The state vectors j, m transform in rotations like D(R) j, m = m j, m j, m D(R) j, m. m m (R) = j, m exp. where. d (j) m m (β) j, m exp ij )

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1 Anguar momentum agebra It is easy to see that the operat J J x J x + J y J y + J z J z commutes with the operats J x, J y and J z, [J, J i ] 0 We choose the component J z and denote the common eigenstate of the operats J and J z by j, m We know QM II that J j, m jj + j, m, j 0,,, 3, J z j, m m j, m, m j, j +,, j, j We define the adder operats J + and J : J ± J x ± ij y They satisfy the commutation reations We see that and [J +, J ] J z [J z, J ± ] ±J ± [ J, J ± ] 0 J z J + j, m J + J z j, m m + J + j, m J J + j, m J + J j, m jj + J + j, m, so we must have J + j, m c + j, m + The fact c + can be deduced from the nmaization condition j, m j, m δ jj δ mm We end up with J ± j, m j mj ± m + j, m ± Matrix eements wi be j, m J j, m jj + δ j jδ m m j, m J z j, m mδ j jδ m m j, m J ± j, m j mj ± m + δ j jδ m,m± We define Wigner s function: Since D j R j, m exp ij ˆnφ j, m [J, DR] [J ij ˆnφ, exp ] 0, we see that DR does not chance the j-quantum number, so it cannot have non zero matrix eements between states with different j vaues The matrix with matrix eements D j R is the j + -dimensiona irreducibe representation of the rotation operat DR The matrices D j R fm a group: The product of matrices beongs to the group: D j m m R R D j m m R D j R, m where R R is the combined rotation of the rotations R and R, the inverse operation beongs to the group: D j R D j mm R The state vects j, m transfm in rotations ike DR j, m m j, m j, m DR j, m m j, m D j R With the hep of the Euer anges D j R where j, m exp ij zα exp ij yβ exp ij zγ j, m e im α+mγ d j β, d j β j, m exp ij yβ j, m Functions d j can be evauated using Wigner s fmua d j β k k m+m j + m!j m!j + m!j m! j + m k!k!j k m!k m + m! cos β j k+m m sin β k m+m Orbita anguar momentum The components of the cassicay anaogous operat L x p satisfy the commutation reations [L i, L j ] iɛ ijkl k Using the spherica codinates to abe the position eigenstates, x r, θ, φ, one can show that x L z α i φ x α x L x α i sin φ cot θ cos φ θ φ x L y α i cos φ cot θ sin φ θ φ x L ± α ie ±iφ ±i θ cot θ φ [ x L α sin θ φ + sin θ θ sin θ ] θ

2 We denote the common eigenstate of the operats L and L z by the ket-vect, m, ie L z, m m, m L, m +, m Since R 3 can be represented as the direct product R 3 R Ω, where Ω is the surface of the unit sphere positiondistance from the igin and direction the position eigenstates can be written crespondingy as x r ˆn Here the state vects ˆn fm a compete basis on the surface of the sphere, ie dωˆn ˆn ˆn We define the spherica harmonic function: Y m θ, φ Y m ˆn ˆn, m The scaar product of the vect ˆn with the equations gives and [ sin θ Y m and D The state L z, m m, m L, m +, m i φ Y m θ, φ my m θ, φ sin θ + θ θ sin θ ˆn θ, φ ] + + Y m 0 φ is obtained from the state ẑ rotating it first by the ange θ around y-axis and then by the ange φ around z-axis: Furtherme so ˆn DR ẑ Dα φ, β θ, γ 0 ẑ,m Dφ, θ, 0, m, m ẑ, m ˆn Y θ, φ m, m ẑ Y m Y m + θ, φ D φ, θ, 0, m ẑ + 0, φ δ m0, D m0 φ, θ, γ 0 D m0 α, β, 0 As a specia case + Y θ, φ β,α D 00 θ, φ, 0 d 00 θ P cos θ Couping of anguar momenta We consider two Hibert spaces H and H If now A i is an operat in the space H i, the notation A A means the operat A A α β A α A β in the product space Here α i H i In particuar, A α β A α β, where i is the identity operat of the space H i Crespondingy A operates ony in the subspace H of the product space Usuay the subspace of the identity operats, even the identity operat itsef, is not shown, f exampe A A A It is easy to verify that operats operating in different subspace commute, ie [A, A ] [A, A ] 0 In particuar we consider two anguar momenta J and J operating in two different Hibert spaces They commute: [J i, J j ] 0 The infinitesima rotation affecting both Hibert spaces is ij ˆnδφ ij ˆnδφ ij + J ˆnδφ The components of the tota anguar momentum J J + J J + J obey the commutation reations [J i, J j ] iɛ ijk J k, ie J is anguar momentum A finite rotation is constructed anaogousy: D R D R exp J ˆnφ exp J ˆnφ Base vects of the whoe system We seek in the product space { j m j m } f the maxima set of commuting operats i J, J, J z and J z

3 Their common eigenstates are simpy direct products j j ; m m j, m j, m If j and j can be deduced from the context we often denote m m j j ; m m The quantum numbers are obtained from the eigenequations J j j ; m m j j + j j ; m m J z j j ; m m m j j ; m m J j j ; m m j j + j j ; m m J z j j ; m m m j j ; m m ii J, J, J and J z Their common eigenstate is denoted as shty j j ; jm jm j j ; jm if the quantum numbers j and j can be deduced from the context The quantum numbers are obtained from the equations J j j ; jm j j + j j ; jm J j j ; jm j j + j j ; jm J j j ; jm jj + j j ; jm J z j j ; jm m j j ; jm [J, J z ] 0, [J, J z ] 0, so we cannot add to the set i the operat J, n to the set ii the operats J z J z Both sets are thus maxima and the cresponding bases compete and thonma, ie j j ; m m j j ; m m j j m m j j ; jm j j ; jm j j jm In the subspace where the quantum numbers j and j are fixed we have the competeness reations j j ; m m j j ; m m m m j j ; jm j j ; jm jm One can go from the basis i to the basis ii via the unitary transfmation j j ; jm m m j j ; m m j j ; m m j j ; jm, so aso the transfmation matrix C jm,mm j j ; m m j j ; jm is unitary The eements j j ; m m j j ; jm of the transfmation matrix are caed Cebsch-Gdan s coefficients Since J z J z + J z, we must have m m + m, so the Cebsch-Gdan coefficients satisfy the condition j j ; m m j j ; jm 0, if m m + m Further, we must have QM II j j j j + j It turns out, that the C-G coefficients can be chosen to be rea, so the transfmation matrix C is in fact thogona: j j ; m m j j ; jm j j ; m m j j ; jm jm δ mm δ m m m m j j ; m m j j ; jm j j ; m m j j ; j m δ jj δ mm As a specia case j j and m m m + m we get the nmaization condition m m j j ; m m j j ; jm Recursion fmuas Operating with the adder operats to the state j j ; jm we get J ± j j ; jm J ± + J ± m m j j ; m m j j ; m m j j ; jm, j mj ± m + j j ; j, m ± m m j m j ± m + j j ; m ±, m + j ± m j ± m + j j ; m, m ± j j ; m m j j ; jm Taking the scaar product on the both sides with the vect j j ; m m we get j mj ± m + j j ; m m j j ; j, m ± j m + j ± m j j ; m, m j j ; jm + j m + j ± m j j ; m, m j j ; jm

4 , - * + The Cebsch-Gdan coefficients are determined uniquey by the recursion fmuas the nmaization condition m m j j ; m m j j ; jm 3 the sign conventions, f exampe j j ; j m J z j j ; jm 0 Note Due to the sign conventions the der of the couping is essentia: Exampe L + S-couping j 0,,, m m, +,,, j s m m s ± { ± j, when > 0, when 0 I j j ; jm ± j j ; jm Graphica representation of recursion fmuas Recursion fmua in m m -pane We fix j, j and j Then m j, m j, m + m j Recursion when j and j s / Using the seection rue m m m, m m s and the shthand notation the J -recursion gives + + m + + m m, +, m + m + m + m +, +, m +, m, +, m + m + + m + 3 m +, +, m + Appying the same recursion repeatedy we have Using recursion fmuas We see that > B H A every C-G coefficient depends on A, the nmaization condition determines the absoute vaue of A, 3 the sign is obtained from the sign conventions m, +, m + m + + m m m + 5 m + 3, +, m + + m + + m m m m m + 7 m + 5, +, m m + +, +, + If j j max j + j and m m max j + j one must have j j ; jm j j ; m j, m j j j ; jm j m j m

5 the nmaization condition j j ; m j, m j j j ; jm and the sign convention give j j ; m j, m j j j ; jm Thus, in the case of the spin-bit couping,, +, +, m, +, m + m + + With the hep of the recursion reations, nmaization condition and sign convention the rest of the C-G coefficients can be evauated, too We get j +, m j, m + m + + m + + m m, m s m m +, m s m m + + Rotation matrices If D j R is a rotation matrix in the base { j m m j,, j } and D j R a rotation matrix in the base { j m m j,, j }, then D j R D j R is a rotation matrix in the j + j + -dimensiona base { j, m j, m } Seecting suitabe superpositions of the vects j, m j, m the matrix takes the fm ike On the other hand we have j j ; m m DR j j ; m m j j ; m m j j ; jm jm j m j j ; jm DR j j ; j m j j ; j m j j ; m m j j ; m m j j ; jm D j mm Rδ jj jm j m j j ; m m j j ; j m We end up with the Cebsch-Gdan series D j m m RD j m m R j m As an appication we have dωy θ, φy m m j j ; m m j j ; jm j j ; m m j j ; jm D j mm R θ, φy m θ, φ ; 00 ; 0 ; m m ; m D j R D j R D j+j 0 D j+j 0 D j j One can thus write D j D j D j+j D j+j D j j As a check we can cacuate the dimensions: j + j + j + j + + j + j j j + The matrix eements of the rotation operat satisfy j j ; m m DR j j ; m m j m DR j m j m DR j m D j m m RD j m m R