Vignerova D matrica

Vignerova D matrica predstavlja matricu ireducibilnih reprezentacija grupa SU(2) i SO(3). Vignerova D matrica je kvadratna matrica operatora rotacija dimenzija $2j+1$ sa opštim elementima:

$D_{m'm}^{j}(\alpha ,\beta ,\gamma )\equiv \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma }.$

Matrica je dobila ime po Eugenu Vigneru, koji ju je prvi uveo 1927. godine.

Definicija D matrice

Generatori Lijevih algebri SU(2) i SO(3) označimo sa $J_{x}$, $J_{y}$, $J_{z}$. Za njih vrede sledeće komutacione relacije:

$[J_{x},J_{y}]=iJ_{z},\quad [J_{z},J_{x}]=iJ_{y},\quad [J_{y},J_{z}]=iJ_{x},$

Operator

$J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}$

predstavlja Kazimirov operator od SU(2) (ili SO(3) ). Operator rotacija može da se prikaže kao:

${\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha J_{z}}e^{-i\beta J_{y}}e^{-i\gamma J_{z}},$

gde su $\alpha ,\;\beta ,$ i$\gamma \;$ Ojlerovi uglovi. Vignerova D matrica je kvadratna matrica dimenzija $2j+1$ sa opštim elementima:

$D_{m'm}^{j}(\alpha ,\beta ,\gamma )\equiv \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma }.$

Pri tome mala Vignerova d- matrica označena je sa:

$d_{m'm}^{j}(\beta )=\langle jm'|e^{-i\beta j_{y}}|jm\rangle$

Mala Vignerova d- matrica

Mala Vignerova d- matrica može da se predstavi kao:

${\begin{array}{lcl}d_{m'm}^{j}(\beta )&=&[(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}\sum \limits _{s}\left[{\frac {(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\right.\\&&\left.\cdot \left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}\right].\end{array}}$

Matrični elementi male d- matrice povezani su sa Jakobijevim polinomima $P_{k}^{(a,b)}(\cos \beta )$ sa nenegativnim $a\,$ i $b\,$. Neka je

$k=\min(j+m,\,j-m,\,j+m',\,j-m').$

Onda je:

${\hbox{Ako je}}\quad k={\begin{cases}j+m:&\quad a=m'-m;\quad \lambda =m'-m\\j-m:&\quad a=m-m';\quad \lambda =0\\j+m':&\quad a=m-m';\quad \lambda =0\\j-m':&\quad a=m'-m;\quad \lambda =m'-m\\\end{cases}}$

Onda uz uslov $b=2j-2k-a\,$ relacija je:

$d_{m'm}^{j}(\beta )=(-1)^{\lambda }{\binom {2j-k}{k+a}}^{1/2}{\binom {k+b}{b}}^{-1/2}\left(\sin {\frac {\beta }{2}}\right)^{a}\left(\cos {\frac {\beta }{2}}\right)^{b}P_{k}^{(a,b)}(\cos \beta ),$

gde su $a,b\geq 0.\,$

Svojstva Vignerove D matrice

Sledećih šest operatora:

${\begin{array}{lcl}{\hat {\mathcal {J}}}_{1}&=&i\left(\cos \alpha \cot \beta \,{\partial \over \partial \alpha }\,+\sin \alpha \,{\partial \over \partial \beta }\,-{\cos \alpha \over \sin \beta }\,{\partial \over \partial \gamma }\,\right)\\{\hat {\mathcal {J}}}_{2}&=&i\left(\sin \alpha \cot \beta \,{\partial \over \partial \alpha }\,-\cos \alpha \;{\partial \over \partial \beta }\,-{\sin \alpha \over \sin \beta }\,{\partial \over \partial \gamma }\,\right)\\{\hat {\mathcal {J}}}_{3}&=&-i\;{\partial \over \partial \alpha },\end{array}}$

${\begin{array}{lcl}{\hat {\mathcal {P}}}_{1}&=&\,i\left({\cos \gamma \over \sin \beta }{\partial \over \partial \alpha }-\sin \gamma {\partial \over \partial \beta }-\cot \beta \cos \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{2}&=&\,i\left(-{\sin \gamma \over \sin \beta }{\partial \over \partial \alpha }-\cos \gamma {\partial \over \partial \beta }+\cot \beta \sin \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{3}&=&-i{\partial \over \partial \gamma },\\\end{array}}$

zadovoljava komutacione relacije:

$\left[{\mathcal {J}}_{1},\,{\mathcal {J}}_{2}\right]=i{\mathcal {J}}_{3},\qquad {\hbox{and}}\qquad \left[{\mathcal {P}}_{1},\,{\mathcal {P}}_{2}\right]=-i{\mathcal {P}}_{3}$

Uz to dva niza uzajamno komutiraju:

$\left[{\mathcal {P}}_{i},\,{\mathcal {J}}_{j}\right]=0,\quad i,\,j=1,\,2,\,3,$

Kvadrati tih operatora su jednaki:

${\mathcal {J}}^{2}\equiv {\mathcal {J}}_{1}^{2}+{\mathcal {J}}_{2}^{2}+{\mathcal {J}}_{3}^{2}={\mathcal {P}}^{2}\equiv {\mathcal {P}}_{1}^{2}+{\mathcal {P}}_{2}^{2}+{\mathcal {P}}_{3}^{2}.$

Eksplicitni oblik je:

${\mathcal {J}}^{2}={\mathcal {P}}^{2}=-{\frac {1}{\sin ^{2}\beta }}\left({\frac {\partial ^{2}}{\partial \alpha ^{2}}}+{\frac {\partial ^{2}}{\partial \gamma ^{2}}}-2\cos \beta {\frac {\partial ^{2}}{\partial \alpha \partial \gamma }}\right)-{\frac {\partial ^{2}}{\partial \beta ^{2}}}-\cot \beta {\frac {\partial }{\partial \beta }}.$

Dejstvo operatora ${\mathcal {J}}_{i}$ na prvi indeks D-matrice je:

${\mathcal {J}}_{3}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=m'\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},$

$({\mathcal {J}}_{1}\pm i{\mathcal {J}}_{2})\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\sqrt {j(j+1)-m'(m'\pm 1)}}\,D_{m'\pm 1,m}^{j}(\alpha ,\beta ,\gamma )^{*}.$

S druge strane dejstvo ${\mathcal {P}}_{i}$ operatora na drugi indeks D-matrice je:

${\mathcal {P}}_{3}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=m\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},$

$({\mathcal {P}}_{1}\mp i{\mathcal {P}}_{2})\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\sqrt {j(j+1)-m(m\pm 1)}}\,D_{m',m\pm 1}^{j}(\alpha ,\beta ,\gamma )^{*}.$

Konačno dobija se:

${\mathcal {J}}^{2}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\mathcal {P}}^{2}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=j(j+1)D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}.$

Relacija ortogonalnosti

$\int _{0}^{2\pi }d\alpha \int _{0}^{\pi }\sin \beta d\beta \int _{0}^{2\pi }d\gamma \,\,D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )^{\ast }D_{mk}^{j}(\alpha ,\beta ,\gamma )={\frac {8\pi ^{2}}{2j+1}}\delta _{m'm}\delta _{k'k}\delta _{j'j}.$

Kronekerov proizvod matrica

Kronekerov proizvod D matrica

$\mathbf {D} ^{j}(\alpha ,\beta ,\gamma )\otimes \mathbf {D} ^{j'}(\alpha ,\beta ,\gamma )$

čini reducibilnu matričnu reprezentaciju specijalnih grupa SO(3) i SU(2). Redukcijom na ireducibilne komponente dobija se:

$D_{mk}^{j}(\alpha ,\beta ,\gamma )D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )=\sum _{J=|j-j'|}^{j+j'}\sum _{M=-J}^{J}\sum _{K=-J}^{J}\langle jmj'm'|JM\rangle \langle jkj'k'|JK\rangle D_{MK}^{J}(\alpha ,\beta ,\gamma )$

Simboli $\langle jmj'm'|JM\rangle$ su Klebš-Gordanovi koeficijenti.

Veza sa sfernim harmonicima i Ležandrovim polinomima

Za celobrojne vrednosti $l$ i za drugi indeks jednak nuli matrični elementi D-matrice proporcionalni su sfernim harmonicima i pridruženim Ležandrovim polinomima:

$D_{m0}^{\ell }(\alpha ,\beta ,0)={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell }^{m*}(\beta ,\alpha )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta })\,e^{-im\alpha }$

Odatle se dobija sledeća relacija za male d-matrice:

$d_{m0}^{\ell }(\beta )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta })$

Ako su oba indeksa jednaka nuli tada su matrični elementi proprcionalni Ležandrovom polinomu:

$D_{0,0}^{\ell }(\alpha ,\beta ,\gamma )=d_{0,0}^{\ell }(\beta )=P_{\ell }(\cos \beta ).$

Tabela male Vignerove d- matrice

Za j=1/2

• $d_{1/2,1/2}^{1/2}=\cos(\theta /2)$
• $d_{1/2,-1/2}^{1/2}=-\sin(\theta /2)$

Za j=1

• $d_{1,1}^{1}={\frac {1+\cos \theta }{2}}$
• $d_{1,0}^{1}={\frac {-\sin \theta }{\sqrt {2}}}$
• $d_{1,-1}^{1}={\frac {1-\cos \theta }{2}}$
• $d_{0,0}^{1}=\cos \theta$

Za j=3/2

• $d_{3/2,3/2}^{3/2}={\frac {1+\cos \theta }{2}}\cos {\frac {\theta }{2}}$
• $d_{3/2,1/2}^{3/2}=-{\sqrt {3}}{\frac {1+\cos \theta }{2}}\sin {\frac {\theta }{2}}$
• $d_{3/2,-1/2}^{3/2}={\sqrt {3}}{\frac {1-\cos \theta }{2}}\cos {\frac {\theta }{2}}$
• $d_{3/2,-3/2}^{3/2}=-{\frac {1-\cos \theta }{2}}\sin {\frac {\theta }{2}}$
• $d_{1/2,1/2}^{3/2}={\frac {3\cos \theta -1}{2}}\cos {\frac {\theta }{2}}$
• $d_{1/2,-1/2}^{3/2}=-{\frac {3\cos \theta +1}{2}}\sin {\frac {\theta }{2}}$

Za j=2

• $d_{2,2}^{2}={\frac {1}{4}}\left(1+\cos \theta \right)^{2}$
• $d_{2,1}^{2}=-{\frac {1}{2}}\sin \theta \left(1+\cos \theta \right)$
• $d_{2,0}^{2}={\sqrt {\frac {3}{8}}}\sin ^{2}\theta$
• $d_{2,-1}^{2}=-{\frac {1}{2}}\sin \theta \left(1-\cos \theta \right)$
• $d_{2,-2}^{2}={\frac {1}{4}}\left(1-\cos \theta \right)^{2}$
• $d_{1,1}^{2}={\frac {1}{2}}\left(2\cos ^{2}\theta +\cos \theta -1\right)$
• $d_{1,0}^{2}=-{\sqrt {\frac {3}{8}}}\sin 2\theta$
• $d_{1,-1}^{2}={\frac {1}{2}}\left(-2\cos ^{2}\theta +\cos \theta +1\right)$
• $d_{0,0}^{2}={\frac {1}{2}}\left(3\cos ^{2}\theta -1\right)$

Literatura

• Abramowitz, Milton; Stegun, Irene A., eds. , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. . New York: Dover. 1965. ISBN 978-0486612720. 
• Wigner E. P., Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, New York: Academic Press (1959)
• Messiah, Albert, Quantum Mechanics (Volume II) (12th ed.). . New York: North Holland Publishing. 1981. ISBN 978-0-7204-0045-8.