Klebš-Gordanovi koeficijenti

Klebš-Gordanovi koeficijenti sa oznakom $\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$ ili $C_{j_{1}m_{1}j_{2}m_{2}}^{JM}$koriste se u matematici i fizici da bi se za Lijeve grupe dekomponovao tenzorski proizvod dve ireducibilne reprezentacije. Koriste se i prilikom sabiranja ugaonih momenata. Imenovani su u čast nemačkih matematičara Alfreda Klebša i Paula Alberta Gordana.

Definicija

Neka Lijeva grupa $G$ ima dve ireducibilne reprezentacije $D^{(\alpha )}$ i $D^{(\beta )}$. Vektori baze u dve reprezentacije pretpostavimo da su $\psi _{\mu }^{(\alpha )}$ i $\psi _{\nu }^{(\beta )}$. Ireducibilni tenzorski operator predstavlja tenzorske komponente ${\hat {F}}_{\chi }^{(k)}$, koje se transformišu po ireducibilnim reprezentacijama grupe, tj. ako zadovoljavaju uslov:

${\tilde {\hat {F}}}_{\chi }^{(k)}=\sum _{\chi '}D_{\chi '\chi }^{(k)}(g){\hat {F}}_{\chi '}^{(k)}.$

Vektori $|{\hat {F}}_{\chi }^{(k)}\psi _{\nu }^{(\beta )}\rangle$, gde $\chi =1,\;2,\;\ldots ,\;f_{k};\;\nu =1,\;2,\;\ldots ,\;f_{\beta }$ obrazuju bazu reprezentacije od $D^{(k)}\times D^{(\beta )}$. U opštem slučaju taj prikaz je reducibilan, pa se dade prikazati pomoću linearnih kombinacija baze ireducibilih reprezentacija. Dobija se:

$|{\hat {F}}_{\chi }^{(k)}\psi _{\nu }^{(\beta )}\rangle =\sum _{\gamma \rho }\langle k\chi ,\;\beta \nu \vert \gamma \rho \rangle \{{\hat {F}}^{(k)}\psi ^{(\beta )}\}_{\rho }^{\gamma }.$

Tako dani koeficijenti $\langle k\chi ,\;\beta \nu \vert \gamma \rho \rangle$ nazivaju se opšti Klebš-Gordanovi koeficijenti grupe $G$.

Operatori ugaonih momenata

Operatori ugaonih momenata su autoadjungirani operatori, koji zadovoljavaju relacije komutacije:

$[{\textrm {j}}_{k},{\textrm {j}}_{l}]={\textrm {j}}_{k}{\textrm {j}}_{l}-{\textrm {j}}_{l}{\textrm {j}}_{k}=i\hbar \sum _{m}\varepsilon _{klm}{\textrm {j}}_{m},\quad \mathrm {gde} \quad k,l,m\in (x,y,z)$

a $\varepsilon _{klm}$ je Levi-Čivita simbol. Tri operatora zajedno čine vektorski operator: $\mathbf {j} =[{\textrm {j}}_{x},{\textrm {j}}_{y},{\textrm {j}}_{z}]$

$\mathbf {j} ^{2}={\textrm {j}}_{x}^{2}+{\textrm {j}}_{y}^{2}+{\textrm {j}}_{z}^{2}.\,$ je primer Kazimirovoga operatora.

${\textrm {j}}_{\pm }={\textrm {j}}_{x}\pm i{\textrm {j}}_{y}.\,$

Stanja ugaonih momenata

Iz gornjih definicija dobija se da $\mathbf {j} ^{2}$ komutira sa ${\textrm {j}}_{x}$, ${\textrm {j}}_{y}$ and ${\textrm {j}}_{z}$:

$[\mathbf {j} ^{2},{\textrm {j}}_{k}]=0\ \mathrm {za} \ k=x,y,z$

Kada dva ermitska operatora komutiraju tada postoji zajednički skup svojstvenih funkcija. Odaberu li se $\mathbf {j} ^{2}$ i ${\textrm {j}}_{z}$ onda nalazimo svojstvena stanja koristeći komutacione relacije:

${\begin{alignedat}{2}\mathbf {j} ^{2}|j\,m\rangle =\hbar ^{2}j(j+1)|j\,m\rangle &\;\;\;j=0,{\frac {1}{2}},1,{\frac {3}{2}},2,\ldots \\{\textrm {j}}_{z}|j\,m\rangle =\hbar m|j\,m\rangle &\;\;\;m=-j,-j+1,\ldots ,j.\end{alignedat}}$

S druge strane operatori ${\textrm {j}}_{+}$ i ${\textrm {j}}_{-}$ menjaju $m$ vrednosti:

${\textrm {j}}_{\pm }|j\,m\rangle =C_{\pm }(j,m)|j\,m\pm 1\rangle$

$C_{\pm }(j,m)={\sqrt {j(j+1)-m(m\pm 1)}}={\sqrt {(j\mp m)(j\pm m+1)}}.$

Stanja ugaonih momenata mora da budu ortogonalana i normalizirana:

$\langle j_{1}\,m_{1}|j_{2}\,m_{2}\rangle =\delta _{j_{1},j_{2}}\delta _{m_{1},m_{2}}.$

Tenzorski proizvod

Neka $V_{1}$ predstavlja $2j_{1}+1$-dimenzionalni vektorski prostor sa bazom određenom stanjima:

$|j_{1}m_{1}\rangle ,\quad m_{1}=-j_{1},-j_{1}+1,\ldots j_{1}$

Drugi prostor $V_{2}$ neka je $2j_{2}+1$-dimenzionalni vektorski prostor sa bazom određenom stanjima:

$|j_{2}m_{2}\rangle ,\quad m_{2}=-j_{2},-j_{2}+1,\ldots j_{2}.$

Tenzorski proizvod tih prostora $V_{12}\equiv V_{1}\otimes V_{2}$ je $(2j_{1}+1)(2j_{2}+1)$ dimenzionalan prostor sa bazom:

$|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \equiv |j_{1}m_{1}\rangle \otimes |j_{2}m_{2}\rangle ,\quad m_{1}=-j_{1},\ldots j_{1},\quad m_{2}=-j_{2},\ldots j_{2}.$

Dejstvo operatora na takvoj bazi može se definisati pomoću:

$({\textrm {j}}_{i}\otimes 1)|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \equiv (j_{i}|j_{1}m_{1}\rangle )\otimes |j_{2}m_{2}\rangle$

i

$(1\otimes {\textrm {j}}_{i})|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \equiv |j_{1}m_{1}\rangle \otimes j_{i}|j_{2}m_{2}\rangle \quad \mathrm {za} \quad i=x,y,z.$

Ukupni ugaoni moment se onda može definisati sa:

${\textrm {J}}_{i}={\textrm {j}}_{i}\otimes 1+1\otimes {\textrm {j}}_{i}\quad \mathrm {za} \quad i=x,y,z.$

Ugaoni momenti zadovoljavaju komutacione relacije:

$[{\textrm {J}}_{k},{\textrm {J}}_{l}]=i\hbar \epsilon _{klm}{\textrm {J}}_{m},\quad \mathrm {} \quad k,l,m\in (x,y,z)$ pa sledi:

${\begin{aligned}\mathbf {J} ^{2}|(j_{1}j_{2})JM\rangle &=\hbar ^{2}J(J+1)|(j_{1}j_{2})JM\rangle \\{\textrm {J}}_{z}|(j_{1}j_{2})JM\rangle &=\hbar M|(j_{1}j_{2})JM\rangle ,\quad \mathrm {za} \quad M=-J,\ldots ,J.\end{aligned}}$

Ukupni ugaoni moment treba da zadovoljava triangularnu relaciju:

$|j_{1}-j_{2}|\leq J\leq j_{1}+j_{2}.$

Ukupan broj svojstvenih stanja jednak je dimenziji $V_{12}$

$\sum _{J=|j_{1}-j_{2}|}^{j_{1}+j_{2}}(2J+1)=(2j_{1}+1)(2j_{2}+1).$

Formalna definicija koeficijenata

Stanja ukupnoga ugaonoga momenta mogu se razviti:

$|(j_{1}j_{2})JM\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1}m_{1}j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$

a koeficijenti $\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$ toga razvoja nazivaju se Klebš-Gordanovi koeficijenti. Ukoliko na obe strane gornjega izraza primenimo operator ${\textrm {J}}_{z}={\textrm {j}}_{z}\otimes 1+1\otimes {\textrm {j}}_{z}$ onda možemo da vidimo da su koeficijenti različiti od nule samo ako je $M=m_{1}+m_{2}.\,$

Rekurzije

Uz pomoć operatora ${\textrm {J}}_{\pm }={\textrm {j}}_{\pm }\otimes 1+1\otimes {\textrm {j}}_{\pm }$ dobijamo:

${\textrm {J}}_{\pm }|(j_{1}j_{2})JM\rangle =C_{\pm }(J,M)|(j_{1}j_{2})JM\pm 1\rangle =C_{\pm }(J,M)\sum _{m_{1}m_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\pm 1\rangle .$

Primenimo li isti operator na desnu stranu prve jednačine iz prošloga poglavlja dobija se:

${\begin{aligned}{\textrm {J}}_{\pm }&\sum _{m_{1}m_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle \\&=\sum _{m_{1}m_{2}}\left[C_{\pm }(j_{1},m_{1})|j_{1}m_{1}\pm 1\rangle |j_{2}m_{2}\rangle +C_{\pm }(j_{2},m_{2})|j_{1}m_{1}\rangle |j_{2}m_{2}\pm 1\rangle \right]\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle \\&=\sum _{m_{1}m_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \left[C_{\pm }(j_{1},m_{1}\mp 1)\langle j_{1}{m_{1}\mp 1}j_{2}m_{2}|JM\rangle +C_{\pm }(j_{2},m_{2}\mp 1)\langle j_{1}m_{1}j_{2}{m_{2}\mp 1}|JM\rangle \right].\end{aligned}}$

Kombinujući te rezultate dobija se rekurzija:

$C_{\pm }(J,M)\langle j_{1}m_{1}j_{2}m_{2}|JM\pm 1\rangle =C_{\pm }(j_{1},m_{1}\mp 1)\langle j_{1}{m_{1}\mp 1}j_{2}m_{2}|JM\rangle +C_{\pm }(j_{2},m_{2}\mp 1)\langle j_{1}m_{1}j_{2}{m_{2}\mp 1}|JM\rangle .$

Uzmemo li $M=J$ dobijamo:

$0=C_{+}(j_{1},m_{1}-1)\langle j_{1}{m_{1}-1}j_{2}m_{2}|JJ\rangle +C_{+}(j_{2},m_{2}-1)\langle j_{1}m_{1}j_{2}m_{2}-1|JJ\rangle .$

Ortogonalnost

$\sum _{J=|j_{1}-j_{2}|}^{j_{1}+j_{2}}\sum _{M=-J}^{J}\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle \langle JM|j_{1}m_{1}'j_{2}m_{2}'\rangle =\langle j_{1}m_{1}j_{2}m_{2}|j_{1}m_{1}'j_{2}m_{2}'\rangle =\delta _{m_{1},m_{1}'}\delta _{m_{2},m_{2}'}$

$\sum _{m_{1}m_{2}}\langle JM|j_{1}m_{1}j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|J'M'\rangle =\langle JM|J'M'\rangle =\delta _{J,J'}\delta _{M,M'}.$

Eksplicitan prikaz koeficijenata

$\langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle =$

$\delta _{m,m_{1}+m_{2}}{\sqrt {\frac {(2j+1)(j+j_{1}-j_{2})!(j-j_{1}+j_{2})!(j_{1}+j_{2}-j)!}{(j_{1}+j_{2}+j+1)!}}}\ \times$

${\sqrt {(j+m)!(j-m)!(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!}}\ \times$

$\sum _{k}{\frac {(-1)^{k}}{k!(j_{1}+j_{2}-j-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(j-j_{2}+m_{1}+k)!(j-j_{1}-m_{2}+k)!}}.$

Specijalni slučajevi

Za $J=0$ Klebš-Gordanovi koeficijenti su:

$\langle j_{1},m_{1};j_{2},m_{2}|00\rangle =\delta _{j_{1},j_{2}}\delta _{m_{1},-m_{2}}{\frac {(-1)^{j_{1}-m_{1}}}{\sqrt {2j_{2}+1}}}.$

Za $J=j_{1}+j_{2}$ i $M=J$ imamo

$\langle j_{1},j_{1};j_{2},j_{2}|j_{1}+j_{2},j_{1}+j_{2}\rangle =1.$

Za $j_{1}=j_{2}=J/2$ i $m_{2}={-m_{1}}$ vredi:

$\langle j_{1},m_{1};j_{1},{-m_{1}}|2j_{1},0\rangle ={\frac {(2j_{1})!^{2}}{(j_{1}-m_{1})!(j_{1}+m_{1})!{\sqrt {(4j_{1})!}}}}.$

Za $j_{1}=j_{2}=m_{1}={-m_{2}}$ vredi:

$\langle j_{1},j_{1};j_{1},{-j_{1}}|J,0\rangle =(2j_{1})!{\sqrt {\frac {2J+1}{(J+2j_{1}+1)!(2j_{1}-J)!}}}.$

Za $j_{2}=1,m_{2}=0$ imamo:

${\begin{aligned}\langle j_{1},m;1,0|j_{1}+1,m\rangle &={\sqrt {\frac {(j_{1}-m+1)(j_{1}+m+1)}{(2j_{1}+1)(j_{1}+1)}}},\\\langle j_{1},m;1,0|j_{1},m\rangle &={\frac {m}{\sqrt {j_{1}(j_{1}+1)}}},\\\langle j_{1},m;1,0|j_{1}-1,m\rangle &=-{\sqrt {\frac {(j_{1}-m)(j_{1}+m)}{j_{1}(2j_{1}+1)}}}.\end{aligned}}$

Simetrije

${\begin{aligned}\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle &=(-1)^{j_{1}+j_{2}-J}\langle j_{1}\,{-m_{1}}j_{2}\,{-m_{2}}|J\,{-M}\rangle \\&=(-1)^{j_{1}+j_{2}-J}\langle j_{2}m_{2}j_{1}m_{1}|JM\rangle \\&=(-1)^{j_{1}-m_{1}}{\sqrt {\frac {2J+1}{2j_{2}+1}}}\langle j_{1}m_{1}J\,{-M}|j_{2}\,{-m_{2}}\rangle \\&=(-1)^{j_{2}+m_{2}}{\sqrt {\frac {2J+1}{2j_{1}+1}}}\langle J\,{-M}j_{2}m_{2}|j_{1}\,{-m_{1}}\rangle \\&=(-1)^{j_{1}-m_{1}}{\sqrt {\frac {2J+1}{2j_{2}+1}}}\langle JMj_{1}\,{-m_{1}}|j_{2}m_{2}\rangle \\&=(-1)^{j_{2}+m_{2}}{\sqrt {\frac {2J+1}{2j_{1}+1}}}\langle j_{2}\,{-m_{2}}JM|j_{1}m_{1}\rangle \end{aligned}}$

Veza sa 3-jm simbolima i D-matricama

Klebš-Gordanovi koeficijenti povezani su sa 3-j simbolima:

$\langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle =(-1)^{j_{1}-j_{2}+m_{3}}{\sqrt {2j_{3}+1}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&-m_{3}\end{pmatrix}}.$

Integracijom tri Vignerove D matrice dobija se Klebš Gordanov koeficijent:

$\int _{0}^{2\pi }d\alpha \int _{0}^{\pi }\sin \beta d\beta \int _{0}^{2\pi }d\gamma D_{MK}^{J}(\alpha ,\beta ,\gamma )^{\ast }D_{m_{1}k_{1}}^{j_{1}}(\alpha ,\beta ,\gamma )D_{m_{2}k_{2}}^{j_{2}}(\alpha ,\beta ,\gamma )={\frac {8\pi ^{2}}{2J+1}}\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle \langle j_{1}k_{1}j_{2}k_{2}|JK\rangle .$

Literatura

• 3j, 6j i 9j simboli
• Abramowitz, Milton; Stegun, Irene A., eds. , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. . New York: Dover. 1965. ISBN 978-0-486-61272-0. 
• Edmonds, A. R., Angular Momentum in Quantum Mechanics, Princeton. . New Jersey: Princeton University Press. 1957. ISBN 978-0-691-07912-7. 
• Messiah, Albert , Quantum Mechanics (Volume II) (12th ed.). . New York: North Holland Publishing. 1981. ISBN 978-0-7204-0045-8.