Some applications of Wigner’s D-matrix to the addition formulas for the Spherical harmonics And to Wigner D-matrix for the SO(4) group

Some applications of Wigner’s D-matrix to

The addition formulas for the Spherical harmonics

And to Wigner D-matrix for the SO(4) group

M. Hage-Hassan

Université Libanaise, Faculté des Sciences Section (1) Hadath-Beyrouth

Abstract

In this work we use the Analytic Hilbert space (Fock space) and Wigner's D-matrix and its generating function as starting point to determine the generating functions of spherical harmonics and the 3-j symbols of SU (2) .We find also the well known addition formula for the spherical harmonics. We find also another addition formula for the spherical harmonics. By applying the homomorphism of SO(4) group with the groups

) 2 ( ) 2 ( SU

SU ⊗ and the generating functions of the 3-j symbols of SU(2) we obtain the generating functions of Wigner D-matrix for the SO(4) group. We calculate also particulars cases of Wigner D-matrix of SO (4). Finally, we find a new derivation of the Generating function for Legendre polynomials. It is important to emphasize that our method is elementary and it is very useful in teaching of angular momentum.

1. Introduction

Many scientists have studied the angular momentum for long time ago by using different methods [1-8] and angular momentum topic is part of the course of quantum mechanics [8-17]. However, the representation matrix elements of finite rotations and its symmetries were determined by Wigner [1] and then by other scientists [8-10]. They find also that the Spherical harmonics are particular case of Wigner’s D-matrix. The spherical harmonics are particularly important for representing physical quantities of interest in physics [11-19]. Nevertheless a simple derivation of addition formula for the spherical harmonics remains to be done from the couplings of angular momentum [20].

On other side the representation theory of SO (4) is also very important in physics [21-23], but the Wigner's D-matrix of SO (4) remains to be find by a simple method [21]. We emphasize that this work is a continuation of a previous work on the symmetries of Wigner's D-matrix and the Gaussian equation of hypergeometric functions [10].

In this work we use the generating function method which consists of working in the analytic Hilbert space, or Fock space, and the generating function, G.F, [24].

I- We compute easily the Wigner’s D-matrix of SO(3) using the representation of SU(2). In addition, we derive the G.F of Spherical harmonics. Moreover after the integration of the product of three G.F of Wigner’s D-matrix we derive the G.F of 3-j symbols of angular momentum [7, 8]. And using the G.F of 3-j symbols and the G.F of Spherical harmonics we find after a simple’s calculation the well known the addition formula for the spherical harmonics and a second addition formula forYlm(r₁ r₂)

r r + .

space we find the G.F of Wigner's D-matrix of SO (4). Furthermore, we determine the expressions of particulars cases.

Finally, we find a new derivation of the Generating function of Legendre polynomials. The plan of this paper is as follows: in section two we expose the derivation of the generating functions of Wigner's D-matrix and the generating function of spherical harmonics. In section three we find the generating functions of 3-j symbols and the spherical harmonics. In section four, the generating function of the addition formula for the spherical harmonics is derived. The well known addition formula for the spherical harmonics is derived in part five. The new addition formula for the spherical harmonics is finding in part six. In section seven we expose the generators and the basis of SO(4). We derive the generating functions of SO(4) in section eight. We calculate in part nine the particular cases. In part ten we expose a new derivation of generating function for Legendre polynomials.

2- The

generating functions of Wigner's D-matrix and

The Spherical

harmonics

We summarize the determination of Wigner's D-matrix and spherical harmonics and its generating functions.

2.1 The generating function of Wigner's D-matrix of SU(2) A- The Analytic Hilbert space

The Analytic Hilbert space or Fock space [8, 24] is defined by F_n =

{

f_n=u1×...×u_n

}

with scalar product is:

      

(

f_n',f_n

)

=

∫

f_n'

( )

u f_n

( )

u dμ_n(u)

       (2.1)

Withdμn

( )

u is the measure of integration, and:

  

( )

n _{i i i i} i i n i i i n n u uu dx dy u x iy d _⎥ = + ⎦ ⎤ ⎢ ⎣ ⎡₋ =

∑

∏

₌ = , exp ) ( ₁ 1 1 π μ (2.2)

Note that for n = 1 the Analytic Hilbert space is a subspace of the cylindrical basis of the harmonic oscillator .

B-The representation of SU(2) in the analytic Hilbert space

By analogy with the orbital angular momentum Lrwe define the generators of SU(2) by:

1 1 , 1 1 1 , ( 1 1 )/2 2 2 1 1 3 1 2 2 1 2 1 2 2 1 1 z z z z J z z z z i J z z z z J ∂ − ∂ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ − ∂ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ − ∂ = (2.3) And ( 1 1 ) 2 1 2 2 1 1 2 z z z z J ∂ + ∂ = r We find that:         ( ) ( 1) ( ), 3 ( ) ( ) 2 u m u J u j j u J ϕjm = + ϕjm ϕjm = ϕjm r        (2.4)    

)! ( )! ( ) ( ) ( 2 ) ( 1 m j m j u u u m j m j jm − + = + − ϕ (2.5) The functions

{

ϕ_jm(u)

}

are orthonormal basis isomorphs to the Schwinger realization of SU(2) in terms of boson operators [7,8], consequently to facilitate the calculations we follow this representation ,ϕ_jm(u), in the rest of this work.

C- The Wigner’s D-matrix of the rotation and it’s generating function

The Wigner’s D-matrix of the rotation _R (Ω)=_eiψJz_eiθJy_eiϕJz

3 is given by:           ( )

(

( )

)

( ,' )( ) , ( ) ' 3 ' } ,' ( ϕ ϕ θ ψθϕ ϕ ψ _Ω= = Ω = Ω R e d e With D jmm im im jm jm j m m (2.6)

By multiplying ϕjm_'(v)ϕjm(u)and summing with respect to j, m, m’ we find the G.F of Wigner's D3-matrix in terms of the representation matrixρR3(Ω)=

( )

Z

(

)

_⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = =

∑

2 1 1 2 2 1 2 1 ' , , '( ) ( ,' }( ) ( ) exp ) , , ( u u z z z z v v u d D v z G _jm m m j j m m jm ϕ ϕ ξ ν (2.7) With , 2 sin , 2 cos ( )/2 2 2 2 2 / ) ( 1 1 1 ϕ ψ ϕ ψ _ρ θ θ ρ + _{= + =} − = + = i i e iy x z e iy x z         (2.8) And D d =ρ Djmm Ω d =ρΩ j j m m,' }( ) ( ,' }( ), ( (2.9)  Moreover, the group SU (2) being unitary thereforeZt.Z = ρ2. (2.10)

2.2 The generating function of spherical harmonics 2.2.1 The spherical harmonics

The rotation group SO(3) leaves invariant the quadratic form

∑

3₌₁ 2

i xi . And it’s

well known in Quantum mechanics that the orbital angular momentum is:         Lr=rr×pr =D23ir+D31 rj+D12kr WithDij =

(

xipj −xjpi

)

.

The commutators of the generators of SO(3) are:

[

L_x,L_y

]

=i_hL_x,

[

L_y,L_z

]

=i_hL_x,

[

L_z,L_x

]

=i_hL_y (2.11) And the Casimir operator 2 2 2 2

z y

x L L

L

Lr = + + commutes with , , .

The spherical harmonics are the eigenfunctions of angular momentum operators

z L and Lr2 . We found that:       LzYlm(θϕ)= mYlm(θϕ), L2Ylm(θϕ)=hl(l+1)Ylm(θϕ) r h        (2.12)

The relationship between ₍l _{,' )}(ψθϕ)

m m

D andYlm(θϕ) is well known [15-19]:

( ), ( ) (cos ) 1 2 4 ) ( _(0,0) 2 / 1 ) , 0 ( θϕ θ π ψθϕ l _l lm l m Y D P l D ⎟ Ω = ⎠ ⎞ ⎜ ⎝ ⎛ + = (2.13) ) (cosθ l P is Laguerre polynomial.

2.2.2 The generating function of spherical harmonics

We put ν₁ =t ν₂ =t in (2.7) then we find the generating function of spherical harmonics.

∫

(

)

_⎥ = ⋅ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 2 ] ) ( exp[ ) ( exp 2 1 1 2 2 1 a r t d u u z z z z t t r r μ (2.15) With ] ( ) ( ) 1 2 4 [ ] 2 ) ( exp[ 2 1 r Y u l r a lm lm lm r r r ϕ π

∑

₊ = ⋅ (2.16) ) , (u₁ u₂ a

ar = r Is a vector of length zero,ar⋅ ar=0 and has the components

a

-

u

u

,

a

i(u

u

),

a

3

2

u

1

u

2 2 2 2 1 2 2 2 2 1 1

=

+

=

−

+

=

(2.17) And r =

(

x,y,z

)

, x=z1z2 +z2 z1, y =i(-z1z2 +z2 z1 ) r _(2.18)        z=z₁z₁-z₂z₂ =rcosθ,With r=ρ2. 

3- The generating functions of 3-j symbols

And the Coupling of angular momentum

 

We shall make a revision of the determination of generating functions of the coupling angular momentum. We use the analytic Hilbert space and Wigner's D-matrix.

3.1 The coupled function of two angular momentums.

The function of two-coupled angular momentum in the Hilbert space of the analytic functions is:    Ψ

( )

, =

∑

, ( ) ( ) ( ) 2 2 1 1 2 1 3 2 1 ) 1 1 2 2 1 2 3 3 (jj jm u v mm jm j m j j j m ϕjm u ϕjm v (3.1) And D_(mj3 _,m)( ) (j₁j₂)j₃m'R(j₁j₂)j₃m 3 ' 3 Ω = (3.2)

The coefficient

j

1

m

1

,

j

2

m

2

(

j

1

j

2

)

j

3

m

3 is the Clebsh-Gordan.

Wigner introduced the 3-j symbols which are very useful for following and are related to the Clebsh-Gordan coefficients by:

1 1 2 2 1 2 3 3 2 / 1 3 3 2 1 3 2 1

₍

₁

₎

_{1 2 2}

₍

₂

₁

₎

_,

₍

₎

m

j

j

j

m

j

m

j

j

m

m

m

j

j

j

_{j j m}

−

+

−

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

_{− − −} (3.3)

3.2 The generating function of 3-j symbols

We will deduce the G.F of 3-j symbols by integration of the product of three G.F of Wigner's D3-matrix (2.7). So we write:

3 3₁[ ( , , )] ( ₁) ( ₂) _{( )}( ) _{( )}( ) 3 2 1 3 2 1 3 2 1 u H y H z d z d u z y G G _{j j j} i j j j j j j i i i

∫ ∏

= =

∑

=

μ

μ

(3.5) With the help of the expression

∫

exp[

α

z+

β

z]d

μ

(z)=eαβwe find:

G3 =

∫

G(t,u)× G(t, y)dμ(t) (3.6) With ( , ) exp

[

[ , ] [ , ] [ 2, 3]

]

3 3 1 2 2 1 1 u u t u u t u u t u t G = + +

The functionG( ut, ) is the G.F of 3-j Symbols.

Then, we find after the development ofG( ut, )the general Van der Wearden invariant is.

) 2 ( )! 2 ( )! 2 ( )! 1 ( ] , [ ] , [ ] , [ ) ( 3 2 1 ) 2 ( 2 1 ) 2 ( 1 3 ) 2 ( 3 2 ) ( 3 2 1 3 2 1 j J j J j J J u u u u u u u H j J j J j J j j j − − − + = − − − (3.7) And ( )

[

( )

]

, 3 2 1 3 2 1 3 1 , ) (123 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =

∑ ∏

₌ m m m j j j u u H i i i m i i m j j j j

ϕ

(3.8)

3.3 The generating function of coupling two angular moments    

We deduce from the above expression and G(u,v,z) the generating function of couplings of two angular moments [7, 8]: 

G

c

(

u

,

v

,

z

)

=

exp[{

γ

[

u

,

v

]

+

ξ

1

(

z

,

v

)

+

ξ

2

(

z

,

u

)}]

=

( )

z

(

)

(

u v

)

j jm j j j jj jm jm j j , 1 2 1 3 2 1 2 1 2 1 ) ( 2 1 Ψ Φ +

∑

ϕ γξ ξ (3.9) With _{j j j}

(

γξ ξ

)

γ p j ϕ _jq

( )

ξ j p j p ₃ 3 3 2 1 ( 1)! _{( )!} 3 ) ( 3 2 1 − + + = Φ − (3.10) And J = j₁+ j₂ + j₃,p= j₁+ j₂,q= j₁− j₂ (3.11)

4. The generating function of the addition formula

For the Spherical harmonics

The function of the couplings of two spherical harmonics is:

Using the formula of spherical harmonics (2.13) and the generating function of the couplings of two angular momentums (3.9) we deduce the generating function of the addition formula for the spherical harmonics:

    I =

∫

{

exp

[

t1t2 +t3t4

]

exp[+{γ[u1v2−u2v1]+ξ1

(

z'1u1+z'2u2

)

+ξ1

(

z'1v1+z'2v2

)

] 

(

)

(

)

( ) 2 1 3 4 4 3 4 3 2 1 1 2 2 1 2 1 d t v v z z z z t t u u z z z z t t

μ

⎪⎭ ⎪ ⎬ ⎫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − +

( ) ( )

_{( )}

₍

₎

_{( )}

_× ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ Ψ Φ + × ⎩ ⎨ ⎧ =

∫

∑

z u v j j t t i t t m j j j m j j j jm jm j j j i , ' 1 2 1 ! ! 123 12 33 2 1 ) ( 2 1 3 4 3 2 1 ϕ γξξ    

{

[

_'

(

₁, ₂

)

( ', )( ₁)

(

₁, ₂

)

]

[

_'

(

₃, ₄

)

( ', )( ₂)

(

₁, ₂

)

]

}

( ) 2 2 2 2 2 2 2 1 1 1 1 1 1 1 t t D d u u t t D m m d jm v v d t j m j m j m m j m j ϕ ϕ ϕ μ ϕ (4.2) With d₁ =

(

ρ₁Ω₁

)

,d₂ =

(

ρ₂Ω₂

)

_(4.3) Taking into account (2.17) we write that:

r1 =

(

x1=z1z2 +z2 z1 ,y1=i(-z1z2 +z2 z1 ),z1 =z1z1-z2z2

)

r

rr₂ =

(

x₂ =z₃z₄ +z₄ z₃ ,y₂ =i(-z₃z₄ +z₄ z₃ ),z₂ =z₃z₃-z₄z₄

)

_(4.4) Two cases arise to study: the first case is j3 = 0, hence j1= j2 =l, and the second case is j₃ = j₁+ j₂orj'3= j1+ j2.

5- The First addition formula for

The spherical harmonics

We consider the case j3 = 0, so we find as result the following expression:

[

]

{

exp _{1 2 3 4} exp[ { [ _{1 2 2 1}]] 2 tt t t u v u v I =

∫

+ +

γ

−

(

)

(

)

( ) ( ) () 2 1 3 4 4 3 4 3 2 1 1 2 2 1 2 1 d u d v d t v v z z z z t t u u z z z z t t μ μ μ ⎪⎭ ⎪ ⎬ ⎫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − × (5.1)

5.1- The integration with respect to u1, u2 and v1, v2 gives:

5.2 The integration with respect to t1 and t2 give: I2 =

∫

{

exp

( )

t3t4 exp[a(t3,t4).Λ]dμ(t3,t4) r r (5.3) With Λr =Xir+Yrj+Zkr, and Z=rr₁.rr_{2 (5.4)}

After developing the quantity under the integral and using the formulas (2.12). We find:

( )

( ) ( )

(

) (

!

)

! ( ) ( , ) 2 ! (0, ) 3 4 4 3 4 3 2 D d t t m j m j t t i t t I j j m jm m j m j j i μ

∫ ∑

∑

⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Λ − + = − + − r (5.5) The integration with respect to t3 and t4 give us the well-known formula for addition

formula the spherical harmonics: ) ˆ ( ) ˆ ( 1 2 4 ) ˆ . ˆ (_{1 2} Y r₁ Y r₂ l r r P _lm l l m lm l

∑

− = ∗ + =

π

(5.6)

6- The second addition formula for

The spherical harmonics

We will prove that the expression Ylm(r₁ r₂) r

r + is the second addition formula. We consider the case j3 = j1 + j2 consequently (4.2) becomes:

[

]

{

(

)

(

)

(

)

(

)

( ) ] ' ' ' ' exp[ exp 2 1 3 4 4 3 4 3 2 1 1 2 2 1 2 1 2 2 1 1 2 2 2 1 1 1 4 3 2 1 3 t d v v z z z z t t u u z z z z t t v z v z u z u z t t t t I

μ

ξ

ξ

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − × + + + + + =

_∫

(

)

(

)(

)(

)

( )

(

)

× ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Φ + + + + =

∑

₌1₊ 2 1 2 2 1 2 1 0 ' 1 2 1 2 1 2 4 2 1 2 1 ξ ξ ϕ π j j j j j jm jm j j z j j j j

[

j₁m₁j₂m₂ (j₁j₂)

(

j₁ j₂

)

mYl₁m₁(r₁)Yl₂m₂(r₂)

]

}

r r + (6.1) With

(

)

(

(

)

)

( ) ( )

_{( ) ( )}

. ! 2 ! 2 ! 1 2 0 2 1 2 2 2 1 2 1 2 1 2 1 2 1 2 1 j j j j j j j j j j j ξ ξ ξ ξ = + + Φ = + (6.2) And _⎥ × ⎦ ⎤ ⎢ ⎣ ⎡ + = + + 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 1 . )! 2 2 ( )! 2 ( )! 2 ( ) ( j j j j m m j j j j m j m j 2 1 2 2 2 2 1 1 1 1 2 1 2 1 2 1 2 1 _. )! ( )! ( )! ( )! ( )! ( )! ( ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + − + − − + + + + m j m j m j m j m m j j m m j j (6.3)  

The calculations of the first part of I3 with the help of (2.15) we find that:

And j=l,l₁ = j₁,l₂ = j₂,l =l₁+l₂.

By putting ξ₁=ξ₂ =1 we find the second addition formula for the spherical harmonics:       

(

(

)

)

(

+

)(

+

)

× + + = + 1 2 1 2 1 2 4 ) ( 2 1 2 1 2 1 j j j j r r Y_lm r r π       . ( ) ( ) )! ( )! ( )! ( )! ( )! ( )! ( 2 1 , 2 1 2 2 2 2 1 1 1 1 2 1 2 1 2 1 2 1 2 2 1 1 2 1 _{j m j m j m j m} Y r Y r m m j j m m j j m l m l m m r r

∑

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + − + − − + + + + (6.5)

7- The generators of SO(4) group and the couplings

Of two angular moments

The group SO (4) has been studied by several authors and since long time [21] we are going to give a summary of these works in the following section.

7.1 Summary on the generators of the rotation SO(4)

The rotation group SO (4) leaves invariant the quadratic form, which implies in analogy to the angular momentum of SO (3) that the six infinitesimal generators of SO (4) are written in the form:

(

)

( )

( )

( )

i i ij t i j j i ij x p a X a p x p x D ∂ ∂ = = − = , (7.1) With

( )

t

a is the transpose of the matrix

( ) (

a = x1 x2 x3 x4

)

. And Lr=D23ir+D31 rj+D12kr,        A D i D j D k r r r r 34 24 14 + + = _(7.2)

The relations of the commutations are thus:

[

Li,Lj

]

=iεijkLk,

[

Li,Aj

]

=iεijkLk,

[

Ai,Aj

]

=iεijkLk, (7.3) In order to construct irreducible representations of SO(4) it is convenient to introduce another set of generators for this group, i.e.:

      

(

)

(

)

, 2 1 , 2 1 A L N A L Mr = r+ r r = r− r        (7.4)  The commutation relations for the generators Mr and Nr can easily be obtained,

using the relations (2.12)

[

Mi,Mj

]

=iεijkMk,

[

Ni,Nj

]

=iεijkNk,

[

Mi,Nj

]

=0 (7.5)

The above commutation relations is equivalent either to SO(3) or to SU(2), ThereforeMr₁ and N2

r

are analogous to kinetic moments and:

2 _{2 2 2 2 2 2} 1 1 1 1 1 1 2 _{( 1) , ( 1)} m l l l m l N m l l l m l Mr = + r = + (7.6)

With L (l1l2)lm l(l 1)(l1l2)lm , Lz (l1l2)lm m(l1l2)lm

2 _{= + = =}

r

(7.7)

7.2 The homomorphism of SO(4) ≅SU(2) ⊗ SU(2)and the coupling function

The rotation R3 of SO (3) is generalized to R4 of SO (4) by an expression of

the form [23]: R₄ =Rr

(

ψ'θ'ϕ'

)

XR₃(0θϕ)with the condition[X,Lz]=0. However in this

work we use the homomorphism of SO (4) withSU(2)⊗SU(2). And we follow the angular momentum notations,Jr₁,Jr₂,Jr₃ = Jr₁+Jr₂ et X =K =

( ) ( )

J_{1 z} − J_{2 z}, with

[

Jz,K

]

=0, instead of: M, NetA3 =M3−N3 r

r

. The R4 rotation in this case is:

_{R e}iψ'Jz_eiθ'Jy_eiϕ'Jz_eiχk_eiθJy_eiϕJz

4 = (7.8) And the elements of the matrix are:

(

)

(

)

=

∑

{

( )

(

' '0

)

× '' '' '' , 3 3 2 1 4 3 ' 3 2 1 ' 3 3 ψ θ m j j m m D m j j j R m j j j

(

)

χ

(

)

j

(

ψθϕ

)

}

m m k i D m j j j e m j j j1 2 '' '' 1 2 ' '' ' '' '',' (7.9) D... (...) are the elements of SU (2).

We will then determine the elements of the matrix i k

eχ using the method of the generating function. However the coupling function of SO(4) ≅ SU(2) ⊗ SU(2)in the Hilbert space of the analytic functions is

( )

1 2

) (j1j2 jm t ,t Ψ with ( , ), ( 3, 4) 2 2 1 1 t t t t t t = = and ti =xi+iyi. Furthermore ( ) , ( ) 2, 2 / 2 1 2 / 1 e t e t e t t eiχk = iχ iχk = −iχ /2 ₄ 4 3 2 / 3) , ( ) (t e t e t e t eiχk = −iχ iχk = +iχ (7.10) And Ψ_{( )}

( )

=

∑

_{1 1}, _{2 2} ( _{1 2}) _{3 3} (₁, ₂) (₃, ₄) 2 2 1 1 2 1 3 3 2 1j jm t mm jm j m j j j m jm t t jm t t j ϕ ϕ (7.11)

It is important to emphasize that the choice ofR₄ is not unique [23]. and j1, j2, j3 are integer or half-integer.

8- Generating function of Wigner D-matrix

Elements of SO(4)

≅

SU(2)

⊗

SU(2)

 

We shall calculate in this part the G.F of the representation matrix of SO (4). Then we will deduce the expressions of particular cases of Wigner D4-matrix.

8.1 The Generating function of Wigner D4-matrix Elements of SO(4)

The expression of the elements of the matrix of i k

In this case the generating function is:        =

∫

{

[

+

(

)

+

(

)

]

× 2 2 1 1 2 1 3 exp '[t ,t ] ' z'.t ' z'.t I γ ξ ξ                exp[ [ , ]+

( ) ( ) }

. + 2 . 2 ( )= 1 1 2 1 t d t z t z t t eiχk γ ξ ξ μ   

(

)

(

)

[

( )

( )

Φ

(

)

Φ

(

)

]

× ⎪⎩ ⎪ ⎨ ⎧ + +

∑

_' 1 2 1 2 3 3 ' ' ' ' 1 2 1 2 1 ' 3 2 1 3 2 1 3 ' 3 3 3 3 ' 3 3 2 1 ξ ξ γ ξ γξ ϕ ϕjm jm jj j _{jj j} m j j j j z z j j       

( )

( )

( )

}

3 3 2 1 3 ' 3 2 1 ) ( ) (jj jm t jj jm t dμ t

∫

Ψ Ψ (8.2) Using Gauss's formula:

[

]

(

det( )

)

exp( ) exp 1 1 1 1dxdy V XV AV V B X A X B t t t t n i i i n − − = − + + = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

∫∏

π (8.3) With

( )

, 4 3 2 1 ⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = t t t t V   ⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − = ₋ 0 0 0 0 0 0 0 ' 0 0 ' 0 0 0 χ χ γ γ γ γ i i e e X ,  ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = − 2 2 1 2 2 2 1 2 1 1 ' ' z z e z e z A i i ξ ξ ξ ξ χ χ ,  ⎟⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = − 2 2 2 2 1 2 2 1 1 1 χ χ ξ ξ ξ ξ i i e z e z z z B         (8.4) 

We find the expression of the generating function of Wigner D4-matrix:

I3 =

(

)

[

]

⎥× ⎦ ⎤ ⎢ ⎣ ⎡ + × − ) det( ' ) sin( 2 exp ) det( 1 2 1 2 ' 2 ' 1 ' 2 ' 1 1 X z z z z i X χ γξξ γ ξ ξ (8.5)

(

)

× ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ' − ' ' /det( ) ' exp ' ₁ 1 2 2 2 1 2 2 2 1 1 1z e z z e X z ix ix γγ ξ ξ ξξ

(

)

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + − ) det( / ' ' ' ' exp ' _{1 2 2} 1 2 1 2 2 2 1 1 1z e z z e X z ix ix γγ ξξ ξ ξ     

With _{det( )=}

(

_{1−2(γγ')cos}

( )

_{χ +(γγ')}2

)

X

9- Expressions of particular cases of Wigner

D-matrix elements of SO (4)

 

We will calculate the Expressions of the particular cases of the Wigner D4-matrix

(

)

' ₃

(

_{1 2}

)

_{3 3} 3

2

1j j m e j j j m

j iχk _(9.1)

I- We consider the case j3 = j3’= 0

If we consider the case j3 = j3 ' = 0 we find that the generating function is:

(

1 2( ')cos

( )

( ')2

)

1 γγ χ γγ + − (9.2) This expression is the generating function for Gegenbauer polynomials [25, 26]:

(

)

) ( 2 1 0 2 t C t _n n n λ λ α α α

∑

∝ = − = + − (9.3) Whether λ=1 we find the first expression of Wigner D4-matrix:

(

)

(

)

_⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = (cos ) 1 2 1 00 00 1 2 1 2 1 2 1 1 χ χ j k i C j j j e j j (9.4)

II- We consider the case j3’= 0

The generating function of

(

j₁j₂

)

00eiχk

(

j₁j₂

)

j₃0 is:

(

( )

)

(

)

( )

(

)

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − + − 2 2 1 2 1 2 _{1 2( ')cos ( ')} sin ' 2 exp ) ' ( cos ) ' ( 2 1 1 γγ χ γγ χ ξ ξ γ γγ χ γγ z z i (9.5) In doing the development of (9.5) and taking into account of the relations:

( ) (

_{( )}

)

(

)

( )

(

)

!

(

( )

)

! , ! ₃ 2 1 3 1 2 1 3 2 1 0 3 3 3 2 1 3 3 j j p j z z z j j p j j j j j ξ ξ γ ξ γξ ϕ − = Φ = + +

(

)

( )

( )

! ' ' ' ' _{1 2} ' 3 2 1 _p p j j j γ ξ ξ γ = Φ _(9.6) We find another expression of Wigner D4-matrix of SO(4):

(

)

(

)

_{(2 1)( )!}

(

sin

)

(cos( )) )! 1 )( 1 ( 2 0 00 1 3 3 3 3 2 1 2 1 3 3 3 3 χ χ χ + − − + + + + = j j p j j k i C j p j j p p j j j e j j _(9.7)

10- Appendix. A new derivation of generating function

For Legendre polynomials

We consider the integral:

exp

[

(

2

)

]

exp

(

)

() ( ₁) ( ₂) 2 1 1 2 2 1 2 1 2 1 2 1 4 d t d u d u u u z z z z t t t t u u I

∫

λ _⎟⎟ μ μ μ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = _(10.1)

  And a r a x a y a3'z ' 2 ' 1 ˆ '⋅ = + + r

Let us remember that: dμ(u1)=exp

(

−u1u1

)

dudv=exp

[

−

(

u2+v2

)

]

dudv Using the Gauss integral:

    

( )

A xx dnx

( )

A n j i j i ij n det / 1 exp 1 , 2 / ₌ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −

∫

∑

= − π (10.4) We find

(

)

(

) (

)

₍

₎

_⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + − − + − = ⋅ + + v u z iy x i x i z iy v u r a v u 1 1 ) ˆ ' ( 2 2 λ λ λ λ λ r (10.5) And det

( )

A =1+λ2 −2λz (10.6) After integration of (10.3) we find the Generating function for Legendre polynomials =

∑

− + l l l P z ) (cos 2 1 1 2 _λ λ θ λ (10.7)

10-References

[1] Wigner E P 1931 Gruppentheorie und ihre Anwendung auf die Quantenmechanik Der atomspektren », (Braunschweig: Vieweg)

[2] V. L. Van der Waerden, Die Gruppentheoreische Method in der Quantenmechanik, Springer, 1931.

[3] Racah, G., Theory of complex spectra II. Phys. Rev. 62,438-462 (1942). [4] Rose, M. E., “Multipole fields,” p. 92. Wiley, New York, 1955.

[5] Cartan, E., « Leçons sur la théorie des spineurs I,» , Hermann, Paris, 1938. [6] Hermann Weyl, “The classical groups”, Princeton University Press 1946

[7] J. Schwinger,”On angular Momentum”, in quantum Theory of Angular Momentum, Ed. L.C. Biedenharn and H. Van Dam (New York: Academic Press)

[8] V. Bargmann, Reviews of modern Physics, V: 34, 4,300 (1962). [9] T. Regge, “Symmetry properties of Clebsch-Gordan’s coefficients”, Nuovo Cimento 10,544 (1958).

[10] M.Hage-Hassan “Deriving Euler’ identity and Kummer’s series of Gauss’s

hypergeometric functions using the symmetries of Wigner d-matrix.

(reaserchGate)

[11] A. Messiah, « Mécanique Quantique Tomes I et II », E. Dunod, Paris (1965)  Richard Feynman, “Lectures on Physics” Vol. III

[12] W. Greiner,“Quantum Mechanics (An introduction) “Ed. Springer (1994) [13] J. Sakurai, “Modern Quantum Mechanics”, Ed. Addison Wesley (1994) [14] L. D. Landau and E. M. Lifshitz. “Quantum Mechanics: Non-relativistic Theory”, Ed. Pergamon Press, Oxford, (1977).

[15] N. J. Vilenkin, “Fonctions spéciales et théorie de la représentation des groupes” Dunod (1969).

[16] M Chaichian and R. Hagedorn, “Symmetries in Quantum Mechanics”.

[17] M.E. Rose, “Elementary Theory of Angular Momentum”, John Wiley and Sons, Inc., New York, 1957

[18] A. R. Edmonds, “Angular Momentum in Quantum Mechanics” Princeton, U.P., Princeton, N.J., (1957)

[20] M. Hage-Hassan, G. Grenet,J-P Gazeau and M. Kibler “Formula for Ylm(r1 r2) r r × ”. J. of phys. A: 13(1980).

[21] L.C. Biedenharn, “Wigner Coefficients for the R4 Groups”, J. of Math. Physics

V: 2, n.3 (1961)

[22] D.E. Littelewood, The Theory of Group Characters, ClarendonPress, oxford,(1950) [23] M.Hage-Hassan “On the Euler angles for the classical groups and Schwinger approach for SU(3) ” Arxiv : 0805.2740.

[24] M. Hage-Hassan, “Generating function method and its applications to Quantum, Nuclear and the Classical Groups, Arxiv : 1203.2892.

[25] I. Erdelyi, ”Higher transcendal functions “Vol. 1, Mac. Graw-Hill, New-York (1953).