohr orbit under the magnetic flux in D space. For the hydrogen atom it specifics the square momentum of the electron in a circular Bohr orbit with the same total quantum number in D space.
4. Conclusions In this paper, we present the solutions of two physical models in momentum space. First, the momentum Coulomb Green’s function in D dimensions is given. From which the energy spectrum and momentum eigenstate are presented. These results may be important in the electron momentum spectroscopy [24] that offers us in the first time to see the probability distribution of a quantum system in momentum representation. Second, the topological effect of a magnetic flux in the Coulomb system is discussed. It is worthy to note that the effect of a magnetic flux defined by a two-dimensional field can cover arbitrary higher dimensional energy spectrum and eigenstate. This result can be attributed to the nature of topological interaction of a magnetic flux. The flux effect just relates to the winding number around the polar axis and physically couples to the angular momentum of a charged particle. Thus the two-dimensional field can have effects in the higher dimensional system. In the past three decades, the topological effect has penetrated to many areas of science. We hope our discuss is helpful to understand the role of topology in physics.
Acknowledgment This work is supported by National Science Council of Taiwan.
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
Appendix A. Representation of AB effect in momentum space In this appendix we prove the momentum representation A(p) of an AB magnetic flux in arbitrary dimensions is equivalent to that of position representation A(x). The proof is divided into two parts. We first prove A(p) has the same representation as A(x) in two dimensions. Second, we generalize the result by mathematical induction to the higher dimensional space. Given a D = 2 plane parametrized by coordinates (x, y), a magnetic flux perpendicular to the 2 plane can be expressed by the following two-dimensional field AðxÞ ¼ 2g
y^ex þ x^ey ; x2 þ y 2
ð69Þ
where g is the coupling constant used to measure the strength of interaction and eˆx,y stand for the unit vector along the x, y axis, respectively. The field produces a magnetic flux tube perpendicular to the 2 plane. To see this let us introduce the azimuthal angle u(x) = tan1(y/x) around the flux tube, the components of the vector field can be in terms of Ai = 2goiu(x). The associated magnetic field lines are confined to an infinitely thin tube along the z-axis, B3 ¼ 2g3ij oi oj uðxÞ ¼ 4pgdðx? Þ;
i; j 2 1; 2;
ð70Þ
vector x^ ” (x, y). Since the magnetic flux U0 is where x^ represents the two-dimensional R defined by the integral U0 ¼ dxdyB3 , the coupling constant g is related to the magnetic flux by g = U0/4p, and thus the strength of interaction. The momentum representation of (69) is given by the Fourier transformation Z 1 dxeipx AðxÞ: AðpÞ ¼ ð71Þ 2p Without losing the generality, taking p along the x-axis, we find the integral in polar coordinates becomes Z 1 AðpÞ ¼ ^eu dqdueipq cos u ; ð72Þ 2p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where u 2 (0, 2p) is the angle between p and x, and q ¼ x2 þ y 2 . By using the formulas [23, p. 517] Z p dueib cos u cos nu ¼ in pJ n ðbÞ; ð73Þ 0
where Jm(x) is the Bessel function, and Z 1 1 dqJ m ðbqÞ ¼ ; b 0
ð74Þ
the vector field of an AB magnetic flux in momentum space is found to be ^eU p ^e1 þ p1^e2 ¼ 2g 2 2 ; ð75Þ p p2 þ p21 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p ¼ p21 þ p22 , eˆU is the unit vector of polar angle U in D = 2 momentum space, and (eˆ1, eˆ2) are the unit vectors of Cartesian coordinates in momentum space. The associated AðpÞ ¼ 2g
D.-H. Lin / Annals of Physics 323 (2008) 317–336
331
magnetic field in momentum space can be discussed as above. One see that the momentum representation of an AB magnetic flux is same as that of in position space. The result is not surprising because the interaction of an AB magnetic flux with a charged particle is purely topological, and thus independent of representations. In D = 3 momentum space, the magnetic flux is given as AðpÞ ¼ 2g
p3^e2 þ p2^e3 : p23 þ p22
ð76Þ
The components of momentum in polar coordinates are parametrized by 8 > < p1 ¼ p cos H1 ; p2 ¼ p sin H1 cos U; > : p3 ¼ p sin H1 sin U;
ð77Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p ¼ p21 þ p22 þ p23 , 0 6 H1 6 p, and 0 6 U 6 2p. From the representation, it can be shown that the interaction of magnetic flux is purely topological. Indeed, AðpÞ dp ¼ dU
ð78Þ
such that the phase factor in NPF is given by Z
pb
ie
AðpÞ dp ¼ ie
pa
Z
kb
_ dkAðpÞ pðkÞ ¼ il0
ka
Z
S
_ dkUðkÞ;
ð79Þ
0
a topological invariant. Similarly, in the D = 4 momentum space, the magnetic flux is defined by AðpÞ ¼ 2g
p4^e3 þ p3^e4 : p24 þ p23
ð80Þ
The coordinates are parametrized by 8 p1 ¼ p cos H2 ; > > > < p ¼ p sin H cos H ; 2 1 2 ð81Þ > p3 ¼ p sin H2 sin H1 cos U; > > : p4 ¼ p sin H2 sin H1 sin U: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here p ¼ p21 þ p22 þ p23 þ p24 , 0 6 H1,2 6 p, and 0 6 U 6 2p. It is easily to check the phase factor in NPF is again given by (79) Z S Z pb Z kb _ _ ie AðpÞ dp ¼ ie dkAðpÞ pðkÞ ¼ il0 dkUðkÞ: ð82Þ pa
ka
0
Thus the interaction is described by the topological winding number around the polar axis in 4 space. For an arbitrary D dimensions, the magnetic flux is defined by AðpÞ ¼ 2g
pD^eD1 þ pD1^eD : p2D þ p2D1
The coordinates are parametrized by
ð83Þ
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
8 p ¼ p cos HD2 ; > > > 1 > > > < p2 ¼ p sin HD2 cos HD3 ; ð84Þ > > > pD1 ¼ p sin HD2 sin HD3 sin H1 cos U; > > > : pD ¼ p sin HD2 sin HD3 sin H1 sin U; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p ¼ p21 þ þ p2D , 0 6 H1,. . .,D2 6 p, and 0 6 U 6 2p. By induction, it can be proved that the phase factor in NPF is given by (82) again and the interaction is just associated with topological winding number in high dimensions. Appendix B. Spherical harmonics in D dimensions To obtain the representation of spherical harmonics in D dimensions given in Eq. (51), let us introduce the polar coordinates (r, h0, h1, . . . , hD2) in D space 8 x1 ¼ r cos hD2 ; > > > > > > < x2 ¼ r sin hD2 cos hD3 ; ð85Þ > > > x ¼ r sin h sin h sin h cos h ; > D1 D2 D3 1 0 > > : xD ¼ r sin hD2 sin hD3 sin h1 sin h0 ; where 0 6 r < 1, 0 6 h0 6 2p, 0 6 hk 6 p, k P 1. The infinitesimal distance between two points is given by ds2 ¼ dx2 ¼ dr2 þ r2 dh2D2 þ r2 sin2 hD2 dh2D3 þ þ r2 sin2 hD2 sin2 hD3 sin2 h1 dh20
ð86Þ
and the Laplace operator in D space reads 1 o 1 o o D2 2 D1 o r r ¼ D1 sin hD2 þ þ r or or ohD2 r2 sin2 hD2 ohD2 1 o o k sin h þ þ k ohk r2 sin2 hD2 sin2 hD3 sin2 hkþ1 sink hk ohk þ
o2 : r2 sin2 hD2 sin2 hD3 sin2 h1 oh20 1
The Schro¨dinger equation 2M 2M 2 r 2 V ðrÞ þ 2 E WðxÞ ¼ 0 h h
ð87Þ
ð88Þ
in D space can be simplified by defining the following variables WðxÞ ¼ RðrÞHðD2Þ ðhD2 Þ HðkÞ ðhk Þ Hð0Þ ðh0 Þ that yields the set of equation
ð89Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336
! d2 þ k0 Hð0Þ ðh0 Þ ¼ 0; dh20 1 d d ð1Þ k0 sin h1 H ðh1 Þ 2 Hð1Þ ðh1 Þ þ k1 Hð1Þ ðh1 Þ ¼ 0; sin h1 dh1 dh1 sin h1
333
ð90Þ ð91Þ
1 d d ðkÞ kk1 k sin hk H ðhk Þ 2 HðkÞ ðhk Þ þ kk HðkÞ ðhk Þ ¼ 0; k dhk sin hk dhk sin hk 1 d d D2 ðD2Þ sin hD2 H ðhD2 Þ dhD2 sinD2 hD2 dhD2 kD3 2 HðD2Þ ðhD2 Þ þ kD2 HðD2Þ ðhD2 Þ ¼ 0; sin hD3
ð92Þ
ð93Þ
and 1 rD1
d 2M kD2 2ME D1 d r R 2 V ðrÞR 2 R þ 2 R ¼ 0: dr dr r h h
ð94Þ
The physical solution of h0 in (90) is given by Hð0Þ ðh0 Þ ¼ eiJ 0 h0
ð95Þ
with k0 ¼ J 20 , and J0 = 0, ± 1, ± 2, . . . With the help of the definition zk = coshk, H(k)(hk), k P 1, satisfies the following equation ð1 z2k Þ
d2 ðkÞ d ðkÞ kk1 HðkÞ þ kk HðkÞ ¼ 0 H ðk þ 1Þzk H 2 dzk dzk ð1 z2k Þ
ð96Þ
in which kk is a constant of separation. The equation can be solved by considering the case kk1 = 0 first, it yields kk ¼ J k ðJ k þ kÞ;
J k ¼ 0; 1; 2; . . .
ð97Þ
if we require that the solution is finite at the values zk = ±1. The corresponding solution ðkÞ H(k) denoted by P J k ðzk Þ (k P 1) is found to be ðkÞ
ðk1Þ=2
P J k ðzk Þ ¼ T J k
ðzk Þ;
ð98Þ
T bn ðzÞ
is the Gegenbauer polynomials [21, p. 781]. When k = 1, the polynomials rewhere ð1Þ duces to that of the Legendre polynomials, i.e. P J k ðzk Þ ¼ P J 1 ðz1 Þ. For the case k = 1 and kk1 „ 0, Eq. (96) is the associated Legendre equation. We have the solution of associated Legendre function ð1ÞJ
P J 1 0 ðz1 Þ ¼ P JJ 01 ðz1 Þ
ð99Þ
if the conditions zk = ±1 are finite is satisfied. For the case k > 1, kk1 have been found by the eigenequation of H(k1)(hk1), and kk1 = Jk1(Jk1 + k 1), Jk1 = 0, 1, 2, . . . Let HðkÞ ðhk Þ ¼ ð1 z2k Þ
J k1 =2
gðzk Þ:
ð100Þ
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
Inserting the result into the Eq. (96), it yields ð1 z2k Þ
d2 g dg ð2J k1 þ k þ 1Þzk þ ðJ k J k1 ÞðJ k J k1 þ kÞg ¼ 0: dz2k dzk
ð101Þ
The solution of (101) is still the Gegenbauer function. If we require H(k)(hk) is finite at zk = ±1, we find that J k J k1 ¼ 0; 1; 2; . . .
ð102Þ
for J k > J k1 J
þðk1Þ=2
and the solution g(zk) is still Gegenbauer polynomials T J k1 k J k1 ðkÞJ (96) denoted by P J k k1 ðzk Þ is given by ðkÞJ k1
P Jk
ðzk Þ ¼ ð1 z2k Þ
J k1 =2
J
þðk1Þ=2
T J k1 k J k1
ðzk Þ. Thus the solution of
ðzk Þ:
ð103Þ
With the help of orthogonal relation of Gegenbauer polynomial [21, p. 783], Z 1 2Cðn þ 2b þ 1Þ b ; dzð1 z2 Þ T bn ðzÞT bm ðzÞ ¼ dnm ð2n þ 2b þ 1ÞCðn þ 1Þ 1
ð104Þ
ðkÞJ
the orthogonal relation of P J k k1 is found to be Z 1 ðkÞJ ðk1Þ=2 ðkÞJ k1 dzð1 z2 Þ P Jk ðzÞP J 0 k1 ðzÞ ¼ dJ k J 0k k
1
2CðJ k þ J k1 þ kÞ : ð2J k þ kÞCðJ k J k1 þ 1Þ
ð105Þ
The spherical harmonics in D dimensions is then given by [22] " #1=2 D 2 Y 1 ð2J k þ kÞCðJ k J k1 þ 1Þ Y J D2 ;...;J 1 ;J 0 ðhD2 ; . . . ; h1 ; h0 Þ ¼ ðD1Þ=2 pffiffiffi CðJ k þ J k1 þ kÞ p k¼1 2 ðD2ÞJ D3
P J D2
ð1ÞJ
ðcos hD2 Þ P J 1 0 ðcos h1 ÞeiJ 0 h0 :
ð106Þ
To discuss the topological effect of a magnetic flux in quantum state, we note that the Gegenbauer polynomials T bn ðzÞ can be in terms of the hypergeometric function as follows [21, p. 783]: Cðn þ 2b þ 1Þ 1z b F n; n þ 2b þ 1; b þ 1; T n ðzÞ ¼ b : ð107Þ 2 2 n!Cðb þ 1Þ With the relation between the hypergeometric function F and the Jacobi polynomials P ðb;bÞ n [13, p. 212] 1z n!Cðb þ 1Þ ðb;bÞ P F n; n þ 2b þ 1; b þ 1; ðzÞ; ð108Þ ¼ 2 Cðn þ b þ 1Þ n the Gegenbauer polynomials T bn can be expressed as T bn ðzÞ ¼
Cðb=2ÞCðb=2 þ 1=2ÞCðn þ 2b þ 1Þ ðb;bÞ pffiffiffi P n ðzÞ: 2 pCðbÞCðn þ b þ 1Þ
Thus we have
ð109Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336 ðkÞJ k1
P Jk
ðzÞ ¼ ð1 z2 Þ
335
J k1 =2
CðJ k1 =2 þ ðk 1Þ=4ÞCðJ k1 =2 þ ðk 1Þ=4 þ 1=2ÞCðJ k þ J k1 þ kÞ pffiffiffi 2 pCðJ k1 þ ðk 1Þ=2ÞCðJ k þ ðk 1Þ=2 þ 1Þ ðJ
þðk1Þ=2;J k1 þðk1Þ=2Þ
P J kk1 J k1
ðzÞ: ð110Þ
The spherical harmonics in D space then has the following convenient representation Y J D2 ;...;J 1 ;J 0 ðhD2 ; .. .; h1 ;h0 Þ ¼
1
"
D2 Y
ð2J k þ kÞCðJ k J k1 þ 1ÞCðJ k þ J k1 þ kÞ
#1=2
pffiffiffi 2ðD1Þ=2 p k¼1 2ðk1Þ C2 ðJ k þ ðk 1Þ=2 þ 1Þ " # J D2 Y hk hk k1 ðJ k1 þðk1Þ=2;J k1 þðk1Þ=2Þ cos sin P J k J k1 ðcos hk Þ eiJ 0 h0 : 2 2 k¼1 ð111Þ
It is easily to check the orthonormal relation Z ^ ðDÞ Y J 0 ;...;J 0 ;J 0 ðhD2 ; ; h1 ; h0 ÞY dX 1
D2
J D2 ;...;J 1 ;J 0 ðhD2 ; . . . ; h1 ; h0 Þ
0
¼ dJ 0D2 J D2 dJ 01 J 1 dJ 00 J 0 :
ð112Þ
Finally, we find the summation of product of spherical harmonics Y J D2 J^ can be expressed as X Y J D2 J^ ð^ xb ÞY J D2 J^ ð^ xa Þ ^ J
¼
J D2 X J D3 ¼0
J1 X
Y J D2 ;...;J 1 ;J 0 ðhD2 Þb ; ...; ðh1 Þb ;ðh0 Þb
J 0 ¼J 1
ðhD2 Þa ;. ..;ðh1 Þa ; ðh0 Þa ( ) J D3 J D2 J2 J1 D 2 Y X X X X 1 ð2J k þ kÞCðJ k J k1 þ 1ÞCðJ k þ J k1 þ kÞ ¼ ðD1Þ 2k1 C2 ðJ k þ ðk 1Þ=2 þ 1Þ p k¼1 J D3 ¼0 J D4 ¼0 J 1 ¼0 J 0 ¼J 1 2 ( J D 2 Y ðhk Þb ðhk Þa ðhk Þb ðhk Þa k1 iJ 0 ½ðh0 Þb ðh0 Þa cos sin sin e cos 2 2 2 2 k¼1 ) Y J D2 ;...;J 1 ;J 0
ðJ
þðk1Þ=2;J k1 þðk1Þ=2Þ
P J kk1 J k1
ðJ
þðk1Þ=2;J k1 þðk1Þ=2Þ
ðcosðhk Þb ÞP J k k1 J k1
ðcosðhk Þa Þ :
ð113Þ
The representation is convenient to discuss the topological effect of a magnetic flux in the quantum system. References [1] H.V. McIntosh, Symmetry and Degeneracy, in: E.M. Loebl (Ed.), Group Theory and its Applications, vol. II, Academic Press, New York, pp. 1968–1975, and references therein. [2] W. Hamilton, On the application of the method of quaternions to some dynamical questions, in: Proc. Roy. Irish Acad. 3 (1847) 441–448; see also (Sir) W.R. Hamilton, Elements of Quaternions, Longmans, Green, New York, 1866.
336 [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
D.-H. Lin / Annals of Physics 323 (2008) 317–336 B. Podolsky, L. Pauling, Phys. Rev. 34 (1929) 109. V. Fock, Z. Phys. 29 (1935) 145. H. Kleinert, Phys. Lett. A 252 (1999) 277. Alternative forms of Coulomb Green’s function in D = 3 momentum space were given by J. Schwinger and L. Hostler by local approach and Fourier transformation see: J. Schwinger, J. Math. Phys. 5 (1964) 1606; L. Hostler, J. Math. Phys. 5 (1964) 591. D.H. Lin, Ann. Phys. (NY) 292 (2001) 95. I.H. Duru, H. Kleinert, Phys. Lett. B 84 (1979) 185. For review see C. Grosche, F. Steiner, Handbook of Feynman Path Integrals, Springer, New York, 1998, and references therein. R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York, 1965. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, third, World Scientific, Singapore, 2005. D.H. Lin, J. Phys. A 30 (1997) 3201. W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems of the Special Function of Mathematical Physics, Springer, Berlin, 1966. D.H. Lin, J. Phys. A 31 (1998) 7577. Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. P.A.M. Dirac, Proc. Roy. Soc., Ser. A 133 (1931) 60. C.N. Yang, Phys. Rev. Lett. 33 (1974) 445. R. Jackiw, Phys. Rev. Lett. 50 (1983) 555; Phys. Rev. D 29 (1984) 2375. D.H. Lin, J. Math. Phys. 41 (2000) 2723. D.H. Lin, J. Phys. A 31 (1998) 4785. P.M. Morse, H. Feshbach, Methods of Theoretical Physics, vol. I, McGraw-Hill, New York, 1953. Y.K. Qian, J.Y. Zeng, Science in China (Series A) 36 (1993) 395. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1994. E. Weigold, I.E. McCarthy, Electron Momentum Spectroscopy, Kluwer Academic, New York, 1999.
Annals of Physics 323 (2008) 317–336 www.elsevier.com/locate/aop
Solution of Coulomb system in momentum space De-Hone Lin
*
Department of Physics, National Sun Yat-sen University, Kaohsiung, Taiwan Received 5 February 2007; accepted 5 April 2007 Available online 12 April 2007
Abstract The solution of D-dimensional Coulomb system is solved in momentum space by path integral. From which the topological effect of a magnetic flux in the system is given. It is revealed that the flux effect represented by the two-dimensional field of Aharonov–Bohm covers any space-dimensions. 2007 Elsevier Inc. All rights reserved. PACS: 03.65.Ge; 03.65Db; 03.65Vf Keywords: Solution of H-atom in momentum representation; Topological effect in the Coulomb system; Coulomb Green’s function
1. Introduction The momentum space representation of Coulomb system is most convenient to reveal its symmetry and degeneracy [1]. It was pointed out that by Hamilton [2] the dynamics of the three-dimensional Coulomb system in momentum representation is equivalent to that of a free particle on the hyperspherical surface in four-dimensional space. The result directly reveals the SO(4) symmetry of the Coulomb system and can be generalized to the system with 1/r potential in D dimensions that exhibits the SO(D + 1) symmetry. Modern discussions about the Coulomb system in momentum representation can be dated back to 1929 in [3] by Podolsky and Pauling who performed the Fourier transformation for the eigenstate of Coulomb system in position space to obtain its momentum representation. Nevertheless, the solution by Schro¨dinger equation in momentum space was still lacking. *
Fax: +886 7 5253709. E-mail address: [email protected]
0003-4916/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2007.04.006
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
It was Fock in 1935 [4] who solved the problem in momentum space. Fock wrote the Schro¨dinger equation in three-dimensional momentum space, and then projected the momentum space onto the surface of a unit sphere in four space. A further transformation of eigenstate leaded to the Hamiltonian for the hydrogen atom was invariant under rotations, in four space, of this S3-sphere around its center and obtained the momentum eigenstate which had been obtained by Podolsky and Pauling. As the global approach of quantum mechanics, path integral approach for the Coulomb system in momentum space is formulated by Kleinert until fairly recently in Ref. [5]. From which he obtains the Coulomb Green’s function in momentum space [6] and its energy spectrum. The corresponding eigenstate and its generalization to include the topological effect are extracted and discussed in [7]. In this paper, we shall generalize Kleinert’s results in two directions: first to the Coulomb system in momentum space in arbitrary dimensions and second to include the topological effect of a magnetic flux. In Section 2 we generalize Kleinert’s results to the D-dimensional Coulomb system in momentum space. The energy spectrum and eigenstate in D dimensions are presented. In Section 3 the topological effect of a magnetic flux in the charged particle of Coulomb system in D dimensions is given in momentum representation. Conclusions are summarized in Section 4. 2. Solution of the D-dimensional Coulomb system in momentum space The momentum Green’s function for a point particle moving in a potential barrier in D space can be represented as the following path integral [8,9]: Z 1 Z Z D Z D p Gðpb ; pa ; EÞ ¼ DD x expfA½p; x; f g; dS Df U½f ð1Þ 2p 0 where the action A½p; x; f ¼
Z
S
dkfip_ x þ f ½Hðp; xÞ Eg
ð2Þ
0
in which H is the classical Hamiltonian, f(k) is an arbitrary dimensionless fluctuating scale variable, and U[f(k)] is some convenient gauge-fixing functional. The only condition on U[f(k)] is that Z Df ðkÞU½f ðkÞ ¼ 1: ð3Þ Eq. (1) reduces to that of Feynman’s original formulation [10] if U[f(k)] is taken as the delta functional d[f 1]. For the Coulomb system under consideration, H ¼ p2 =2 a=r, where natural units with h = c = M = 1 are used. The path integral can be performed if the gauge fixing functional U[f] is chosen as [5] ( 2 2 ) N N Y X 1 n 2 pn E U½f fn xn : ð4Þ exp rn 2r2n 2 n¼0 n¼0 With which the gauge condition measure is given by
R
Df U½f ¼ 1 is automatically satisfied when the sliced
D.-H. Lin / Annals of Physics 323 (2008) 317–336
Z
Df
"Z N Y n¼0
1
1
# dfn pffiffiffiffiffiffiffiffiffiffiffiffi ; 2p=n
319
ð5Þ
where the approximation ‘‘’’ becomes equality ‘‘=’’ while limNfi1. Inserting the representation of U[f] into (1), we have the action # 2 Z S " 1 2 p2 f2 fa A½p; x; f ¼ E þ 2 dk ip_ x þ x : ð6Þ 2 r 2 2r 0 The path integrations over f and x now are Gaussian and can be performed. The former gives us the factor ( ) N X n a2 exp ; ð7Þ 2 n¼0 and the latter associated with x variable has the result ( ) D=2 2 N N Y 2D ð2p=n Þ 1X ð2Dpn Þ exp : 2 2 D 2 n¼0 n ðp2n þ j2 Þ2 n¼0 ðpn þ j Þ
ð8Þ
Here we have defined (2E) = j2. The representation of Green’s function in Eq. (1) now reduces to that of the path integral in D-dimensional momentum space as follows: Z 1 D 2D ð2pÞ dS Gðpb ; pa ; EÞ D=2 D ð2pa Þ ðp2a þ j2 Þ 0 "Z # N 1 Y 2D d D p n ð9Þ exp ANE : D=2 2 D ðpn þ j2 Þ 1 ð2pn Þ n¼1 The sliced action reads " # N X 1 ð2Dpn Þ2 n a2 N AE ¼ : 2 n ðp2n þ j2 Þ2 2 n¼0
ð10Þ
The sliced integrations can be simplified by defining a stereographic projection onto a unit sphere in (D + 1) space as follows: z¼
2jp ; 2 p þ j2
z¼
p 2 j2 ; p 2 þ j2
z2 þ z2 ¼ 1:
Then the action in Eq. (10) becomes " # N X 1 ðD^zn Þ2 n a2 N AE ½^z ¼ 2 n j2 2 n¼0 and the measure in Eq. (9) turns into "Z # Z 1 D N 1 Y ð2pÞ dDþ1^zn dS ; ð2pj2 a ÞD=2 n¼1 1 ð2pj2 n ÞD=2 0
ð11Þ
ð12Þ
ð13Þ
320
D.-H. Lin / Annals of Physics 323 (2008) 317–336
where ^z ¼ (z, z) is the (D + 1)-dimensional unit vector. With the help of the definition 1/j2 = l, the momentum path integral becomes the standard representation on D sphere in (D + 1) space [11, Ch. 8] Z 1 Gðpb ; pa ; EÞ ¼ ð2pÞD dS Kðpb ; pa ; SÞ; ð14Þ 0
where the pseudo-propagator Kðpb ; pa ; SÞ
"Z N Y
1 ð2pa =lÞ
D=2 n¼1
1
1
dDþ1^zn ð2pn =lÞ
#
D=2
exp ANE ^zð1=j2 ! lÞ :
ð15Þ
The propagator is solvable and corresponds to a particle moving on a sphere in (D + 1)dimensional space. Since the stereographic projection is from the flat to the curved space, a measure correction described by the Riemannian scalar curvature R must be included that is given by (e.g. [11, p. 742]) Z S R Ameasure ¼ : ð16Þ dk 6l 0 Here R = D(D 1)/r2 for a D sphere of radius r. Thus the measure correction gives a contribution expfA measure g ¼ expfj2 DðD 1ÞS=6g to the propagator that leads to the accurate energy spectrum. With this correction, the pseudo-propagator in momentum space is found to be [9,11,12] 1 X 2J D1 þ ðD þ 1Þ 2 CððD þ 1Þ=2Þ ðDþ1 2 1Þ Kðpb ; pa ; SÞ ¼ C J D1 ðcos #Þ ðDþ1Þ=2 ðD þ 1Þ 2 2p ¼0 J D1
e f j
2 ½ðJ
2 2 D1 þ1ÞþðD3Þ=2 a
gS=2 ;
ð17Þ
where C an ðzÞ is the Gegenbauer polynomials [13, p. 218] and # is the angular between the (D + 1) vectors ^zb and ^za : cos # ¼ ^zb ^za ¼
ðp2b j2 Þðp2a j2 Þ þ 4j2 pb pa : ðp2b þ j2 Þðp2a þ j2 Þ
ð18Þ
By performing the integration on S, the momentum Green’s function of the Coulomb system in D space is found to be Gðpb ; pa ; EÞ ¼ ð2pÞD
1 X
2
2 2 J D1 ¼0 2E½ðJ D1 þ 1Þ þ ðD 3Þ=2 þ a 2J D1 þ D 1 CððD þ 1Þ=2Þ ðDþ1 1Þ 2 C J D1 ðcos #Þ: D1 2pðDþ1Þ=2
ð19Þ
The energy spectrum are determined by the poles and given by EJ D1 ¼
a2 2½ðJ D1 þ 1Þ þ ðD 3Þ=22
;
that agrees with the result given in Ref. [14]. Particularly, when D = 3, we have
ð20Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336
321
1 X n2 2 P ðcos #Þ ; 2 2 n 2p 2En þ a2 n¼1
Gðpb ; pa ; EÞ ¼ ð2pÞ3
ð21Þ
where we have defined n = (JD1 + 1) and used the equalities C 1J D1 ðcos #Þ ¼ sinðJ D1 þ 1Þ#= sin #
ð22Þ
P n ðcos #Þ ¼ sin n#=n sin #:
ð23Þ
and
The momentum representation (21) was first given by Kleinert in [5]. Their poles display the correct energy levels of hydrogen atom En ¼
a2 ; 2n2
n ¼ 1; 2; 3; . . . :
ð24Þ ðDþ11Þ
2 To extracting the momentum eigenstate of Coulomb system, we note that C J D1 has the expansion [13, p. 223]
ðDþ11Þ
2 C J D1
ðDþ11Þ
2 ðcos #Þ ¼ C J D1
¼
ðuv þ ð1 u2 Þ
CðD 2Þ C2 ðD=2 1=2Þ
J D1 X
1=2
ð1 v2 Þ
1=2
ðcos #Þ
cos D#Þ
4J D2 ðD þ 2J D2 2Þ
J D2 ¼0
CðJ D1 J D2 þ 1ÞC2 ðJ D2 þ ðD 1Þ=2Þ ð1 u2 ÞJ D2 =2 CðD þ J D1 þ J D2 1Þ
ð1 v2 Þ
J D2 =2
J
þðD1Þ=2
D2 C J D1 J D2
J
þðD1Þ=2
D2 ðuÞC J D1 J D2
D=21
ðvÞC J D2 ðcos D#Þ; ð25Þ
where we have defined u¼
ðp2b j2 Þ ; ðp2b þ j2 Þ
v¼
ðp2a j2 Þ ; ðp2a þ j2 Þ
ð26Þ
^b , p^a in D space. Since the polynomial and D# is the angular between the unit vectors p D=21 C J D2 ðcos D#Þ has the expansion into D-dimensional ultra-spherical harmonics [11, p. 667] D=21
C J D2 ðcos D#Þ ¼
D2 2pD=2 X Y pb ÞY J D2 J^ ð^pa Þ; ^ ð^ 2J D2 þ D 2 CðD=2Þ ^ J D2 J
ð27Þ
J
^ ¼ ðJ D3 ; J D4 ; . . . ; J 0 Þ with the values Jk = 0, 1, 2, . . . (Jk P Jk1; k P 1) and for where J k = 1, J0 < 0 applicable, we obtain
322
D.-H. Lin / Annals of Physics 323 (2008) 317–336 J D1 X 1 X X
Gðpb ; pa ; EÞ ¼ ð2pÞD
2
2E½ðJ D1 þ 1Þ þ ðD 3Þ=22 þ a2 CðD 1Þ 2J D1 þ D 1 C Dþ1 2 D pffiffiffi 2 D1 ðD 1Þ p C 2 C 2 J D2 J =2 4 CðJ D1 J D2 þ 1ÞC2 ðJ D2 þ ðD 1Þ=2Þ 1 u2 D2 ð1 v2 ÞJ D2 =2 CðD þ J D1 þ J D2 1Þ J D1 ¼0 J D2 ¼0
þðD1Þ=2
J
D2 C J D1 J D2
^ J
J
þðD1Þ=2
D2 ðuÞC J D1 J D2
ðvÞY J D2 J^ ð^ pb ÞY J D2 J^ ð^ pa Þ:
ð28Þ
From which the spectrum representation of momentum Green’s function can be expressed as D
Gðpb ; pa ; EÞ ¼ ð2pÞ
1 X
J D1 X X
2
J D1 ¼0 J D2 ¼0
C E½ðJ D1 þ 1Þ þ ðD 3Þ=2 þ a2 =2
^ J WJ D1 ;J D2 ;J^ ðpb ÞWJ D1 ;J D2 ;J^ ðpa Þ
þ ;
ð29Þ
where C is an independent constant. The momentum eigenstate of Coulomb system in D dimensions is found to be " #1=2 ð2J D1 þ D 1Þð2jÞ3D4 WJ D1 ;J D2 ;J^ ðpÞ ¼ 2p J D2 1=2 4 CðJ D1 J D2 þ 1ÞC2 ðJ D2 þ ðD 1Þ=2Þ 1 2 CðD þ J D1 þ J D2 1Þ ðp þ j2 ÞðD1Þ J D2 2 2 2jp J D2 þðD1Þ=2 p j 2 C pÞ: ð30Þ Y ^ ð^ J J D1 D2 p þ j2 p2 þ j2 J D2 J The normalization condition Z dD pWJ D1 ;J D2 ;J^ ðpÞ WJ D1 ;J D2 ;J^ ðpÞ ¼ 1
ð31Þ
can be checked by using the recursion formula zC mn ðzÞ ¼
1 ½ðn þ 1ÞC mnþ1 ðzÞ þ ðn þ 2m 1ÞC mn1 ðzÞ; 2ðn þ mÞ
and the orthogonality relations of Gegenbauer polynomials [13, p. 378] Z 1 p212k Cðn þ 2kÞ k1=2 dx C kn ðxÞC km ðxÞð1 x2 Þ ¼ dm;n ; n!ðk þ nÞC2 ðkÞ 1
ð32Þ
ð33Þ
where the volume element in momentum space is given by ^ ðDÞ dD p ¼ pD1 dp dX ¼ pD1 dp sinD2 HD2 sink Hk sin H1 dHD2 dHk dH1 dU
ð34Þ
with the ranges of coordinates 0 6 p 6 1, 0 6 U 6 2p, and 0 6 Hk 6 p (k P 1). Particularly, when D = 3, the result in (30) reduces to that of
D.-H. Lin / Annals of Physics 323 (2008) 317–336
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 2j5 l 4pnCðn lÞ 2 Cðl þ 1Þ Wn;l;m ðpÞ ¼ Cðn þ l þ 1Þ p l 2 1 2jp p j2 lþ1 C nl1 2 Y lm ðH; UÞ: p þ j2 ðp2 þ j2 Þ2 p2 þ j2
323
ð35Þ
Here we have identified n = (J2 + 1) and l = J1 with n = 1, 2, 3, . . . and l = 0, 1, 2, . . . The eigenstate was first obtained by Podolsky and Pauling by carrying out the Fourier transformation on the eigenstate in position space in [3]. For D = 2, the Coulomb eigenstate is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2J 0 þ 1Þð2jÞ 4J 0 CðJ 1 J 0 þ 1ÞC2 ðJ 0 þ 1=2Þ WJ 1 ;J 0 ðpÞ ¼ CðJ 1 þ J 0 þ 1Þ 2p J 0 2 2 1 2jp J 0 þ1=2 p j 2 C J 1 J 0 ð36Þ Y J 0 ð^pÞ; ðp þ j2 Þ p2 þ j2 p 2 þ j2 pffiffiffiffiffiffi where Y J 0 ð^ pÞ ¼ expfiJ 0 Ug= 2p. It is of interest to evaluate the average values of the square of the momentum in the various quantum states: Z 1 Z p Z 2p 2 ^ ðDÞ W hp2 i ¼ pD1 dp dX ð37Þ ^ ðpÞp WJ ^ ðpÞ: J D1 ;J D2 ;J D1 ;J D2 ;J 0
0
0
With the help of (33), one find 2 a 2 2 hp i ¼ j ¼ ; n þ ðD 3Þ=2
n ¼ 1; 2; 3; . . .
ð38Þ
This quantity characterizes the circular Bohr orbit of Coulomb system in D space. The average of square momentum of the electron in 3 space becomes
a2 : ð39Þ hp2 i ¼ j2 ¼ n 3. Magnetic flux effect in the eigenstate of Coulomb system in D dimensions In this section we discuss the topological effect of an Aharonov–Bohm (AB) magnetic flux [15] in the momentum eigenstate of a charged particle moving in the Coulomb field in D dimensions. Introducing the polar coordinates (r, h0, h1, . . . , hD2) in D space, the Cartesian coordinates are related by 8 x1 ¼ r cos hD2 ; > > > > > > < x2 ¼ r sin hD2 cos hD3 ; ð40Þ > > > xD1 ¼ r sin hD2 sin hD3 sin h1 cos h0 ; > > > : xD ¼ r sin hD2 sin hD3 sin h1 sin h0 ; where 0 6 r < 1, 0 6 h0 6 2p, 0 6 hk 6 p, k P 1. The magnetic flux in D space located at the force center is defined by
324
D.-H. Lin / Annals of Physics 323 (2008) 317–336
AðxÞ ¼ 2g
xD^eD1 þ xD1^eD ; x2D þ x2D1
ð41Þ
where eˆD1,D stand for the unit vector along the D 1, D axis. The field generalizes the effect of a magnetic flux in 2 space to arbitrary D space. It defines an interaction of field with a charged particle in D space that just relates to the topological winding number around the polar axis in D space (See Appendix A). The topological effect is conveniently considered by a global nonintegrable phase factor (NPF) [16–18]. The NPF represents the interaction of a charged particle with the magnetic field by the phase modulation as follows: ( Z ) pb
Gflux ðpb ; pa ; EÞ ¼ Gðpb ; pa ; EÞ exp ie
AðpÞ dp ;
ð42Þ
pa
where Gflux is the new Green’s function that has included the effect of the magnetic flux defined by the vector potential A(p). For an AB magnetic flux, the A(p) is the same as A(x) and given by (see Appendix A) AðpÞ ¼ 2g
pD^eD1 þ pD1^eD : p2D þ p2D1
ð43Þ
Using the polar coordinates (p, U, H1, . . . , HD2) in momentum pace pD1 ¼ p sin HD2 sin HD3 sin H1 cos U; pD ¼ p sin HD2 sin HD3 sin H1 sin U
ð44Þ
1=2
with p ¼ ðp21 þ p22 þ þ p2D Þ and the ranges of coordinates 0 6 p < 1, 0 6 U 6 2p, 0 6 Hk 6 p, k P 1, one find Z S Z pb Z kb _ _ ie AðpÞ dp ¼ ie dk AðpÞ pðkÞ ¼ il0 d kUðkÞ; ð45Þ ka
pa
0
where U(k) = U(p(k)), U_ ¼ dU=dk, and l0 is the dimensionless number l0 = 2eg. Given two paths C1 and C2 in momentum space connecting pa and pb, the integral only differs by an integer multiple of 2p. The winding number k is thus given by Z S 1 _ k¼ dk UðkÞ: ð46Þ 2p 0 The flux effect is purely topological, its value is then given by ( Z ) pb exp ie AðpÞ dp ¼ expfil0 ð2kp þ Ub Ua Þg:
ð47Þ
pa
To consider the topological effect, it is very convenient to use another representation of the spherical harmonics in D space in (27). Let us first consider the D = 3 case. The wellknown representation of spherical harmonics gives l X k¼l
Y lk ð^ pb ÞY lk ð^pa Þ ¼
l X 2l þ 1 Cðl k þ 1Þ k P ðcos Hb ÞP kl ðcos Ha ÞeikðUb Ua Þ : 4p Cðl þ k þ 1Þ l k¼l
ð48Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336
325
Here (H, U) are the angular of the polar coordinates of the total momentum vector in D = 3 momentum space. With the help of the relation between the associated Legendre polynomial P kl ðxÞ and the Jacobi function P ðk;kÞ ðxÞ [19] n P kl ðcos HÞ ¼ ð1Þk
Cðk þ l þ 1Þ ðk;kÞ ðcos H=2 sin H=2Þk P lk ðcos HÞ; Cðl þ 1Þ
ð49Þ
the summation of product in (48) becomes l l X X 2l þ 1 Cðl k þ 1ÞCðl þ k þ 1Þ Y lk ð^ pb ÞY lk ð^ pa Þ ¼ 4p C2 ðl þ 1Þ k¼l k¼l k Hb Ha Hb Ha cos sin sin cos 2 2 2 2 ðk;kÞ
ðk;kÞ
P lk ðcos Hb ÞP lk ðcos Ha ÞeikðUb Ua Þ :
ð50Þ
Generalizing the procedure to higher dimensions we find that (27) in D space has the representation X Y J D2 J^ ð^pb ÞY J D2 J^ ð^pa Þ ^ J
¼
J D2 X J D3 ¼0
J1 X
Y J D2 ;...;J 1 ;J 0 ððHD2 Þb ; . . . ; ðH1 Þb ; Ub Þ
J 0 ¼J 1
Y J D2 ;...;J 1 ;J 0 ððHD2 Þa ; . . . ; ðH1 Þa ; Ua Þ ( ) J D3 J D2 J2 J1 D 2 Y X X X X 1 ð2J k þ kÞCðJ k J k1 þ 1ÞCðJ k þ J k1 þ kÞ ¼ ðD1Þ 2k1 C2 ðJ k þ ðk 1Þ=2 þ 1Þ p k¼1 J D3 ¼0 J D4 ¼0 J 1 ¼0 J 0 ¼J 1 2 ( J D 2 Y ðHk Þb ðHk Þa ðHk Þb ðHk Þa k1 iJ 0 ðUb Ua Þ cos sin sin cos e 2 2 2 2 k¼1 ðJ þðk1Þ=2;J k1 þðk1Þ=2Þ ðJ k1 þðk1Þ=2;J k1 þðk1Þ=2Þ : ðcosðH Þ ÞP cosðH Þ ð51Þ P J kk1 k b k a J k1 J k J k1
The detail derivation of the result is given in Appendix B. The correctness of the representation can be proved by the following orthogonality relations of the Jacobi function [13, p. 383] Z 1 2aþbþ1 a b dxð1 xÞ ð1 þ xÞ P nða;bÞ ðxÞP ða;bÞ m ðxÞ ¼ a þ b þ 2n þ 1 1 Cða þ n þ 1ÞCðb þ n þ 1Þ dm;n ð52Þ n!Cða þ b þ n þ 1Þ which leads to Eq. (111) in Appendix B has the orthonormal relation Z ^ ðDÞ Y J 0 ;...;J 0 ;J 0 ðHD2 ; . . . ; H1 ; UÞY dX J D2 ;...;J 1 ;J 0 ðHD2 ; . . . ; H1 ; UÞ D2 1 0 ¼ dJ 0D2 J D2 dJ 01 J 1 dJ 00 J 0 ;
ð53Þ
where the solid angle in D space is given by ^ ðDÞ ¼ sinD2 HD2 sink Hk sin H1 dHD2 dHk dH1 dU: dX
ð54Þ
326
D.-H. Lin / Annals of Physics 323 (2008) 317–336
To include the flux effect by the NPF, we change summation indices by defining Jk Jk1 = nk for the k = 2, . . . , (D 1), and replaces J1 with n1 by defining J1 J0 = n1. Owing to the true equality, j 1 X X j¼0
f ðj; j lÞ ¼
l¼0
1 X 1 X n¼0
f ðn þ l; nÞ;
ð55Þ
l¼0
we obtain the new representation of (28) "
D 1 Y
1 X
#
1 X
2 i2 D1 k¼1 nk ¼0 J 0 ¼1 2E n þ J þ 1 þ ðD 3Þ=2 þ a2 k 0 k¼1 2 PD1 3 Dþ1 2 k¼1 nk þ J 0 þ D 1 C CðD 1Þ 2 5 D pffiffiffi 4 2 D1 ðD 1Þ p C 2 C 2 PD2
P . D2 n þJ 0 k¼1 k 4 n þ J þ ðD 1Þ 2 CðnD1 þ 1ÞC2 k 0 k¼1
P D2 CðnD1 þ 2 k¼1 nk þ J 0 þ ðD 1ÞÞ PD2 PD2 n þJ 0 =2 n þJ 0 =2 k¼1 k k¼1 k ð1 u2 Þ ð1 v2 Þ PD2 PD2 D
Gðpb ; pa ; EÞ ¼ ð2pÞ
h P
nk þJ 0 þðD1Þ=2
nk þJ 0 þðD1Þ=2
C nD1 k¼1 ðuÞC nD1 k¼1 ðvÞ h P i
P 8 k k1 2 2
P k : k¼1 2k1 C2 ~k¼1 n~k þ J 0 þ ðk 1Þ=2 þ 1 Pk1 n~ þJ 0 ~ k¼1 k ðHk Þb ðHk Þa ðHk Þb ðHk Þa cos cos sin sin 2 2 2 2 Pk1 Pk1 P nk ~k¼1 P k 1
n~k þJ 0 þðk1Þ=2;
n~k þJ 0 þðk1Þ=2;
P nk
~k¼1 n~k þJ 0 þðk1Þ=2
Pk1
~k¼1
~k¼1 n~k þJ 0 þðk1Þ=2
ðcosðHk Þb Þ 9 > = 1 ðcosðHk Þa Þ eiJ 0 ðUb Ua Þ : ðD1Þ > p ;2
ð56Þ With the help of Poisson’s summation formula [11, p. 143] 1 X k¼1
f ðkÞ ¼
Z
1
dy 1
1 X
e2pnyi f ðyÞ;
n¼1
the topological effect in the Coulomb Green’s function is given by
ð57Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336
" Gflux ðpb ;pa ; EÞ ¼ ð2pÞ
D
D1 1 YX
#
h P
1 X J 0 ¼1
k¼1 nk ¼0
327
2
. i2 2E þ jJ 0 þ l0 j þ 1 þ ðD 3Þ 2 þ a2 2 PD1 3 Dþ1 2 k¼1 nk þ jJ 0 þ l0 j þ D 1 C CðD 1Þ 2 5 D pffiffiffi 4 ðD 1Þ p C 2 C2 D1 2 PD2
P . D2 4 k¼1 nk þjJ 0 þl0 j CðnD1 þ 1ÞC2 n þ jJ þ l j þ ðD 1Þ 2 k 0 0 k¼1
P D2 CðnD1 þ 2 k¼1 nk þ jJ 0 þ l0 j þ ðD 1ÞÞ PD2 PD2 ð1 u2 Þ k¼1 nk þjJ 0 þl0 j =2 ð1 v2 Þ k¼1 nk þjJ 0 þl0 j =2 PD2 PD2 D1 k¼1 nk
nk þjJ 0 þl0 j þðD1Þ=2
nk þjJ 0 þl0 j þðD1Þ=2
C nD1 k¼1 ðuÞ C nD1 k¼1 ðvÞ i
P 8 h P k k1
P k : k¼1 2k1 C2 ~ n~ þ jJ þ l j þ ðk 1Þ=2 þ 1 k¼1 k
ðHk Þb ðHk Þa ðHk Þb ðHk Þa cos cos sin sin 2 2 2 2 Pk1 Pk1 n~ þjJ 0 þl0 jþðk1Þ=2;
0
0
Pk1
n~ þjJ 0 þl0 jþðk1Þ=2
~k¼1 n~k þjJ 0 þl0 j
~k¼1 k P nk ~k¼1 k ðcosðHk Þb Þ ) Pk1 Pk1 k þjJ 0 þl0 jþðk1Þ=2; k þjJ 0 þl0 jþðk1Þ=2 ~k¼1 n~ ~k¼1 n~ ðcosðHk Þa Þ P nk
1 ðD1Þ
2
p
eiJ 0 ðUb Ua Þ :
ð58Þ
From which we find the spectrum representation of Green’s function " Gflux ðpb ; pa ; EÞ ¼ ð2pÞ
D
D 1 Y
1 X
#
k¼1 nk ¼0
1 X J 0 ¼1
C h P i2 D1 E þ a2 =2 k¼1 nk þ jJ 0 þ l0 j þ 1 þ ðD 3Þ=2 WnD1 ;...;n1 ;J 0 ðpb ÞWnD1 ;...;n1 ;J 0 ðpa Þ þ :
ð59Þ
The energy spectrum is determined by the poles and reads a2 EnD1 ;...;n1 ;J 0 ¼ h P i2 : D1 2 k¼1 nk þ jJ 0 þ l0 j þ 1 þ ðD 3Þ=2 The momentum eigenstate in D space is given by
ð60Þ
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
"
D1 X
WnD1 ;;n1 ;J 0 ðpÞ ¼ 2
!
#1=2
nk þ jJ 0 þ l0 j þ D 1
k¼1
PD2
P 31=2 D2 n þjJ 0 þl0 j 2 k¼1 k 4 þ 1ÞC n þ jJ þ l j þ ðD 1Þ=2 Cðn D1 k 0 0 k¼1 j 62 7
P 4 5 D2 p2 n þ jJ þ l j þ ðD 1ÞÞ CðnD1 þ 2 0 0 k¼1 k i
P 2 h Pk 31=2 k1 D2 Y 2 ~k¼1 n~k þ jJ 0 þ l0 j þ k Cðnk þ 1ÞCðnk þ 2 ~k¼1 n~k þ jJ 0 þ l0 j þ kÞ 4 5
P k 2k1 C2 k¼1 ~k¼1 n~k þ jJ 0 þ l0 j þ ðk 1Þ=2 þ 1 PD2 nk þjJ 0 þl0 j =2 PD2 2 2 k¼1 n þjJ 0 þl0 j þðD1Þ=2 p j 1 2jp k¼1 k 2 C n D1 ðD1Þ p þ j2 p2 þ j2 ðp2 þ j2 Þ 8 k 1 P >DY n þjJ þl j 2 < Hk Hk ~k¼1 ~k 0 0 cos sin > 2 2 : k¼1 2
2D4 3D4
P k 1
n~k þjJ 0 þl0 jþðk1Þ=2;
~
k 1 P
n~k þjJ 0 þl0 jþðk1Þ=2
~k¼1
P nk k¼1
9 > = ðcosHk Þ eiJ 0 U : > ;
The normalization condition Z dD pWnD1 ;...;n1 ;J 0 ðpÞWnD1 ;...;n1 ;J 0 ðpÞ ¼ 1
ð61Þ
ð62Þ
can be proved by Eqs. (33) and (52). When D = 3, we obtain the energy spectrum En2 ;n1 ;J 0 ¼
a2 2ðn2 þ n1 þ jJ 0 þ l0 j þ 1Þ
ð63Þ
2
and the momentum eigenstate pffiffiffiffiffiffiffi 2 2j5 ðn1 þjJ 0 þl0 jÞ 1=2 2 ðn2 þ n1 þ jJ 0 þ l0 j þ 1Þ Wn2 ;n1 ;J 0 ðpÞ ¼ p 1=2 ½2ðn1 þ jJ 0 þ l0 jÞ þ 1Cðn2 þ 1ÞCðn1 þ 1ÞCðn1 þ 2jJ 0 þ l0 j þ 1Þ Cðn2 þ 2ðn1 þ jJ 0 þ l0 jÞ þ 2Þ n1 þjJ 0 þl0 j 2 2 1 2jp n1 þjJ 0 þl0 jþ1 p j C n2 p 2 þ j2 ðp2 þ j2 Þ2 p2 þ j2 P nðjJ1 0 þl0 j;jJ 0 þl0 jÞ ðcos HÞ ðcos H=2 sin H=2Þ which has been obtained in [7]. For D = 2, one get 2 1 a : En1 ;J 0 ¼ 2 n1 þ jJ 0 þ l0 j þ 1=2
jJ 0 þl0 j
eikU
ð64Þ
ð65Þ
The result agrees with that in Ref. [20]. The normalized momentum eigenstate in D = 2 reads
D.-H. Lin / Annals of Physics 323 (2008) 317–336
"
j2 4jJ 0 þl0 j ½2ðn1 þ jJ 0 þ l0 jÞ þ 1Cðn1 þ 1ÞC2 ðjJ 0 þ l0 j þ 1=2Þ Wn1 ;J 0 ðpÞ ¼ p2 Cðn1 þ 2jJ 0 þ l0 j þ 1Þ jJ 0 þl0 j 2 2 1 2jp jJ 0 þl0 jþ1=2 p j 2 C eiJ 0 U : n1 ðp þ j2 Þ p2 þ j2 p 2 þ j2
329
#1=2
ð66Þ
Due to the important development of anionic theory in the past 3 decade, the effect of a magnetic flux in the Coulomb eigenstate in D = 2 is of interest for many researcher and maybe have important applications in the cosmic string, D = 3 gravity theories, quantum Hall effect, superconductivity, and repulsive Bose gases. Before finalizing the paper, let us evaluate the average values of the square of the momentum in the various quantum states: Z 2 ^ ðDÞ Wn ;...;n ;J ðpÞp2 W hp i ¼ pD1 dpdX ð67Þ D1 1 0 nD1 ;...;n1 ;J 0 ðpÞ: With the help of Eqs. (37) and (52), one find 2 32 a . 5: hp2 i ¼ j2 ¼ 4 P D1 k¼1 nk þ jJ 0 þ l0 j þ 1 þ ðD 3Þ 2
ð68Þ
This quantity characterizes the modified circular Bohr orbit under the magnetic flux in D space. For the hydrogen atom it specifics the square momentum of the electron in a circular Bohr orbit with the same total quantum number in D space.
4. Conclusions In this paper, we present the solutions of two physical models in momentum space. First, the momentum Coulomb Green’s function in D dimensions is given. From which the energy spectrum and momentum eigenstate are presented. These results may be important in the electron momentum spectroscopy [24] that offers us in the first time to see the probability distribution of a quantum system in momentum representation. Second, the topological effect of a magnetic flux in the Coulomb system is discussed. It is worthy to note that the effect of a magnetic flux defined by a two-dimensional field can cover arbitrary higher dimensional energy spectrum and eigenstate. This result can be attributed to the nature of topological interaction of a magnetic flux. The flux effect just relates to the winding number around the polar axis and physically couples to the angular momentum of a charged particle. Thus the two-dimensional field can have effects in the higher dimensional system. In the past three decades, the topological effect has penetrated to many areas of science. We hope our discuss is helpful to understand the role of topology in physics.
Acknowledgment This work is supported by National Science Council of Taiwan.
330
D.-H. Lin / Annals of Physics 323 (2008) 317–336
Appendix A. Representation of AB effect in momentum space In this appendix we prove the momentum representation A(p) of an AB magnetic flux in arbitrary dimensions is equivalent to that of position representation A(x). The proof is divided into two parts. We first prove A(p) has the same representation as A(x) in two dimensions. Second, we generalize the result by mathematical induction to the higher dimensional space. Given a D = 2 plane parametrized by coordinates (x, y), a magnetic flux perpendicular to the 2 plane can be expressed by the following two-dimensional field AðxÞ ¼ 2g
y^ex þ x^ey ; x2 þ y 2
ð69Þ
where g is the coupling constant used to measure the strength of interaction and eˆx,y stand for the unit vector along the x, y axis, respectively. The field produces a magnetic flux tube perpendicular to the 2 plane. To see this let us introduce the azimuthal angle u(x) = tan1(y/x) around the flux tube, the components of the vector field can be in terms of Ai = 2goiu(x). The associated magnetic field lines are confined to an infinitely thin tube along the z-axis, B3 ¼ 2g3ij oi oj uðxÞ ¼ 4pgdðx? Þ;
i; j 2 1; 2;
ð70Þ
vector x^ ” (x, y). Since the magnetic flux U0 is where x^ represents the two-dimensional R defined by the integral U0 ¼ dxdyB3 , the coupling constant g is related to the magnetic flux by g = U0/4p, and thus the strength of interaction. The momentum representation of (69) is given by the Fourier transformation Z 1 dxeipx AðxÞ: AðpÞ ¼ ð71Þ 2p Without losing the generality, taking p along the x-axis, we find the integral in polar coordinates becomes Z 1 AðpÞ ¼ ^eu dqdueipq cos u ; ð72Þ 2p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where u 2 (0, 2p) is the angle between p and x, and q ¼ x2 þ y 2 . By using the formulas [23, p. 517] Z p dueib cos u cos nu ¼ in pJ n ðbÞ; ð73Þ 0
where Jm(x) is the Bessel function, and Z 1 1 dqJ m ðbqÞ ¼ ; b 0
ð74Þ
the vector field of an AB magnetic flux in momentum space is found to be ^eU p ^e1 þ p1^e2 ¼ 2g 2 2 ; ð75Þ p p2 þ p21 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p ¼ p21 þ p22 , eˆU is the unit vector of polar angle U in D = 2 momentum space, and (eˆ1, eˆ2) are the unit vectors of Cartesian coordinates in momentum space. The associated AðpÞ ¼ 2g
D.-H. Lin / Annals of Physics 323 (2008) 317–336
331
magnetic field in momentum space can be discussed as above. One see that the momentum representation of an AB magnetic flux is same as that of in position space. The result is not surprising because the interaction of an AB magnetic flux with a charged particle is purely topological, and thus independent of representations. In D = 3 momentum space, the magnetic flux is given as AðpÞ ¼ 2g
p3^e2 þ p2^e3 : p23 þ p22
ð76Þ
The components of momentum in polar coordinates are parametrized by 8 > < p1 ¼ p cos H1 ; p2 ¼ p sin H1 cos U; > : p3 ¼ p sin H1 sin U;
ð77Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p ¼ p21 þ p22 þ p23 , 0 6 H1 6 p, and 0 6 U 6 2p. From the representation, it can be shown that the interaction of magnetic flux is purely topological. Indeed, AðpÞ dp ¼ dU
ð78Þ
such that the phase factor in NPF is given by Z
pb
ie
AðpÞ dp ¼ ie
pa
Z
kb
_ dkAðpÞ pðkÞ ¼ il0
ka
Z
S
_ dkUðkÞ;
ð79Þ
0
a topological invariant. Similarly, in the D = 4 momentum space, the magnetic flux is defined by AðpÞ ¼ 2g
p4^e3 þ p3^e4 : p24 þ p23
ð80Þ
The coordinates are parametrized by 8 p1 ¼ p cos H2 ; > > > < p ¼ p sin H cos H ; 2 1 2 ð81Þ > p3 ¼ p sin H2 sin H1 cos U; > > : p4 ¼ p sin H2 sin H1 sin U: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here p ¼ p21 þ p22 þ p23 þ p24 , 0 6 H1,2 6 p, and 0 6 U 6 2p. It is easily to check the phase factor in NPF is again given by (79) Z S Z pb Z kb _ _ ie AðpÞ dp ¼ ie dkAðpÞ pðkÞ ¼ il0 dkUðkÞ: ð82Þ pa
ka
0
Thus the interaction is described by the topological winding number around the polar axis in 4 space. For an arbitrary D dimensions, the magnetic flux is defined by AðpÞ ¼ 2g
pD^eD1 þ pD1^eD : p2D þ p2D1
The coordinates are parametrized by
ð83Þ
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D.-H. Lin / Annals of Physics 323 (2008) 317–336
8 p ¼ p cos HD2 ; > > > 1 > > > < p2 ¼ p sin HD2 cos HD3 ; ð84Þ > > > pD1 ¼ p sin HD2 sin HD3 sin H1 cos U; > > > : pD ¼ p sin HD2 sin HD3 sin H1 sin U; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p ¼ p21 þ þ p2D , 0 6 H1,. . .,D2 6 p, and 0 6 U 6 2p. By induction, it can be proved that the phase factor in NPF is given by (82) again and the interaction is just associated with topological winding number in high dimensions. Appendix B. Spherical harmonics in D dimensions To obtain the representation of spherical harmonics in D dimensions given in Eq. (51), let us introduce the polar coordinates (r, h0, h1, . . . , hD2) in D space 8 x1 ¼ r cos hD2 ; > > > > > > < x2 ¼ r sin hD2 cos hD3 ; ð85Þ > > > x ¼ r sin h sin h sin h cos h ; > D1 D2 D3 1 0 > > : xD ¼ r sin hD2 sin hD3 sin h1 sin h0 ; where 0 6 r < 1, 0 6 h0 6 2p, 0 6 hk 6 p, k P 1. The infinitesimal distance between two points is given by ds2 ¼ dx2 ¼ dr2 þ r2 dh2D2 þ r2 sin2 hD2 dh2D3 þ þ r2 sin2 hD2 sin2 hD3 sin2 h1 dh20
ð86Þ
and the Laplace operator in D space reads 1 o 1 o o D2 2 D1 o r r ¼ D1 sin hD2 þ þ r or or ohD2 r2 sin2 hD2 ohD2 1 o o k sin h þ þ k ohk r2 sin2 hD2 sin2 hD3 sin2 hkþ1 sink hk ohk þ
o2 : r2 sin2 hD2 sin2 hD3 sin2 h1 oh20 1
The Schro¨dinger equation 2M 2M 2 r 2 V ðrÞ þ 2 E WðxÞ ¼ 0 h h
ð87Þ
ð88Þ
in D space can be simplified by defining the following variables WðxÞ ¼ RðrÞHðD2Þ ðhD2 Þ HðkÞ ðhk Þ Hð0Þ ðh0 Þ that yields the set of equation
ð89Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336
! d2 þ k0 Hð0Þ ðh0 Þ ¼ 0; dh20 1 d d ð1Þ k0 sin h1 H ðh1 Þ 2 Hð1Þ ðh1 Þ þ k1 Hð1Þ ðh1 Þ ¼ 0; sin h1 dh1 dh1 sin h1
333
ð90Þ ð91Þ
1 d d ðkÞ kk1 k sin hk H ðhk Þ 2 HðkÞ ðhk Þ þ kk HðkÞ ðhk Þ ¼ 0; k dhk sin hk dhk sin hk 1 d d D2 ðD2Þ sin hD2 H ðhD2 Þ dhD2 sinD2 hD2 dhD2 kD3 2 HðD2Þ ðhD2 Þ þ kD2 HðD2Þ ðhD2 Þ ¼ 0; sin hD3
ð92Þ
ð93Þ
and 1 rD1
d 2M kD2 2ME D1 d r R 2 V ðrÞR 2 R þ 2 R ¼ 0: dr dr r h h
ð94Þ
The physical solution of h0 in (90) is given by Hð0Þ ðh0 Þ ¼ eiJ 0 h0
ð95Þ
with k0 ¼ J 20 , and J0 = 0, ± 1, ± 2, . . . With the help of the definition zk = coshk, H(k)(hk), k P 1, satisfies the following equation ð1 z2k Þ
d2 ðkÞ d ðkÞ kk1 HðkÞ þ kk HðkÞ ¼ 0 H ðk þ 1Þzk H 2 dzk dzk ð1 z2k Þ
ð96Þ
in which kk is a constant of separation. The equation can be solved by considering the case kk1 = 0 first, it yields kk ¼ J k ðJ k þ kÞ;
J k ¼ 0; 1; 2; . . .
ð97Þ
if we require that the solution is finite at the values zk = ±1. The corresponding solution ðkÞ H(k) denoted by P J k ðzk Þ (k P 1) is found to be ðkÞ
ðk1Þ=2
P J k ðzk Þ ¼ T J k
ðzk Þ;
ð98Þ
T bn ðzÞ
is the Gegenbauer polynomials [21, p. 781]. When k = 1, the polynomials rewhere ð1Þ duces to that of the Legendre polynomials, i.e. P J k ðzk Þ ¼ P J 1 ðz1 Þ. For the case k = 1 and kk1 „ 0, Eq. (96) is the associated Legendre equation. We have the solution of associated Legendre function ð1ÞJ
P J 1 0 ðz1 Þ ¼ P JJ 01 ðz1 Þ
ð99Þ
if the conditions zk = ±1 are finite is satisfied. For the case k > 1, kk1 have been found by the eigenequation of H(k1)(hk1), and kk1 = Jk1(Jk1 + k 1), Jk1 = 0, 1, 2, . . . Let HðkÞ ðhk Þ ¼ ð1 z2k Þ
J k1 =2
gðzk Þ:
ð100Þ
334
D.-H. Lin / Annals of Physics 323 (2008) 317–336
Inserting the result into the Eq. (96), it yields ð1 z2k Þ
d2 g dg ð2J k1 þ k þ 1Þzk þ ðJ k J k1 ÞðJ k J k1 þ kÞg ¼ 0: dz2k dzk
ð101Þ
The solution of (101) is still the Gegenbauer function. If we require H(k)(hk) is finite at zk = ±1, we find that J k J k1 ¼ 0; 1; 2; . . .
ð102Þ
for J k > J k1 J
þðk1Þ=2
and the solution g(zk) is still Gegenbauer polynomials T J k1 k J k1 ðkÞJ (96) denoted by P J k k1 ðzk Þ is given by ðkÞJ k1
P Jk
ðzk Þ ¼ ð1 z2k Þ
J k1 =2
J
þðk1Þ=2
T J k1 k J k1
ðzk Þ. Thus the solution of
ðzk Þ:
ð103Þ
With the help of orthogonal relation of Gegenbauer polynomial [21, p. 783], Z 1 2Cðn þ 2b þ 1Þ b ; dzð1 z2 Þ T bn ðzÞT bm ðzÞ ¼ dnm ð2n þ 2b þ 1ÞCðn þ 1Þ 1
ð104Þ
ðkÞJ
the orthogonal relation of P J k k1 is found to be Z 1 ðkÞJ ðk1Þ=2 ðkÞJ k1 dzð1 z2 Þ P Jk ðzÞP J 0 k1 ðzÞ ¼ dJ k J 0k k
1
2CðJ k þ J k1 þ kÞ : ð2J k þ kÞCðJ k J k1 þ 1Þ
ð105Þ
The spherical harmonics in D dimensions is then given by [22] " #1=2 D 2 Y 1 ð2J k þ kÞCðJ k J k1 þ 1Þ Y J D2 ;...;J 1 ;J 0 ðhD2 ; . . . ; h1 ; h0 Þ ¼ ðD1Þ=2 pffiffiffi CðJ k þ J k1 þ kÞ p k¼1 2 ðD2ÞJ D3
P J D2
ð1ÞJ
ðcos hD2 Þ P J 1 0 ðcos h1 ÞeiJ 0 h0 :
ð106Þ
To discuss the topological effect of a magnetic flux in quantum state, we note that the Gegenbauer polynomials T bn ðzÞ can be in terms of the hypergeometric function as follows [21, p. 783]: Cðn þ 2b þ 1Þ 1z b F n; n þ 2b þ 1; b þ 1; T n ðzÞ ¼ b : ð107Þ 2 2 n!Cðb þ 1Þ With the relation between the hypergeometric function F and the Jacobi polynomials P ðb;bÞ n [13, p. 212] 1z n!Cðb þ 1Þ ðb;bÞ P F n; n þ 2b þ 1; b þ 1; ðzÞ; ð108Þ ¼ 2 Cðn þ b þ 1Þ n the Gegenbauer polynomials T bn can be expressed as T bn ðzÞ ¼
Cðb=2ÞCðb=2 þ 1=2ÞCðn þ 2b þ 1Þ ðb;bÞ pffiffiffi P n ðzÞ: 2 pCðbÞCðn þ b þ 1Þ
Thus we have
ð109Þ
D.-H. Lin / Annals of Physics 323 (2008) 317–336 ðkÞJ k1
P Jk
ðzÞ ¼ ð1 z2 Þ
335
J k1 =2
CðJ k1 =2 þ ðk 1Þ=4ÞCðJ k1 =2 þ ðk 1Þ=4 þ 1=2ÞCðJ k þ J k1 þ kÞ pffiffiffi 2 pCðJ k1 þ ðk 1Þ=2ÞCðJ k þ ðk 1Þ=2 þ 1Þ ðJ
þðk1Þ=2;J k1 þðk1Þ=2Þ
P J kk1 J k1
ðzÞ: ð110Þ
The spherical harmonics in D space then has the following convenient representation Y J D2 ;...;J 1 ;J 0 ðhD2 ; .. .; h1 ;h0 Þ ¼
1
"
D2 Y
ð2J k þ kÞCðJ k J k1 þ 1ÞCðJ k þ J k1 þ kÞ
#1=2
pffiffiffi 2ðD1Þ=2 p k¼1 2ðk1Þ C2 ðJ k þ ðk 1Þ=2 þ 1Þ " # J D2 Y hk hk k1 ðJ k1 þðk1Þ=2;J k1 þðk1Þ=2Þ cos sin P J k J k1 ðcos hk Þ eiJ 0 h0 : 2 2 k¼1 ð111Þ
It is easily to check the orthonormal relation Z ^ ðDÞ Y J 0 ;...;J 0 ;J 0 ðhD2 ; ; h1 ; h0 ÞY dX 1
D2
J D2 ;...;J 1 ;J 0 ðhD2 ; . . . ; h1 ; h0 Þ
0
¼ dJ 0D2 J D2 dJ 01 J 1 dJ 00 J 0 :
ð112Þ
Finally, we find the summation of product of spherical harmonics Y J D2 J^ can be expressed as X Y J D2 J^ ð^ xb ÞY J D2 J^ ð^ xa Þ ^ J
¼
J D2 X J D3 ¼0
J1 X
Y J D2 ;...;J 1 ;J 0 ðhD2 Þb ; ...; ðh1 Þb ;ðh0 Þb
J 0 ¼J 1
ðhD2 Þa ;. ..;ðh1 Þa ; ðh0 Þa ( ) J D3 J D2 J2 J1 D 2 Y X X X X 1 ð2J k þ kÞCðJ k J k1 þ 1ÞCðJ k þ J k1 þ kÞ ¼ ðD1Þ 2k1 C2 ðJ k þ ðk 1Þ=2 þ 1Þ p k¼1 J D3 ¼0 J D4 ¼0 J 1 ¼0 J 0 ¼J 1 2 ( J D 2 Y ðhk Þb ðhk Þa ðhk Þb ðhk Þa k1 iJ 0 ½ðh0 Þb ðh0 Þa cos sin sin e cos 2 2 2 2 k¼1 ) Y J D2 ;...;J 1 ;J 0
ðJ
þðk1Þ=2;J k1 þðk1Þ=2Þ
P J kk1 J k1
ðJ
þðk1Þ=2;J k1 þðk1Þ=2Þ
ðcosðhk Þb ÞP J k k1 J k1
ðcosðhk Þa Þ :
ð113Þ
The representation is convenient to discuss the topological effect of a magnetic flux in the quantum system. References [1] H.V. McIntosh, Symmetry and Degeneracy, in: E.M. Loebl (Ed.), Group Theory and its Applications, vol. II, Academic Press, New York, pp. 1968–1975, and references therein. [2] W. Hamilton, On the application of the method of quaternions to some dynamical questions, in: Proc. Roy. Irish Acad. 3 (1847) 441–448; see also (Sir) W.R. Hamilton, Elements of Quaternions, Longmans, Green, New York, 1866.
336 [3] [4] [5] [6]
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