On Wigner's D-Matrix and Angular Momentum

On Wigner’s D-matrix and Angular Momentum

Mehdi Hage-Hassan Lebanese University, Faculty of Sciences - Section 1 Hadath - Beirut

Abstract Using the Analytic Oscillator space (Fock Bargmann space) and the generating function of Wigner’s D-matrix we will determine the generating functions of spherical harmonics and the 3-j symbols of SU(2). We will find also a relation between the Gegenbauer Polynomials and Wigner’s d-matrix. We drive the 3-j symbols and we find simply the expression of 3-j symbols with magnetic momentum nulls. We will find the generating function the 3 − lm symbols of SO(3) and a new relation between the 3-j symbols. We will derive also the generating function of 6-j symbols. We will apply our method for the determination of the generating function and the spherical basis of four dimensional R4 Harmonic oscillators.

1 Introduction

The Harmonic oscillator and the Angular Momentum are parts of the course of quantum mechanics and has been studied since time ago by many scientists using different methods [1,7]. Racah has made an important contribution to the study of coefficients of 3-j and 6-j symbols. Furthermore Wigner fined the representation matrix of rotation. Then Schwinger studied the rotation groups starting from oscillator basis and found the generating functions of 3-j and 6-j. By analogy with the Schwinger work, Bargmann used the Fock space to study the angular momentum.

We observe that many spaces of the oscillators that have wave functions are function of complex numbers and it is conjugated. We will use this property to make a new simple study of angular momentum. For example, the two-dimensional isotropic harmonic oscillator with plane polar coordinates and the R4 wave function in terms of Euler-angles, and the angular part is the Wigner D-matrix [7].

This work starts with the determination of the generating function of Wigner D-matrix and the exploitation of its properties which are very useful for the study of the angular momentum. Then we deduce the generating function of spherical harmonics and we ﬁnd a relationship between the Gegenbauer Polynomials and Wigner’s d-matrix.

Starting from the integration of the product of three Wigner D-matrix and three spherical harmonics we will ﬁnd the 3-j symbols and the 3-lm symbols of SO (3) [8, 17]. We

1 will simply calculate 3-j symbols with magnetic momentum nulls [6, 10].

We ﬁnd a new relation between the 3-j symbols and we derive also the generating function of 6-j symbols [6, 16]. Moreover, we will apply our method for the determination of the generating function and the spherical basis for four dimension R4 Harmonic oscillators [7, 17].

It is important to emphasize that our method is simple and elementary. This paper is organized as follows: in section 2 the derivation of the spherical harmonics and the representation of SU(2) is illustrated. In sections 3 and 4 we will derive The generating function of Wigner’s D-matrix of SU(2), the generating functions of spherical harmonics and Legendre polynomials. In section 5 the relation between Gegenbauer polynomial and Wigner’s D-matrix will be presented. The well known Coupling of two angular momenta and the generating function of 3-j symbols are exposed in parts six and seven. In section 8 we expose the new derivation of the 3-j with magnetic moments mi = 0. We will derive the generating functions of 3−lm in section 9. Then we present the derivation of the new relation between the 3-j symbols in section 10. In section 11 we expose the derivation of the generating function of Racah coeﬃcients. In section 12 we will ﬁnd the generating function and the spherical basis of four dimensional of harmonic oscillator.

2 The spherical harmonics and the representation of SU(2)

We make a summary of the determination of spherical harmonics and the representation of SU(2) using the analytic Hilbert space [11, 16].

2.1 The spherical harmonics

The rotation group SO(3) leaves invariant the quadratic form: x2 + y2 + z2. And it is well known in the theory of angular momentum [1]:

−→ −→ −→ −→ −→ −→ L = r × p = Lx i + Ly j + Lz k (2.1)

And

Lx = ypx − zpy,Ly = zpx − xpz,Lz = xpy − ypx (2.2)

Where −→r the position is vector of the particle and −→p is the momentum. The commu- tators of the generators of SO(3) are:

bLx,Lyc = iηLz, bLy,Lzc = iηLx, bLz,Lxc = iηLy (2.3)

−→2 2 2 2 The Casimir operator L = L + L + L commutes with Lx,Ly,Lz. x y z −→ The spherical harmonics are the eigenfunctions of angular momentum operators L 2

2 and Lz. We found that

−→2 2 LzYlm(θϕ) = ηmYlm(θϕ), L Ylm(θϕ) = η l(l + 1)Ylm(θϕ) (2.4) In Dirac notation [1] we write:

Ylm(θϕ) = hθϕklmi (2.5)

2.2 The Analytic Hilbert space

The Analytic Oscillator space (or Fock space) deﬁned by: zn un(z) = √ (2.6) n!

With scalar product is: Z z¯n0 zn (un0 , un) = √ √ dµ(z) = δn,n0 (2.7) n0! n!

And dµ(z) is the measure of integration: 1 dµ(z) = ( )exp[−zz¯ ]dxdy , z = x + iy (2.8) π i i i Furthermore, it is easy to check the useful relationship: Z exp[αz + βz¯]dµ(z) = eαβ (2.9)

2.3 The representation of SU(2) in the analytic Hilbert space −→ By analogy with the angular momentum L we deﬁne the generators of SU(2) in the analytic Hilbert space, ui, i = 1, 2, by: 1 1 1 1 1 1 1 1 J1 = u1 − u2 ,J2 = u1 − u2 ,J3 = u1 − u2 ∂u2 ∂u1 i ∂u2 ∂u1 2 ∂u1 ∂u2 (2.10) And −→2 1 1 1 J = N(N + 1),N = u1 − u2 (2.11) 2 ∂u1 ∂u2 We ﬁnd that:

−→2 J ϕjm(u) = j(j + 1)ϕjm(u),J3ϕjm(u) = mϕjm(u) (2.12)

With ϕjm(u) is the representation of SU(2) in the analytic Hilbert space with:

(j+m) (j−m) u1 u2 ϕjm(u) = (2.13) p(j + m)!(j − m)!

3 The functions ϕjm(u) are orthonormal basis isomorphs to the Schwinger realization of SU(2) in terms of boson operators [6]:

(a+)j+m(a+)j−m |jmi = 1 2 |00i (2.14) p(j + m)!(j − m)!

−zz/2 It is important to note that the functions un(z)e are the wave functions of subspaces of the cylindrical basis of R2 of the Harmonics oscillators [18, 19, 22].

3 The generating function of Wigner’s D-matrix of SU(2)

We will determine the Wigner’s D-matrix and its generating function. We Show that the orthogonality of D-matrices is performed simply [11].

3.1 The Wigner’s D-matrix

The Wigner’s D-matrix of the rotation R(Ω) = eiψJz eiθJy eiϕJz is given in Dirac notations by: j im0ψ j imϕ D(m0,m}(Ω) = (jm|R(Ω)|jm) = e d(m0,m)(θ)e (3.1) We have [11] also

j+m j−m 2j (z1u1 + z2u2) (−z2u1 + z1u2) ρ R(Ω)ϕjm(u) = (3.2) p(j + m)!(j − m)!

with θ θ z = x + iy = ρcos ei(ψ+ϕ)/2, z = x + iy = ρsin ei(ψ−ϕ)/2 (3.3) 1 1 1 2 2 2 2 2 and

j 2j j D(m0,m}(z) = ρ D(m0,m}(Ω), Ω = (ψθϕ) (3.4)

n j o with (ψ, θ, ϕ) are the angles of Euler and D(m0,m}(z) is a subspace of spherical basis of R4 Harmonic oscillators [7, 22]. 2j j For m = j we ﬁnd that: ϕjm0 (u) = ρ D(m0,0}(Ω)

3.2 The generating function of Wigner’s D-matrix

0 Multiplying (2.10) by ϕjm0 (v) and summing with respect to j, m, m we ﬁnd the generating function of Wigner’s D-matrix

z1 z2 u1 X j G(v, z, u) = exp v1 v2 = ϕjm0 (v)D(m0,m}(z)ϕjm(u) (3.5) −z2 z1 u2 j,m,m0

4 3.3 Orthogonality of D-Matrices

The Orthogonality of D-Matrices is:

Z 0 0 0 z1 z2 u1 z1 z2 u1 exp v1 v2 0 + v1 v2 dµ(z) = −z2 z1 u2 −z2 z1 u2 2 Z X Y ji i ji i ji i ji i j1 j2 ϕ(mi)(v )ϕ(m0i)(v ) ϕ(mi)(v )ϕm0i(v ) D(m0,m}(z)D(m0,m}(z)dµ(z) (3.6) jm i=

with dµ(z) = 2exp(−ρ2)ρ3dρdµ(Ω) and 1 dµ(Ω) = dψsinθdθdϕ (3.7) 8π2 In addition the integration of the ﬁrst member of (3.6) is:

0 0 0 0 exp [(u2v2 + v1u1)(v1u1 + u2v2)] (3.8)

And, the integration of the second member on the variable ρ of (3.6), using the formula: Z ∞ 2n+1 −ρ2 ρ e dρ = n!/2, n = j1 + j2 + 1 (3.9) 0 After development of (3.8) and identiﬁcation with (3.6) we ﬁnd that: Z 2π Z π Z 2π 1 l l 1 D 0 (ψθϕ)D 0 (ψθϕ)dΩ = δm ,m δm0 ,m0 δj ,j (3.10) 2 (m1,m1) (m2,m2) 1 2 1 2 1 2 8π 0 0 0 2j1 + 1

4 The generating functions of spherical harmonics And Legendre polynomials

We will determine the generating function of spherical harmonics and the generating function of Legendre polynomials [20, 21].

4.1 The generating function of spherical harmonics

l The relationship between Dm0,m(ψθϕ) and Ylm(θϕ) is well known [7, 10, 11]:

1 4π 2 Dl (ψθϕ) = , (0,m) 2l + 1 and

l D(0,0)(Ω) = Pl(cosθ) (4.1)

Ylm(θϕ) is the spherical harmonics and is Laguerre polynomial. By putting m0 = 0 in (3.1) we ﬁnd a very useful expression for the rest of this work:

5 l hϕjm(v1, v1)|R(Ω)|ϕjm(t1, t2)i/j! = D(0,m)(ψθϕ) (4.2)

Then we ﬁnd

Z z z u (−→a .−→r ) exp v v 1 2 1 dµ(v) = exp (4.3) −z2 z1 u2 2

After integration we ﬁnd the generating function of spherical harmonics.

−→ −→ 1 ( a . r ) X 4π 2 exp = ϕ (u)Y (−→r ) (4.4) 2 2l + 1 lm lm lm

−→a = −→a (a) is a vector of length zero, −→a .−→a = 0 and has the components

2 2 2 2 ax = −u1 + u2, ay = −i(u1 + u2), az = 2u1u2 (4.5)

and −→ r = (x, y, z), x = z1z2 + z2z1, y = i(−z1z2 + z2z1), (4.6)

2 z = z1z1 − z2z2 = rcosθ, with r = ρ .

4.2 The generating function of Legendre polynomials

Put u2 = u1 in the expression (4.4) and u1 = v1 + iv2 we ﬁnd that:

Z 1 X exp x(−u2 + u2) − iy(u2 + u2) + 2z(u × u ) dµ(u) = rlP (cosθ) (4.7) 2 1 1 1 1 1 1 l l

A- We write the ﬁrst member of (4.7) in matrix form:

−i(y + iz) −ix −2ixv v + (−iy + z)v2 + (iy + z)v2 = v v v v 1 2 1 1 1 2 −ix i(y − iz) 1 2 (4.8) Let r = λ, we ﬁnd: Z −i(y + iz) −ix 1 exp λ v1 v2 v1 v2 d µ(u) = √ −ix i(y − iz) 1 − 2λcosθ + λ2 (4.9) B- We ﬁnd the generating function of Legendre polynomials [20, 21] after comparing the two sides of (4.7):

∞ 2 −1/2 X n (1 − 2λcosθ + λ ) = Pn(cosθ)λ (4.10) n=0

6 5 The relation between Gegenbauer Polynomial and the Wigner’s D-matrix

We will determine a relationship between Gegenbauer Polynomials [20, 21] and the Wigner’s D-matrix. We suppose that in (3.1) and we replace in the generating function (3.5), v1 by u1, v2 by u2 and r = λ, we thus ﬁnd: z1 z2 u1 X j exp u1 u2 = ϕjm0 (u)D(m0,m}(z)ϕjm(u) (5.1) −z2 z1 u2 j,m,m0 After the integration of the expression Z Z z1 z2 X 2j j exp u u dµ(u) = λ D(m0,m}(Ω) ϕjm0 (u)ϕjm(u)dµ(u) −z2 z1 j,m,m0 (5.2) Using the Gauss integral: n Z n 1 Y t −1 du dv exp(−V XV ) = (det(X)) (5.3) π i i i=1

With: V = (v1, v2, . . . , vn), vi = xi + iyi. We ﬁnd that: 1 X = λ2jDj (Ω) (5.4) θ ψ+ϕ 2 (m,m) 1 − 2λcos 2 cos( 2 ) + λ jm The development of the ﬁrst member using generating function formula of the Gegen- β bauer Polynomials Cn (t): ∞ 2 −β X n β (1 − 2tλ + λ ) = λ Cn (t) (5.5) n=0

We ﬁnd the following relation:

θ (ψ + ϕ) X C1 cos cos = Dj (Ω) (5.6) 2j 2 2 (m,m) m

If we put (ψ + ϕ) = 0 we ﬁnd also the new relation:

θ X C1 (cos ) = dj (θ) (5.7) 2j 2 (m,m) m

6 The Coupling of two angular momenta

We make a simple revision of the couplings of two angular moments [10, 11]. We deduce from the orthogonality of the D-Matrices the integration formulas of the product of three D-matrices and the product of three spherical harmonics.

7 6.1 The coupled function of two angular moments −→ −→ −→ The function of two coupled angular momentum, J = j 1 + j 2, in the Hilbert space of the analytic functions is:

1 2 X 1 2 Ψ(j1j2)j3m(u , u ) = hj1m1, j2m2k(j1j2)j3m3iϕj1m1 (u )ϕj2m2 (u ) (6.1) m1m2

with Dj3 (Ω) = Ψ (u1, u2),R(Ω)Ψ (u1, u2) (6.2) (m3,m3) (j1j2)j3m (j1j2)j3m

i i i And (u ) = (u1, u2)

The coeﬃcient

hj1m1, j2m2k(j1j2)j3m3i = h(j1j2)j3m3kj1m1, j2m2i

is the Clebsh-Gordan [6].

Conversely of (6.1) we can write:

1 2 X 1 2 ϕj1m1 (u )ϕj2m2 (u ) = hj1m1, j2m2k(j1j2)j3m3iΨ(j1j2)j3m(u , u ) (6.3) m1m2

6.2 The relationships of Wigner 3-j symbols and the Clebsh- Gordan coeﬃcients

The relationships of 3-j symbols are related to the Clebsh-Gordan coeﬃcients by [10, 11]: j1 j2 j3 j1−j2−m2 −1/2 = (−1) (2j3 + 1) hj1m1, j2m2k(j1j2)j3 − m3i (6.4) m1 m2 m3

Wigner introduced the 3-j symbols for symmetries purpose.

6.3 The integral of the product of three D-matrices and three spherical harmonics

Using the relation (6.3) and (6.4) we ﬁnd that: j1 j2 X j1 j2 j3 j3 j1 j2 j3 D 0 (Ω)D 0 (Ω) = (j + 1) D 0 (Ω) (m ,m1) (m ,m2) 3 0 0 0 (m ,m3) 1 2 m1 m2 m3 3 m1 m2 m3 jmm0 (6.5) We deduce from the orthogonality of matrices D (.) the relations which are very important for the calculation of the symbols of 3-j and 3 − lm.

8 Z 2π Z π Z 2π 1 j1 j2 j3 j1 j2 j3 j1 j2 j3 D 0 (Ω)D 0 (Ω)D 0 (Ω)dΩ = 2 (m ,m1) (m ,m2) (m ,m3) 0 0 0 8π 0 0 0 1 2 3 m1 m2 m3 m1 m2 m3 (6.6) 0 0 In posing m1 = m,2 = m3 = 0 and using (6.6) we ﬁnd the second relation that interests us for the determination of 3 − lm. Z 2π Z π

[Yl1m1 (θ, ϕ)Yl2m2 (θ, ϕ)Yl3m3 (θ, ϕ)] sinθdθdϕ 0 0 1/2 1 l1 l2 l3 l1 l2 l3 = (2l1 + 1)(2l2 + 1)(2l3 + 1) (6.7) 4π 0 0 0 m1 m2 m3

With l1, l2, and l3 are integers, and J = l1 + l2 + l3 is even.

7 The generating function of 3-j symbols

We will determine the generating function of the 3-j symbols [6, 16]. In addition we will give the generating function, G.F, of four coupling as an example of the generalization.

7.1 The generating function of 3-j symbols

We will deduct simply the G.F of 3-j symbols by using the integration of the product of three G.F of Wigner’s D-matrix (6.6) and the formulas (3.9) and (3.10). We multiply the two members of (6.6) by

2(J+1)+1 r dr, J = j1 + j2 + j3 (7.1) And we carry out the integration and then we identify the two members.

A- The integral of the ﬁrst member or the product of three generating functions is: Z 3 3 X i i i G = G(u , z , v ) dµ(z1)dµ(z2) = i=1 exp v3, v2 u3, u2 + v3, v1 u3, u1 + v1, v2 u1, u2 (7.2)

B-The integration of the second member with the help of (2.9) and (3.9) we ﬁnd: Z Z exp[det(t, v)] × exp[det(t, u)]dµ(t) = G3j(t, v) × G3j(t, u)dµ(t) (7.3)

C- The function G3(t, u) is the generating function of the 3-j symbols [6].

3 3 3 1 1 2 G3j(t, u) = exp[det(λ, u)] = exp t1[u , u ] + t2[u , u ] + t3[u , u ] = X 1 2 3 j1 j2 j3 ϕj1m1 (u )ϕj2m2 (u )ϕj3m3 (u )Φ(t1t2t3) (7.4) m1 m2 m3 jm

9 And J−2j1 J−2j2 J−2j3 1/2 t1 t2 t3 Φ(t1t2t3) = [(J + 1)!] 1/2 (7.5) [(J − 2j1)!(J − 2j2)!(J − 2j3)!]

i j i j i j With bu , u c = u1u2 − u2u1 And i j i j i j (u .u ) = u1u2 + u1u2 (7.6) We can write also:

t1 t2 t3 1 2 3 det(t, u) = u1 u1 u1 (7.7) 1 2 3 u2 u2 u2 Note that Regge has studied the symmetries of the 3-j symbols starting from the invariance of the determinant det(t, u)[13]. 3 3 Moreover, if we put in (7.4) u1 = u2 = 0 or j3 = 0, j1 = j2 we ﬁnd that:

1 j1 j1 0 j1−m1 − = (−1) (2j1 + 1) 2 (7.8) m1 −m1 0

7.2 The relation between the generating functions of the coupling and the uncoupling

Using the expression (7.1) we deduce the generating function of the couplings:

1 2 3 1 3 2 expbt3[u , u ] + t2(u .u ) + t1(u .u )c = X −1/2 3 1 2 (2j3 + 1) ϕj3m3 (u )Φ(t1t2t3)Ψ(j1j2)j3m(u , u ) (7.9) jm

And it is very important to observe that [6]:

1 2 3 1 3 2 expbt3[u , u ] + t2(u .u ) + t1(u .u )c = ∂ ∂ ∂ ∂ exp t [ , ] + t u3. + t u3. exp (v1.u1) + (v2.u2) (7.10) 3 v1 v2 2 v1 1 v2

So the function (7.9) is simply deduced from the generating function of the uncoupling functions. Moreover the expression (7.10) is very interesting for the couplings of the several angular moments and for the calculations of the 6-j symbols and Racah coeﬃcients.

7.3 The generating function of four coupling

The generalization of (7.2) is simple and we are going to give the generating function of the couplings of four angular moments. Indeed Z 5 Z 5 X i i i −→ G = [G(u , z , v )]dµ(z1)dµ(z2) = G5j( λ , v) × G5j(λ, v)dµ(λ) (7.11) i=1

10 We follow the same method of calculations of (7.2) and we ﬁnd:

" 4 # X i j G5j(t, u) = exp λij u , u (7.12) i=1,j

8 A new derivation of the 3j with magnetic moments mi = 0

The coefficients 3-j with mi = 0 are frequently found in kinetic momentum calculations but the proposed methods are difficult [6, 11]. In following section, I will determine these coefficients by a very simple method of calculation. 1 1 2 2 3 3 By posing u2 = u1, u2 = u1, and u2 = u1 in (7.4), we find that:

     1 t1 t2 t3 0 −t3 t2 u1 1 2 3 1 2 3 2 t u1 u1 u1 = u1 u1 u1  t3 0 −t1 u1 = (X)A(X) (8.1) 1 2 3 3 u1 u1 u1 −t2 t1 0 u1 We ﬁnd after integration that Z t 3 i 1 exp (X)A(X)Xi=1dµ(u ) = 2 2 2 2 (8.2) 1 + λ (t1 + t2 + t3)

And consequently:

1 X j j j = Φ(t t t ) 1 2 3 (8.3) 1 + λ2(t2 + t2 + t2) 1 2 3 0 0 0 1 2 3 j

After development and identiﬁcation of two members of (8.3) we ﬁnd the 3-j symbols with magnetic moments mi = 0, i = 1, 2, 3:

1 " 3 # 2 J j j j Q (J − 2j )! ! 1 2 3 = (−1)J/2 i=1 i × 2 (8.4) 0 0 0 (J + 1)! Q3 J i=1( 2 − ji)!

If J is even.

9 The generating function of the 3 − lm symbols of SO(3)

We will determine for the ﬁrst time the generating function of the 3 − lm symbols. The generating function of Spherical Harmonics is:

−→ −→ 1 a (u). r X 4π 2 exp = ϕ (u)(r)lY (θϕ) (9.1) 2 2l + 1 lm lm lm

11 Therefore we write:

(−→a (u1) + −→a (u2) + −→a (u3)) .−→r exp = 2 (" 3 1 # $ 3 % ) 4π 2 X Y l1+l2+l3 Y 1 2 3 (r) × (Ylim i(θ, ϕ)) ϕl1m1 (u )ϕl1m1 (u )ϕl1m1 (u ) 2li + 1 lm i=1 i=1 (9.2)

Multiplying the two members by: 1 exp(−r2)dxdydz (9.3) π3/2

We ﬁnd after integration of the ﬁrst member:

Z ∞ −→ 1 −→ 2 −→ 3 −→ 1 ( a (u ) + a (u ) + a (u )) . r −→2 3/2 exp exp(− r )dxdydz = π −∞ 2 1 exp − [u1, u2]2 + [u1, u3]2 + [u2, u3]2 (9.4) 4

And with the help of the integral:

∞ Z 2 n + 1 xne−x dx = Γ /2 (9.5) 0 2

Moreover, after the integration of the second member (9.2), using (6.7) and (9.5), we deduce the generating function of the 3 − lm symbols of SO(3):

1 exp − [u1, u2]2 + [u1, u3]2 + [u2, u3]2 = 4 X 4 J + 3 l1 l2 l3 l1 l2 l3 √ Γ ϕl1m1 (u)ϕl1m1 (v)ϕl1m1 (w) (9.6) π 2 0 0 0 m1 m2 m3 lm

With the help of (8.4) and Gamma duplication formula:

1 √ Γ z + Γ(z) = 21−2z π(2z) (9.7) 2

And the developments of the ﬁst side of (9.6) we ﬁnd the expression of 3 − lm symbols for SO (3).

10 The new relationship between 3-j symbols

We will determine a new relation between the symbols 3-j starting from the expression (9.6) and using the generating function of the symbols 3-j, (7.4).

12 We note that the generating function of the 3 − lm symbols (9.6) could be written as:

1 exp − [u1, u2]2 + [u1, u3]2 + [u2, u3]2 = 4 Z 1 exp − t [u1, u2] + t [u1, u3] + t [u2, u3] exp t [u1, u2] + t [u1, u3] + t [u2, u3] dµ(t) 4 3 2 1 3 2 1 (10.1)

After the development of (10.1), using (7.4) and the identiﬁcation of the second member of (9.6) we ﬁnd the new relation:

" 3 #1/2 J X Y [(ji + mi)!(ji − mi)!] + 1 ! × 2 Q3 0 0 0 0 0 00 0 00 0 i=1(ji + mi)!(ji − mi)![(ji + mi )!(ji − mi )!] m1,m2 i=1 1 " 3 # 2 l1 l2 l3 l1 l2 l3 2 2 2 2 2 2 Y l1 l2 l3 0 0 0 00 00 00 = (J + 1)! (J − 2ji)! (10.2) m m m m m m m1 m2 m3 1 2 3 1 2 3 i=1

With:

0 0 0 00 J = J/2, ji = ji/2, mi + mi = mi, i = 1, 2, 3 (10.3)

11 The generating function of 6-j symbols

We followed at the beginning the simple method proposed by Schwinger [6]. This method starts with the coupling of four angular moments instead of three. But we propose a new simpler method for the calculation of the G.F of 6-j symbols. We note that it is important to introduce the 6-j symbols instead of the Racah coeﬃ- cients for symmetry reasons. The couplings of the four moments are: −→ −→ −→ −→ −→ −→ −→ −→ −→ J = J 1 + J 2 + J 3 + J 4 = J 1 + J 3 + J 2 + J 4 (11.1)

Taking into account of the two coupling states and the relation (7.8), the overlap of the states is written then:

0 0 00 00 h((j1j2)j (j3j4)j ) j = 0, m = 0k ((j1j3)j (j2j4)j ) j = 0, m = 0i = 0 00 j +j −j1−j2 0 00 1/2 0 00 (−1) [(2j + 1)(2j + 1)] W (j1j2j3j4; j j ) (11.2)

With 0 00 j12 = j34 = j , j13 = j24 = j (11.3)

But our method that we used in part three makes the calculation of the overlap elementary much easier than the Schwinger method that we will present later.

13 11.1 The coupling of the four angular moments

The uncoupled functions are: Ψ(t, u) = exp (t1u1) + (t2u2) + (t3u3) + (t4u4) (11.4) −→ −→ −→ −→ −→ −→ We will make the couplings J 12 = J 1 + J 2, J 34 = J 3 + J 4 with the help of (7.10)

∂ ∂ ∂ ∂ Ψ0 = exp α [ , ] + α x. + α x. × 3 t1 t2 2 t1 1 t2 (11.5) ∂ ∂ ∂ ∂ exp β [ , ] + β y. + β y. Ψ(t, u) 3 t3 t4 2 t3 1 t3 −→ −→ −→ We do the coupling again J = J 12 + J 34: ∂ ∂ ∂ ∂ exp γ [ , ] + γ z. + γ z. Ψ0 (11.6) 3 x y 2 x 1 y

11.2 The expression of the coupling of the four angular moments

Then we put z1 = γ1 = z2 = γ2 = 0, γ3 = 1 and so we get the generating function L 0 0 of the function Ψ() of the coupling ((j1j2)j (j3j4)j ) j = 0, m = 0 and are of the form (7.12): " 4 # L X i j Gs (t, u) = exp λij[u , u ] (11.7) i=1,j>i with λ = α λ = α β λ = α β 12 3 13 2 2 14 2 1 (11.8) λ23 = α1β2 λ24 = α1β1 λ34 = β4

R 00 00 Similarly, we get Ψ() the generating function of the coupling ((j1j3)j (j2j4)j ) j = 0, m = 0, which has the form (7.12): " 4 # R X i j Gs (t, u) = exp µij[u , u ] (11.9) i=1,j>i With 0 0 0 0 0 µ12 = α2β2 µ13 = α1β1 µ14 = β3 0 0 0 0 0 (11.10) µ23 = −α1β2 µ24 = α2β1 µ34 = β3 We will ﬁnd the generating function of 6-j symbols by calculating the expression: Z " 4 # " 4 # 4 X i j X i j Y I I = exp µij u , u exp λij u , u dµ(u ) = i=1,j>i i=1,j>i i=1 (11.11) 0 00 X j +j −j1−j2 0 00 0 0 00 00 (−1) ϕj1j2j (α)ϕj3j4j (β)ϕj1j3j (α)ϕj2j4j (β)W (j1j2j3j4; j j ) 00 j1...j We will propose a simple method for calculating the ﬁrst member of (11.11).

14 11.3 The generating function of 6-j symbols

i i In the ﬁrst member of (11.11) we make the change of the variables u2 by u2 j = 1 ... 8 1 2 3 4 5 6 7 8 and we denote u1 u1 u1 u1 u2 u2 u2 u2 by (X) we write then:   0 0 0 0 0 λ12 λ13 λ14  0 0 0 0 −λ12 0 λ23 λ24    0 0 0 0 −λ13 −λ23 0 λ34    0 0 0 0 −λ14 −λ24 −λ34 0  (A) =   (11.12)  0 −µ12 −µ13 −µ14 0 0 0 0    µ12 0 −µ23 −µ24 0 0 0 0    µ13 µ23 0 −µ34 0 0 0 0  µ14 µ24 µ34 0 0 0 0 0 As a result we write (11.11) in the form:

4 Z Y I = exp t X (A)(X) dµ(uI ) (11.13) i=1 The integration of this expression is carried out using Gauss’s formula (5.3) so we ﬁnd:

4 ! !−2 X I = 1 − λijµij + (µ12µ34 + µ13µ24 + µ14µ23) × (λ12λ34 + λ13λ24 + λ14λ23) i=1,j

By replacing λij, µij by their expressions (11.7), (11.9) and by changing the signs of α1 0 and β1, We ﬁnd the generating function of 6-j symbols [6]:

X 0 00 0 0 00 00 ϕj1j2j (α)ϕj3j4j (β)ϕj1j3j (α)ϕj2j4j (β)W (j1j2j3j4; j j ) = (11.15) 00 j1..j

0 0 0 0 0 0 0 0 0 0 0 0 −2 (1−α3α2β2 −β3α1β1 −α3α2β2 −β3α1β1 −α1β2α1β2 −α2β1α2β1 +α3β3α3β3) (11.16) The development of (11.16) and its comparisons with (11.15) give the expression of the Racah coeﬃcients [6, 16].

12 The generating function and the basis of the four dimensional spherical basis of the Harmonic oscillators

The resolution of the Schrdinger equation for an isotropic harmonic oscillator in R4 has been performed by many others [7,8, 18, 22] using the method of separations of variables. However as a simple application of angular momentum couplings we will propose a new method for the determination of the wave function of R4 in terms of Euler-angles (r, θ, ϕ, ψ). We give the orthogonal transformation of the operators of the creations of Dirac which allows us to determine the eigenfunctions of the kinetic 4 moment Lz . Then using the generating method we will ﬁnd the wave functions of R .

15 12.1 The basis of two dimensions Harmonic oscillator

The basis of two dimensions Harmonic oscillator in Dirac notations [1, 22] is:

+ n1 + n1 |n1, n2i = (ax ) (ay ) |0, 0i (12.1)

This basis is not eigenfunctions of Lz. So to obtain the basis which has this property we must take the transformation [1]: √ √ 2 2 A+ = (a+ − ia+),A+ = (a+ + ia+), 1 2 x y 2 2 x y + + (12.2) N1 = A1 A1,N2 = A2 A2

Lz = (N1 − N2),N = N1 + N2

The new basis |N1,N2i can be written in the form: A+j+mA+j−m |jmi = |j + m, j − mi = 1 2 |0, 0i (12.3) p(j + m)!(j − m)!

This basis is eigenfunctions of Lz and N with the values 2m and 2j.

12.2 The generating function of the two dimensions oscillator

The generating function of the two dimensions oscillator may be written in the form: j+m +j−m + + X t1 t2 |Fjm(t1, t2)i = exp[t1A1 + t2A2 ]|0, 0i = p |j, mi jm (j + m)!(j − m)! (12.4) √ √ + + = exp[ax 2(t1 + t2)/2 + iay 2(−t1 + t2)/2]|0, 0i But the generating function of one dimension of harmonic oscillator is: ∞ x2 √ t2 X tn π−1/4exp − + 2x − = √ hxkni, hxkni = u (x) (12.5) 2 2 n n=0 n! Put z in term of Cartesian coordinates z = x + iy, x = ρcosϕ, y = ρsinϕ and using (12.5) we ﬁnd: 1 x2 + y2 X F (t , t , z) = √ exp − + t (z) + t (z) − t t = ϕ (t)ϕ (ρ, ϕ) jm 1 2 π 2 1 2 1 2 jm jm jm (12.6)

12.3 The cylindrical basis of Harmonic oscillator and the An- alytic Hilbert space

The wave function of the cylindrical base in term of Cartesian coordinates is very well known [18, 19], so we put λ = pmω/η = 1 and we write: s 2p! 2 Φ (r, ϕ) = e−r /2L|m|(r)r|m|eimϕ (12.7) nm (p + |m|)! p

16 With n = n1 + n2, p = (n − |m|)/2 (12.8)

|m| Lp (r) Are the Associates Laguerre polynomial: If we put p = 0 in (12.7) we ﬁnd the relation of un(z), (2.6), with the wave function of the cylindrical basis:

n −zz/2 z −zz/2 Φnm(r, ϕ) = un(z)e = √ e (12.9) n!

12.4 The generating function of the spherical base of R4

A- We consider the product of the two uncoupled basis (7.10) by:

j1+m1 j2−m2 j2+m2 j1−m1 t1 t4 t3 t2 p × p (12.10) (j1 + m1)!(j2 − m2)! (j2 + m2)!(j1 − m1)!

And the generating function of 3-j of this basis is:

Gs ((α, ξ), t) = exp [γ(t1t2 − t3t4) + ξ1(t1u1 + t4u2) + ξ2(t3u1 + t2u2)] (12.11)

B- We will derive the generating function of the spherical base of R4.

12.4.1 The generating function of R4

To determine the generating function of the spherical base of R4 we can perform the integration of the below expression using Gauss formula

4 Z Y Gc(s, ξu, λp) = GS ((γξ), t)F2c ((s, t), λρ1, λρ2) dµ(ti) i=1 (12.12)

and F2c(t1, t2, λz) = Fj1m1 (t1, t2, λz1)Fj2m2 (t3, t4, λz2)

The Gauss formula is: n Z n 1 Y t t −1 du dv exp(−V XV + AtV + V B) = (det(X)) exp(AtX−1B) (12.13) π i i i=1

With V = (v1, v2, . . . , vn, vi = xi + iy − i After integration and putting γ = −1 we ﬁnd the generating function of the basis 4 Ψ(j1j2)j(m1m2)m(λρ, ψθϕ) of R :

λ 1 s2 G (s, ξu, λp) = Exp −λρ2 − 2λρ2+ c π (1 − s2)2 (1 − s2) (12.14) s √ z z ξ ξ 1 2 u u 2 λ 1 2 1 2 1 − s −z2 −z1

17 12.4.2 The spherical basis of R4

We ﬁnd the spherical base R4 after development (12.14) and using the generating function of the Associates Laguerre polynomial [20, 21]:

∞ X p (α) 1 − t x t L (x) = e 1−t , x = λr p (1 − t)β+1 p=0 n (12.15) X (−1)kΓ(n + α + 1) With L(α)(x) = xk p k!(n − k)!Γ(k + α + 1) k=0

We write the development of (12.14) in the form:

n+j X s 2j+1 2 j ϕj(j −j )(ξ)ϕjm(u)L (2λρ )D (z) (12.16) (1 − s)n+2+2j 1 2 j1+j2−j ((j1−j2),m)

The spherical base of R4 is therefore

−ρ2/2 j Ψ 0 (ρ, ψθϕ) = N 0 e R (ρ)D (ψθϕ) (12.17) njm,m njm,m nk (j1−j2),m)

The radial part is: k (k+(d−d)2) 2 −ρ2/2 Rnk(ρ) = Nnkρ Ln (ρ )e (12.18)

With n = j1 + j2 − j3, and k = 2j3, d = 4, 2λ = 1. The normalization factor is:

 2Γ(n + 1) N = (12.19) nk Γ(n + k + d/2)

12.5 Generalization of the transformation of R2 basis to R4 and R8 basis.

We observe that (12.2) analogue to the polar transformation [22] written in the matrix form by: √ + + A1 2 ax 1 −1 + = + ,H2(1) = (12.20) A2 2 iay 1 1

With H2(1) is an orthogonal matrix. A- The transformation for R4 basis With Hurwitz orthogonal matrices [22] we can easily generalize transformations (12.2) to oscillator R4 using the polar cylindrical transformation [22]. So we write:

A+  a+  1 −1 −1 −1 1 √ x1 √ + 2 + 2 A2  iay1  1 1 −1 1   + = H4(1)  +  ,H4(1) =   (12.21) A3  4 iay2  4 1 1 1 −1 A+ + 1 −1 1 1 4 ax2

18 B- The transformation for R8 basis The generalization of the transformation generalization for n = 8 is:

 +  +    A1 a1 1 −1 −1 −1 −1 −1 −1 −1 + + A2  ia2  1 1 −1 1 −1 1 1 −1  +  +   A3  ia3  1 1 1 −1 −1 −1 1 1   +  +   A4  1 ia4  1 −1 1 1 −1 1 −1 1   + = H8(1)  + ,H8(1) =   A5  4 ia5  1 1 1 1 1 −1 −1 −1  +  +    A6   a6  1 −1 1 −1 1 1 1 −1  +  +    A7   a7  1 −1 −1 1 1 −1 1 1  + + A8 a8 1 1 −1 −1 1 1 −1 1 (12.22)

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