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Jordan-Schwinger Map in the Theory of Angular Momentum

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Jordan-Schwinger Map in the Theory of Angular Momentum Jordan-Schwinger map in the theory of angular momentum V´aclav Zatloukal∗ Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Bˇrehov´a7, 115 19 Praha 1, Czech Republic (Dated: February 21, 2020) The Jordan-Schwinger representation of the su(2) algebra utilizes ladder operators to efficiently handle su(2) representations with arbitrary spin. In these notes we point out the usefulness of this technique for calculating the Clebsch-Gordan coefficients when two angular momenta are being composed. I. JORDAN-SCHWINGER REPRESENTATION: GENERIC CASE Let the n × n matrices A1;:::; AN form a representation (typically fundamental) of a Lie algebra g: k [Ai; Aj] = cijAk; (1) k where cij are the structure constants of g, and summation over k = 1;:::;N is implied. y y Consider n pairs of creation and annihilation operatorsa ^1;:::; a^n anda ^1;:::; a^n with usual bosonic commutation relations y y y [^aα; a^β] = δαβ ; [^aα; a^β] = 0 ; [^aα; a^β] = 0: (2) Then, the operators ^ y Ai =a ^α(Ai)αβa^β; (3) where (Ai)αβ is the (α; β)-th entry of the matrix Ai, and summation over α; β = 1; : : : ; n is implied, form again a representation of the Lie algebra g. Indeed, ^ ^ y y [Ai; Aj] = (Ai)αβ(Aj)γδ [^aαa^β; a^γ a^δ] y y y y = (Ai)αβ(Aj)γδ (^aα[^aβ; a^γ ]^aδ +a ^γ [^aα; a^δ]^aβ) y y = (Ai)αβ(Aj)βδ a^αa^δ − (Ai)αβ(Aj)γα a^γ a^β y = (AiAj − AjAi)αδ a^αa^δ k y = cij(Ak)αβa^αa^β k ^ = cijAk: (4) ^ The map Ai 7! Ai from matrices to operators (on an abstract Hilbert space) is referred to as the Jordan-Schwinger map [1]. ∗Electronic address: [email protected]; URL: http://www.zatlovac.eu 2 II. JORDAN-SCHWINGER REPRESENTATION: su(2) In particular, we will be concerned with the angular momentum Lie algebra su(2), whose σi fundamental representation is spanned by the 2 × 2 matrices Ji = 2 , i = 1; 2; 3, which fulfil the commutation relations [Ji; Jj] = i "ijk Jk: (5) Here σi denote the standard Pauli matrices 0 1 0 −i 1 0 σ = ; σ = ; σ = : (6) 1 1 0 2 i 0 3 0 −1 For the case of su(2), the Jordan-Schwinger map yields the following operators: 1 J^ = (^aya^ +a ^ya^ ); 1 2 2 1 1 2 i J^ = (^aya^ − a^ya^ ); 2 2 2 1 1 2 1 J^ = (^aya^ − a^ya^ ): (7) 3 2 1 1 2 2 From now on we shall omit the `hat', writing simply Ji instead of J^i, and aα instead ofa ^α. The angular momentum ladder operators J± = J1 ± iJ2 assume a particularly simple form y y J+ = a1a2 ;J− = a1a2: (8) ~2 2 2 2 The angular momentum squared, J = J1 + J2 + J3 , reads 1 J~2 = J 2 + (J J + J J ) 3 2 + − − + 1 1 = (aya − aya )2 + (aya a ay + aya a ay) 4 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 = (N 2 − 2N N + N 2) + (N N − N + N N − N ) 4 1 1 2 2 2 1 2 1 1 2 2 N N = + 1 ; (9) 2 2 where, in passing, we have denoted by N1, N2, and N the number operators y y N1 = a1a1 ;N2 = a2a2:;N = N1 + N2: (10) 1 (Note that now J3 = 2 (N1 − N2).) Normalized states with occupation numbers n1, n2 (i.e., the simultaneous eigenstates of oper- ators N1, N2) read y n1 y n2 (a1) (a2) jn1; n2i = p p j0i ; (11) n1! n2! where j0i is the abstract vacuum state, and n1; n2 = 0; 1; 2;:::. These are also eigenstates jj; mi 2 of J3 and J~ : 2 J3 jj; mi = m jj; mi ; J~ jj; mi = j(j + 1) ; jj; mi = jn1 = j + m; n2 = j − mi : (12) 3 y 1 Therefore, adding a `quantum' with a1 increases the spin j and the spin projection m by 2 , y 1 1 whereas adding a `quantum' with a2 increases j by 2 , but decreases m by 2 (see Fig.). Note that the operators Ji of Eq. (7) preserve the total occupation number n1 + n2, and hence y y also the value of spin j. The Fock space generated by a1, a2 then decomposes into subspaces, 1 1 labelled by j = 2 (n1 + n2) = 0; 2 ; 1;:::, which are invariant under the Ji (and, of course, under 2 the derived operators J± and J~ ). Moreover, within the spin-j subspace, m = −j; −j − 1; : : : ; j, as follows from the inequalities j + m = n1 ≥ 0 and j − m = n2 ≥ 0. Let us remark that one can realize the abstract Fock space as a space of functions in two complex variables f(z1; z2), and the abstract creation and annihilation operators as multiplicative and differential operators y y @ @ a1 ' z1 ; a2 ' z2 ; a1 ' ; a2 ' : (13) @z1 @z2 The vacuum state j0i is identified with 1, and the scalar product can be defined via a two-fold integral over the complex plane 1 Z 2 2 ∗ −|z1j −|z2j ∗ ∗ hfjgi = 2 f (z1; z2) g(z1; z2) e dz1dz1 dz2dz2 : (14) π 2 C The states jj; mi are then realized by the polynomials zj+m zj−m jj; mi ' 1 2 : (15) p(j + m)! p(j − m)! A. Spin coherent states Let us define a spin coherent state by the formula µJ jj; µi = e − jj; m = ji ; µ 2 C: (16) y 2j µaya (a1) jj; µi = e 2 1 j0i p(2j)! y 2j µaya (a1) −µaya = e 2 1 e 2 1 j0i p(2j)! 1 µaya y −µaya 2j = (e 2 1 a e 2 1 ) j0i p(2j)! 1 (ay + µay)2j = 1 2 j0i : (17) p(2j)! To prove the last equality, observe that d µaya y −µaya y µaya y −µaya y (e 2 1 a e 2 1 ) = a e 2 1 [a ; a ]e 2 1 = a : (18) dµ 1 2 1 1 2 In passing we note that Eq. (17) implies (J )` 1 2j 2j1=2 − jj; ji = (ay)2j−`(ay)` j0i = jj; j − `i : (19) `! p(2j)! ` 1 2 ` 4 III. COMPOSITION OF TWO ANGULAR MOMENTA The Jordan-Schwinger representation for a system of two independent angular momenta (la- belled a and b) utilizes four pairs of creation and annihilation operators, and identifies 1 1 J a = (σ ) ay a ;J b = (σ ) by b ;J tot = J a + J b: (20) i 2 i αβ α β i 2 i αβ α β i i i a b (Explicit expressions are analogous to those of Eq. (7).) Since [Ji ;Jj ] = 0 for all i; j = 1; 2; 3, the tot composed angular momentum operators Ji satisfy the su(2) commutation relations, Eq. (5). Our task is now to build out of the tensor product states jja; mai jjb; mbi = jja; mai ⊗ jjb; mbi ~a 2 a ~b 2 b (i.e., eigenstates of the operators (J ) ;J3 ; (J ) ;J3 ) linear combinations that are eigenstates of ~a 2 ~b 2 ~tot 2 tot operators (J ) ; (J ) ; (J ) ;J3 . We shall denote the latter states by jja; jb; jtot; mtoti, and look for their expansion in terms of jja; mai jjb; mbi. The coefficients in this expansion are the Clebsch-Gordan coefficients. First, we realize that y 2ja y 2jb (a1) (b1) jja; jb; ja + jb; ja + jbi = jja; jai jjb; jbi = p p j0i (21) (2ja)! (2jb)! are common eigenstates for both sets of operators. From these we will generate all the other tot eigenstates jja; jb; jtot; mtoti using the ladder operator J− , which lowers the eigenvalue mtot, and the operator [2] y y y y y S = a2b1 − a1b2; (22) which fulfils the following commutation relations: a;b y a;b tot y tot y [N ;S ] = N ; [J3 ;S ] = 0 ; [J± ;S ] = 0: (23) Moreover, by the last two relations, and the first line in Eq. (9), [(J~tot)2;Sy] = 0: (24) y 1 That is, the operator S raises (simultaneously) the eigenvalues ja and jb by 2 , while preserving jtot and mtot: y 1 1 S jja; jb; jtot; mtoti = α ja + 2 ; jb + 2 ; jtot; mtot : (25) To determine the factors α, we realize that the operator SSy can be cast as N a + N b N a + N b SSy = + 1 + 2 − (J~tot)2: (26) 2 2 Hence, choosing α real and positive, we find p α(ja; jb; jtot) = (ja + jb + 1)(ja + jb + 2) − jtot(jtot + 1) p = (ja + jb + 1 − jtot)(ja + jb + 2 + jtot); (27) and after repeated application of Sy we obtain y k 1=2 (S ) 2(ja + jb) + k + 1 k k jja; jb; ja + jb; mtoti = ja + ; jb + ; ja + jb; mtot : (28) k! k 2 2 5 0 0 Now, Eqs. (28) and (19) give (writing for the moment ja; jb instead of ja; jb) (J tot)` (Sy)k − jj0 ; j0 ; j0 + j0 ; j0 + j0 i = `! k! a b a b a b 0 0 1=2 0 0 1=2 2(ja + jb) + k + 1 2(ja + jb) 0 k 0 k 0 0 0 0 = j + ; j + ; j + j ; j + j − ` : (29) k ` a 2 b 2 a b a b At the same time, the left-hand side is equal to y k ` (S ) 1 d a b µ(J−+J−) 0 0 0 0 ` e jja; jai jjb; jbi = k! `! dµ µ=0 y y y y y y 0 y y 0 k ` 2ja 2jb (a2b1 − a1b2) d (a1 + µa2) (b1 + µb2) = j0i ; (30) k! `! dµ` p 0 p 0 µ=0 (2ja)! (2jb)! where we have made use of Eq.
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