VIII. Addition of Angular Momenta

Transcription

1 VIII Addition of Anguar Momenta a Couped and Uncouped Bae When deaing with two different ource of anguar momentum, Ĵ and Ĵ, there are two obviou bae that one might chooe to work in The firt i caed the uncouped bai Here the bai ket are eigentate of both operator: Ĵ j, m ; j, m j ( j + ) j, m ; j, m Ĵ z j, m ; j, m m j, m ; j, m Ĵ j, m ; j, m j ( j + ) j, m ; j, m Ĵ z j, m ; j, m m j, m ; j, m In the cae of pin orbit couping, thi woud mean that our bai tate woud be imutaneou eigenfunction of orbita and pin anguar momentum, and each tate woud have a particuar vaue for {, m } and m Note that thi i ony poibe if (a we aume): [J ˆ, Jˆ ] 0 otherwie, the two operator woud not have imutaneou eigenfunction Thi i ceary true for L ˆ and Ŝ ince the two operator act on different pace Thi reay define what we mean by different anguar momenta, ince operator that do not commute wi, in ome ene, define overapping and thu not competey ditinct form of anguar momentum Thi bai i appropriate if Ĵ and Ĵ do not interact However, when the Hamitonian contain an interaction between thee two anguar momenta (uch a Ĵ Ĵ ), the eigentate wi be mixture of the uncouped bai function and thi bai become omewhat awkward In thee cae, it i eaiet to work in the couped bai, which we now deveop Firt, note that the tota anguar momentum i given by:

2 J ˆ J ˆ + J ˆ It i eay to how that thi i, in fact, an anguar momentum (ie [J ˆ, Jˆ y ] i Jˆ x z ) We can therefore aociate two quantum number, j and m, with the eigentate of tota anguar momentum indicating it magnitude and projection onto the z axi The couped bai tate are eigenfunction of the tota anguar momentum operator Thi pecifie two quantum number for our bai tate ( j and m ) However, a we aw above, the uncouped bai tate were pecified by four quantum number ( j, j, m and m ) and we therefore need to pecify two more quantum number to fuy pecify the couped tate To pecify thee at two quantum number, we note that [J ˆ, Jˆ ˆ ˆ ˆ z ] [(J + J ˆ J + J ), J ˆ ] [ J ˆ J ˆ, Ĵ ] [J ˆ, Jˆ z ] J ˆ z z 0 and imiary for Ĵ z Thu Ĵ z and Ĵ z do not hare common eigenfunction with Ĵ To put it another way, to obtain a definite tate of the tota anguar momentum, one mut generay mix tate with different m and m A thu ead to the concuion that neither m nor m can be one of the other quantum number that pecify the couped bai What about j and j? We, [ J ˆ, J ˆ ] [(J ˆ + J ˆ ˆ ˆ J ˆ + J ˆ ), J ] [ J ˆ J ˆ, J ˆ ˆ ] [ˆ J, J ] J 0 and imiary for J ˆ Further ˆ ˆ ˆ [Ĵ, J ˆ z ] [Ĵ z + Ĵ z, J ] [J z, J ] 0 Hence, Ĵ, Ĵ z, J ˆ and J ˆ hare common eigenfunction, and thee eigenfunction define the couped bai To ay it another way, the eigentate of Ĵ do not mix tate with different j and j Thi i rather profound in the cae of pin-orbit couping it mean that tate with different vaue of wi not be mixed by the couping The appropriate quantum number for the couped bai are j, m, j and j and we have:

3 J ˆ j,m; j, j j( j + ) j,m; j, j Ĵ j,m; j, j m j,m; j, j z J ˆ j,m; j, j j ( j + ) j,m; j, j J ˆ j,m; j, j j ( j + ) j,m; j, j Typicay, certain matrix eement wi be eaier to compute in the couped bai, whie other wi be eaier to compute in the uncouped bai Thu, we wi often need to tranform from one bai to the other Since the couped and uncouped bae are both eigenfunction of Hermitian operator, each form a compete bai for the anguar momentum and therefore we can write: j,m; j, j j ', m ; j ', m j ', m ; j ', m j,m; j, j j ', m j, ' m However, we have aready concuded that Ĵ doe not mix tate with different j and j o: j ', m ; j ', m j,m; j, j j,m ; j,m j,m; j, j δ j, j ' δ j, j ' A a reut the um over j ' and j ' coape to deta function and we get: j,m; j, j j,m ; j,m j,m ; j,m j,m; j, j m,m The tranformation coefficient j,m ; j,m j,m; j, j are known a the Cebch-Gordon (CG) coefficient (or the vector couping coefficient) The CG matrix i unitary (ince it jut tranform a vector from one bai to another) and by convention it eement are choen rea (reca that the phae of j,m; j, j i arbitrary) There i one additiona ymmetry that the CG coefficient poe Notice that: ˆ ˆ ˆ 0 (m J ) j,m; j, j (m J z J z ) j,m; j, j z 0 j,m ; j,m (m Jˆ z J ˆ z ) j,m; j, j 0 (m m m ) j,m ; j,m j,m; j, j Thi impie that either the CG coefficient i zero, or

4 m m + m Thu, for a the non-zero CG coefficient, the index m i actuay redundant; it i away given by the um of m and m b Recurion Reation There are evera way to determine the CG coefficient Perhap the eaiet i to ook them up in a book (they have been extenivey tabuated) However, thi method i the mot prone to error, une you are very carefu to foow a of the ign convention of the text at hand (which may not be the ame a the ign convention in, ay CTDL or thee ecture note) Another route i to impy view the whoe thing a an eigenvaue probem: one impy wihe to determine the eigentate of Ĵ in the uncouped bai The coefficient of the different eigenvector are the CG coefficient However, thi mie out on what i probaby the mot important apect of anguar momentum couping the abiity without any ignificant computation to predict the aowed quantum number and their degeneracie We wi foow a third route to obtain the CG coefficient Thi note that the coefficient are eaiy obtained by recurion, in a manner imiar to what we ued for the pherica harmonic Firt we note that there i ony one non-zero coefficient for m m + m max max max : j, m max ; j, m max j, m max ; j, j No other combination of m and m wi give the correct tota m Thu, the tate j, m max ; j, j and j, m max ; j, m max are equa up to an unimportant contant By convention, thi contant i choen to be We can generate the other coefficient by ucceive appication of the owering operator and judiciou ue of orthogonaity contraint To ee how thi i appied in practice, it i bet to ue pin-orbit couping a an exampe What i the couped bai for a pin-/ eectron with orbita anguar momentum? We can identify the uncouped quantum number:

5 j m m j m m And we want to determine the eigentate of Ĵ (L ˆ + S) ˆ and J ˆ L ˆ + S ˆ z z z We know now that the two tate with maximum m are equa: Couped: Uncouped: +, m + ;,, m ;, + Appying the owering operator, we have J ˆ +, m + ;, (J ˆ + J ˆ ), m ;, + )( ( + + ;, + + ( + )( + ), ),, ;, ;, + +, ;, However, we ao know that J ˆ +, m + ;, ( )( + + ) +, m ;, + +, m ;, Combining thee at two expreion, +, m ;,, ;, + +, ;, + + Thi give u the expreion for the tate with the ame j, and but m m max One can check that thi tate i normaized We can ceary appy thi recurivey to obtain the tate with m m max, m m max 3, etc Ceary thi wi ceae when m m mi n To get ome inight into what thee tate ook ike, we need to make our notation a itte e expicit, temporariy Firt we wi deete the indice for and from a the bra and ket tate ince thee quantum number are the ame throughout the cacuation So, for exampe, j, m ;, j, m and, m ;, m m ; m Second, in term of the abbreviated tate abe, we wi write the above reationhip ymboicay +, +, ; + + ; where mean roughy negecting any contant that are not reevant for the point I want to make In thi cae, they are factor

6 invoving and that wi be very important for doing cacuation but impede our undertanding at the outet Uing thi notation, we can write the next owered tate: 3 +, Ĵ +, (Jˆ J ˆ ; + + Jˆ + Jˆ ; ) ( ) + ; + + ; In fact, if we make row of each of the couped tate we can make a fow chart for the different component: ; +, + + +, ; + + ; + 3, ; + + ; +, + ; ; +, ; So we ee that the characteritic action of the owering operator i to connect upper tate in (m, m ) pace to tate that are beow and to the right of the origina tate Thi action i imited by the fact that m cannot be e than (thi determine the height of the adder) and m cannot be e than (thi determine the width of the rung) c The triange rue So, have we now created a the couped tate? We, the tota number of couped tate in the adder i j + ( + ) +

7 Meanwhie, the number of uncouped tate i ( + )( + ) ( + ) 4 + Since the number of couped and uncouped tate mut be equa, we are miing tate To find the miing tate, notice that a of the above tate have j + That i, we have preumed that and point in the ame direction (equivaenty, we have preumed that they are rotating in the ame pane an in the ame ene cockwie or countercockwie) Thi i ceary an unneceary retriction Hence, we expect there to be another poibiity for j - pecificay, in order to account for a the miing tate, we expect j +, or j To buid thee miing tate, we note that there are two uncouped function with m m max : m ; m and m ; m Meanwhie, we have ony found one couped tate m ; m + : j + ; m Baed on the above argument, we predict there wi ao be a tate j ; m To find it we note that ) thi tate mut be a inear combination of m ; m and m ; m, ince other tate do not conerve m ) the new tate mut be orthogona to the j + ; m, ince both tate are eigentate of the ame operator ( Ĵ ) Uing our expicit expreion for j + ; m (above), it i eay to how that the normaized tate atifying ) and ) i:, m, ;,, ;, +, ; + + where we have once again made an arbitrary choice of phae, o that the coefficient of the firt term i poitive Starting from thi tate, we can appy the owering operator recurivey to generate the tate, m;, with a other poibe vaue of m How do thee adder generaize to arbitrary m and m? We, one find that the poibe vaue of j fa between two imit: j j j j + j Thi may be famiiar to ome of you a the triange rue of anguar momentum couping Phyicay, thi come from the fact that the maximum number of tate we can have with a given m i j +

8 (auming j > j ) Pictoriay, thi correpond to the maximum width of the rung on our adder Now, when we write out a tabe of the m vaue for each j, we find: jj +j jj +j - jj +j - jj +j -3 m j +j X mj +j - X X mj +j - X X X mj +j -3 X X X X X X X X m-j -j +3 X X X X m-j -j + X X X m-j -j + X X m-j -j X We ee that having n different vaue for j require u to have n different tate with a given vaue of m For exampe, in the chart above, we woud need four tate with mj +j -3 to upport the four vaue of j that are ited Since the maximum degeneracy of each m eve i j +, there are j + poibe vaue of j and we concude that they are j j j j + j We can ao check that the number of tate predicted by the triange rue agree with what we know from the uncouped bai For each vaue of j we have j + tate, o the tota number of tate i (if we aume that j > j ): ( ( j + j ) + ) + ( ( j + j ) + ) + + ( ( j j + ) + ) + ( ( j j ) + ) Or, rearranging the term, ( j + ) + j + ( j + ) + j + + ( j + ) j + + ( j + ) j The term outide the parenthee ceary um to zero Meanwhie, there are exacty j + copie of the term in parenthee Hence, the number of tate i ( j + )( j + ) which i preciey the number of tate in the uncouped repreentation