Notas de Aula Cap 3 - Teoria do Momento Angular_210724_103836

ϕ\n\nx' = r (cos ϕ - r sin ϕ)\n{ x = r cos θ\nβ = r sin θ\n\nx' = r cos(θ + ϕ)\n\ny' = r (sin ϕ - r cos ϕ) = x cos ϕ - y sin ϕ\n \n( x' \n y' ) = ( cos ϕ -sin ϕ ) ( x )\n ( sin ϕ cos ϕ ) ( y )\n\nR(ϕ)\n\nEm 3D,\n\n( cos ϕ -sin ϕ 0 )\n( sin ϕ cos ϕ 0 )\n( 0 0 1 ) -> rotação em torno do eixo z\n\nR(ϕ)\n\nRotação em torno de um eixo qualquer\n\nr' = r1 + θ1 + r2i'\n\nx1 = r1k\n\nx2 = r2k\n\nr = cos(θ1x2 - θ2) + ρ sin(θ1Xn2)( p\n\nx1 = R(ϕ)\n\nx2 = R(ϕ)\n\nx3 = R(ϕ)\n\nx4 = R(ϕ)\n\nx5 = R(ϕ)\n\nx6 = R(ϕ) r' = (r')k\nx' = (θr2)r'nk\nRij = (Rij)\n(Rij)\n= cos(ϕ) δij + (1 - cos(ϕ)) n_i n_j - εijk n_k sin(ϕ)\n\ningredientes\nfijando a = a = \n\ná\naa\n(á)\nR(ı)\n= R(ʽ)\n ||\\mathbf{r}||^2 = x_k^i x_k^j (R_{ij} R_{kl}) = (R_{ij} R_{kl}) x_i^k x_j^l =||\\mathbf{r}||^2 = x_k x_k \\rightarrow R_{ij} R_{kl} = \\delta_{kl} \n\nR^T K R_{ij} = \\delta_{ij} \\text{ em } R^T R = I \\rightarrow R^T = R^{-1} \n\\alpha R^i R^l = RR^l = I \\rightarrow R^T R = I \n\\text{det } R^2 = 1 \\Rightarrow \\text{det } R = \\pm 1 \n\\n\\text{Retomemos: } \\left\\{ \\begin{array}{c} \\text{det } R = 1 \\\\ R^T R = I \\rightarrow \\text{ grupo SO(3)} \\end{array} \\right.\n \n\\text{Se } R_i R_2 \\text{ não é rotação, } \text{det } (R_1 R_2) = \\text{det } R_1 \\text{ det } R_2 = 1 \n\n\\Rightarrow \\text{ então } R_i R_2 \\text{ é uma rotação.} \n\\text{Dado } R, \\text{ temos } R^T = R \\text{ logo } 3 \\text{ inversa } \n\\text{R}^T R = I, \\\\ I \\text{ é uma rotação} \n\\text{As rotações formam um grupo chamado de SO(3)}\n\\nR_{ij} (\\vec{a}^2) = \\cos \\theta \\delta_{ij} + (1 - \\cos \\theta) a_i a_j + \\epsilon_{ijk} a_k \\sin \\theta \n\\tau \\text{ where } \\tau R(\\vec{a}) = R(\\vec{a})^2, R(\\vec{a}) A^2 = R(\\vec{a})^2 \\tau^a \\tau^2 Aula dia 08/07/2021\n\\mathbf{r}^2 = R^T r^2 \\rightarrow x_i^k = R_{ij} x_j \nR_{ij} = \\cos \\phi \\delta_{ij} + (1 - \\cos \\phi) a_i a_j + \\sum_{k} \\epsilon_{ijk} a_k \tan \\phi \n\\text{reconhecendo } \\hat{n} = \\frac{\\vec{a}}{a};\\; \\frac{l^2}{1} = \\phi^2\n\\Rightarrow R_{ij} (\\vec{a}) = \\cos \\theta \\delta_{ij} + (1 - \\cos \\theta) a_i a_j + \\sum_{k} \\epsilon_{ijk} a_k \\sin \\theta\nP(\\vec{a}^2) = \\vec{a}^2 - R_{ij}(\\vec{b}^2) \\\\ = R(\\vec{b}^2) \n\\text{em geral } R(\\vec{a}) R(\\vec{b}) \\neq R(\\vec{b}) R(\\vec{a}) \n\n\\text{Como as rotações formam um grupo, } R(\\vec{a}) R(\\vec{b}) = R(\\vec{c})(\\vec{c}^2,\\vec{b}^2) \\n\\Rightarrow (\\vec{b}^2) = \\vec{a}^2 \n\\Rightarrow (\\vec{c},\\vec{a^2})^2 = (\\vec{a}^2)\\rightarrow d = -\\uparrow \\left( a^2\\vec{b}^2 \\right)/(\\vec{v}) \\rightarrow R definindo (2) com relação a \\xi^{j} \n\\partial^{2}R(B) = \\partial^{2} \\xi^{j} \\partial^{2}B^{k} = \\partial^{2}c^{m} \\partial\\xi^{j} \\partial b^{k} \\partial^{2} \\frac{\\partial^{2}b^{k}}{\\partial^{2}b^{l}} = C^{k}_{ij} \n\\rightarrow M_{i}M_{j} = M_{j}M_{i} = M_{k}(C^{k}_{ij} - C^{k}_{ji}) \n\\Rightarrow [M_{i},M_{j}] = f^{k}_{ij} M_{k} \n\\quad Algébra de Lie dos Operadores \n\nf^{k}_{ij} = constante de estrutura de grupos \n\nSe f^{k}_{ij} = 0 \\forall i,j,k \\Rightarrow grupo é chamado de grupos abelianos, \ncomo consequência o grupo é não-abeliano \n\nComo [M_{i}M_{j}]_{mn} = (M_{i}M_{j})_{mn} - (M_{j}M_{i})_{mn} = (M_{i}M_{j})_{mn} - (M_{j}M_{i})_{mn} \n\n= \\epsilon_{mjb} \\epsilon^{b}_{ij} = \\epsilon_{ijm} \n= \\epsilon_{mni} = \\epsilon_{mni} \n\n\\Rightarrow [M_{i},M_{j}]_{mn} = \\epsilon_{mnk} M_{k} \n\n\\Rightarrow [M_{i},M_{j}] = \\epsilon_{ijk} M_{k} \n\nse [M_{i},M_{j}] = \\epsilon_{ijk} M_{k} # Teorema de Lie P1 50(32) \n\nR(\\xi)R(\\lambda\\hat{\\xi}) = R((1+\\lambda)\\hat{\\xi}) = R(\\hat{\\xi})R(\\hat{\\xi}) \n\n\\partial_{\\hat{\\xi}} \\partial R(\\hat{\\xi}) = \\partial R(\\hat{\\xi}) \\partial R(\\hat{\\xi}) = \\partial R(\\hat{\\xi})\\partial \\lambda \n\n\\rightarrow R(\\hat{\\xi'}) = \\frac{\\partial M_{i}^j R(\\hat{\\xi'})}{\\partial \\lambda} \\rightarrow R(\\hat{\\xi}) \\rightarrow R(\\hat{\\xi}) = R(\\hat{\\xi}) comuta com M_{i} A_{i}^0 \n\nR(\\hat{\\xi})= f(\\overline{M} \\cdot \\hat{\\xi}) \n\nf(\\alpha M_{i}^r) f(\\beta \\hat{\\xi}^j) = f(\\alpha \\beta \\hat{\\xi} \\cdot M_{i}) \n\n\\Rightarrow R(\\hat{\\xi}) = e^{\\overline{M}} \\hat{\\xi} \n\nR^{t}(\\hat{\\xi}) = e^{\\overline{M}} \\hat{\\xi} \n\n\\Rightarrow {R}^{n} = \\sum_{i=1}^{N_{j}} \\prod_{i=1}^{N}(\\lambda - \\lambda_{i}) \n\nos autovalores de N são 0, -r(a), i(a), a = (a_{1}+a_{2}+a_{3})^{1/2} \n\n \\Rightarrow R(\\hat{\\xi})^{n} \\neq 1, e^{i\\alpha} e^{i\\lambda} \n\n\\Rightarrow R^{n} = \\theta(0,0) \n\n\\Rightarrow \\hat{n} = R(\\xi) \n\n\\hat{n} = R^{-1} \\rightarrow \\rightarrow e^N = \\sum_{j} \\lambda_{j} \\prod(\\lambda - \\lambda_{j}) \n\\quad i = 1 \\rightarrow \\lambda - \\lambda_{j} \n\nos autovalores de N são 0, -r(a), i(a), a = (a_{1}+a_{2}+a_{3})^{1/2} \n\nT_{\\xi}(R(\\hat{\\xi})) = 1 + e^{-i \\alpha} + e^{i \\alpha} = (1 + 2\\cos [a]) \n\n\\rightarrow \\hat{n} = R(\\hat{\\xi}) \n\nR' = R e^{\\alpha_{k} M^{k} R^{-1}} = e^{\\hat{\\xi}} \n\nR^{n} = R^{n}(R^{-1}) \n\n\\Rightarrow R' = \\sum e^{\\alpha{\\overline{M}} \\hat{n}} \n\n\\quad \\Rightarrow R_{\\phi}^{n} \\cdot \\hat{n} = \\hat{n} \n\n\\Rightarrow R_{ij}R^{-1} = R_{ij}N_{i} \n\nSe R = R(\\xi) \\rightarrow \\partial M_{j}R^{-1} + R_{M} \\partial^{2} R_{b} = 0 \n\n\\rightarrow M_{i}N_{j} - M_{j}N_{i} = \\epsilon_{ij} M_{k} \n\n\\Rightarrow [M_{i},M_{j}] = \\epsilon_{ijk} M_{k} \\Rightarrow Aula dia 13/07/2021\n\n(1) R(\\bar{a}) = \\bar{a}^{2}.M^{*} ; \\ x_{k}^{*} = R_{ij}^{x_{j}}\n\\ \\bar{a}^{*}M_{k}^{*} = a_{m}M_{k}M_{ij}^{*} = \\epsilon_{kij}\n(2) M_{k}^{T} = M_{k}\n(3) [M_{i},M_{j}] = \\epsilon_{ijk}M_{k}\n\n(4) M_{i} = -J_{i}^{k}\nM_{i}^{T} = (\\frac{1}{k}J_{i}^{j})^{T} - M_{i} = i\\frac{1}{k}J_{i}\n\\ J_{i}^{T} = J_{i}^{*} \\implies J_{k} = J_{k}^{T} \\ (hermitiano)\n\n(5) R(\\bar{a}) = e^{-k^{2}}\n\\ d(3) = -\\frac{1}{2} R_{ij}J_{j} = -\\frac{1}{k} \\epsilon_{ijk} \\implies [J_{i},J_{3}] = i\\epsilon_{ijk}M_{k}\n\\ R(\\bar{a})R(B) = R(\\bar{a}^{2}\\bar{b})\nD(\\bar{a})D(\\bar{b}) = D(\\bar{b}^{2}\\bar{a}) , D(\\bar{a}) = e^{-i\\bar{a}.M}\n\nD^{*}(\\bar{a})J_{j}D(\\bar{a}) = R_{ij}(\\bar{a})J_{j} \\, i,j = 1,2,3\n\nObs. Para o SO(N), temos N(N-1)/2 gerados, (i_{j} = 1,2,...,N(N-1)/2)\n\nD^{*}(\\bar{a})M_{i}D(\\bar{a}) = N_{j}(\\bar{a})M_{j};\nM_{i} -> geradores de SO(N)\nN é uma matriz N(N-1)/2\n