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On Wigner's D-Matrix and Angular Momentum

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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 gener- ating 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 momen- tum 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 quan- tum 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 rota- tion. 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 har- monic 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 spher- ical 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 func- tion 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(θ') = ηmYlm(θ'); L Ylm(θ') = η l(l + 1)Ylm(θ') (2.4) In Dirac notation [1] we write: Ylm(θ') = hθ'klmi (2.5) 2.2 The Analytic Hilbert space The Analytic Oscillator space (or Fock space) defined by: zn un(z) = p (2.6) n! With scalar product is: Z z¯n0 zn (un0 ; un) = p p 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 'jm(u) = j(j + 1)'jm(u);J3'jm(u) = m'jm(u) (2.12) With 'jm(u) is the representation of SU(2) in the analytic Hilbert space with: (j+m) (j−m) u1 u2 'jm(u) = (2.13) p(j + m)!(j − m)! 3 The functions 'jm(u) are orthonormal basis isomorphs to the Schwinger realization of SU(2) in terms of boson operators [6]: (a+)j+m(a+)j−m jjmi = 1 2 j00i (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) = ei Jz eiθJy ei'Jz is given in Dirac notations by: j im0 j im' D(m0;mg(Ω) = (jmjR(Ω)jjm) = e d(m0;m)(θ)e (3.1) We have [11] also j+m j−m 2j (z1u1 + z2u2) (−z2u1 + z1u2) ρ R(Ω)'jm(u) = (3.2) p(j + m)!(j − m)! with θ θ z = x + iy = ρcos ei( +')=2; z = x + iy = ρsin ei( −')=2 (3.3) 1 1 1 2 2 2 2 2 and j 2j j D(m0;mg(z) = ρ D(m0;mg(Ω); Ω = ( θ') (3.4) n j o with ( ; θ; ') are the angles of Euler and D(m0;mg(z) is a subspace of spherical basis of R4 Harmonic oscillators [7, 22]. 2j j For m = j we find that: 'jm0 (u) = ρ D(m0;0g(Ω) 3.2 The generating function of Wigner's D-matrix 0 Multiplying (2.10) by 'jm0 (v) and summing with respect to j; m; m we find the gen- erating function of Wigner's D-matrix z1 z2 u1 X j G(v; z; u) = exp v1 v2 = 'jm0 (v)D(m0;mg(z)'jm(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 '(mi)(v )'(m0i)(v ) '(mi)(v )'m0i(v ) D(m0;mg(z)D(m0;mg(z)dµ(z) (3.6) jm i= with dµ(z) = 2exp(−ρ2)ρ3dρdµ(Ω) and 1 dµ(Ω) = d sinθdθd' (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 1 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 ( θ')D 0 ( θ')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].
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