Angular momentum. Reminder: orbital momentum Angular momentum operator : orbital momentum and spin

Transcription

1 Remnder: orbal momenum operaor : orbal momenum and spn In hs lecures we show how properes of angular momenum can be derved from properes of angular momenum operaors -angular momenum operaors and marx represenaon -spn, spn ½ saes -magnec momen and Paul equaon

2 Remnder, orbal momenum [ L, L j ] [ L j, L k ] jk L L Y lm, l l Y lm, L z m= -l, -l +,,, l-, l m L l,m l l l,m e m m neger L z l,m m l,m We have derved he form of egenvalues of orbal momenum by solvng he egenvalue equaons and mposng reasonable condon We obaned ha l=neger and m= neger and changes from -l o l

3 # "!!! # #! #! $ $!! % ) ' ( + & * Every operaor whch fulflls commuaon rules of orbal momenum operaor wll be called angular momenum operaor We wll see wha we can learn abou egensaes from commuaon rules alone, we wll see ha here are only wo possble physcally meanngful ypes of operaors lke ha- orbal momenum (L) and spn (S) [ J, J j ] j x, y, z [ J j, J k ] jk J J J y [, J y ] J [, ] z [ J y, ] J y J j,m j j j,m j,m m j,m

4 -, / O N K J G L I M H X S Q U R W V T P : < ; 5 _ ] b ` ^ Z Y a \ [ j g f = E C D A F B j,m J + j,m hus: [ j j (m) Rsng and lowerng operaors J + J y J y <jm J + j,m <jm J + j,m ed J y and J y J y ] <jm J <jm J J y J y J y [ J y, ] [ J y, ] jm jm [ j j J J J + h [ j j [ j j ] ] ] J j,m J jm j j j,m Egensae wh he same j j,m jm m j,m Egensae wh m- m= -j, -j +,,, j-, j j=, /,,3/,, 5/

5 q l o r m k s n u v x w z } { y ˆ ƒ Ž Š Œ š œ ž Marx represenaon for J= J + J y and J y p J + ; J y J + j,m [ j j ] j,m- J + j,m [ j j ] j,m+ <,n,m ~ [ ] n, m [ ] n, m <,n J y,m [ ] n, m [ ] n, m J y

6 Ÿ ª «± ¹ ³ ¼ µ» º ½ Å È Â Á ¾ Ç Æ Ä À Ã É Î Ï Ì Í Ê Ñ Marx represenaon for S=/ J + J y and J y J + ; J y J + j,m J + j,m S x [ j j [ j j </,n S x,m <,n S y,m Ë ² [ 3 4 [ 3 4 S y ] j,m- ] j,m+ ] ] n, m n, m m= +/, -/ :S=J [ 3 4 [ 3 4 S z Ð ] ] n, m n, m

7 Ø Ö Ô Ò Õ Ó ë æ á Ù ß í è ã éê äå à Ú ì ç â Þ ô ó ò ð ñ î ï ö ø õ ÿ ú þ ù ý û ü Spn ½ projecons Imagne un vecor n he drecon {x,y,z} ={ } sn cos,sn sn,cos n As wh usual vecors we projec spn vecor on a un vecor by a scalar produc S n S ÛÝÜ n S x sn cos S y sn sn S z cos S n cos sn e sn e cos The egenvecors of hs marx wll have well defned spn projecon along n S n cos sn e cos sn e S n sn e cos sn e cos

8 Spn projecons Now remember ha egenvecors of S_z, wh well defned spn projecon along z are S z S z cos sn e cos sn e Thus he probably o fnd ou egenvecor of S_z- measure projecon of hbar/ n he spn dreced over n s cos

9 ' & ( % $ 8 67 Magnec Momen Curren (charge) gong n a loop has magnec momen Energy of magnec momen n magnec feld s ( classcal) s B Classcally, magnec momen of a loop of curren s proporonal o I*A = curren*area If we consder a parcle wh charge q gong around n a loop hen I #q r v ; A r ; I A! " )qvr *ql m +!,- q L m ; /! g q J m The mysery explaned only by Drac equaon s ha for an elemenary spn=/ parcle gro-magnec facor g= Whle for orbal momenum we have g= Magnec momen s a ool o sudy srucure of parcles Neuron whch has no charge has a magnec momen 354 N 383 e S m N

10 M N O Z Y W T Q P UV S R X \ [? K ; H F 9 A I SJ = E G C L < Paul equaon Unlke orbal momenum, spn s no relaed o poson varable ( space), hus spn and space varables are oally ndependen [ S, r ] we can wre a wave funcon of spn=/ parcle n he followng way : r, s z + r - r + - r r ampludes o fnd he parcle wh approprae values of s_z ( /, -/) : : + - m V rb q m L B g S q m B + -

11 d ^ a b Sc ` e _ p r m g k o s n h q z y x Ž Œ Spn precesson ]d d + - g S e m B + - q=-e If parcle s no supposed o move n space- change s poson varables Paul equaon reduces o he one above Les assume B feld s along z f d d + - g S e m Bj Sl + - g S e m B z + - d d uwv g S e 4 m B - {w d d Šw }~ A g S e 4 m B - exp ƒ ˆ

12 š œ ž Spn precesson, con Les assume ha he sae s an egensae of S_x a =, wh projecon+ h/, s x In hs case: exp exp We can calculae expecaon values of s_x and s_y operaors o see f heyoscllae wh me < S x Ÿ exp, exp exp exp cos

13 ³ ± ² «ª ¹ Spn precesson < S y exp, exp µ sn exp exp Whle he z componen ( expecaon value of S_z) s zero o sar wh and does no change (why?) Spn precesses wh he frequency g S e m B ºeB - m Seng n elecron mass and magnec feld esla we ge he frequency of 8 bllon roaons per second (n muon expermen we are seng here a he deparmen spn precesson wll be used o measure pary volaon n he muon decay)

14 À ¾ ¼» ½ Addon of angular momena In mos of lfe problems we have many parcles, and each of hs parcles can be have boh orbal angular momenum (L) and spn (S) From classcal mechancs we know ha s he oal angular momenum of he sysem whch s conserved J, J Quanuechancally here are he followng quesons o be asked ) Wha are he allowed saes of he oal angular momenum? Is oal angular momenum sll angular momenum- does have all he properes of an angular momenum operaor? ) If so: wha s he relaon beween he saes of he oal angular momenum and he saes of ndvdual angular momena J_ L S

15 Ç Å Ã Â Á Æ Ä È Ò Ð Î Ì É Ô Ó Ö Ñ Ï Ë Ê Õ Í Ý Ø ß Þ Ú Û Ù Ü ã æ Addon of J_ and J_ Les magne we have wo parcles, wh angular momena J, J, J, L, S, We do no ye know how hey are composed of spn and orbal momenum, les assume now ha J_ and J_ are smply angular momena and ha quanuechancally hey have all he properes of angular momenum operaors, hus boh for J_ and J_ we have [ J,, J, j ] j x, y, z J, J, x J, y J, z [, J y ] ec J, j,m, j j j,m, J, z j,m, m j,m, m= -j, -j +,,, j-, j ) Queson: Is an angular momenum operaor? Jàâá J äâå çâè J

16 ì ï þ ü û ù ö ó ò ý ø õ ÿ ú ô $ & " % #! Properes of J éëê J íëî J ðëñ The be s, ha J s also an angular momenum operaor Les check an example commung rule : [, J y ]? J x J x, J z J z, J y J y J y [ J x J x, J y J y ] J x, J y J x, J y J z J z Snce J x, J y, J x, J x because he parcles, have nohng o do wh each oher and her respecve angular momenum operaors operae on saes of dfferen parcles Thus we mus have : J j,m j j j,m j,m m j,m m= -j, -j +,,, j-, j ) The queson now s : wha are j,m n relaon o j_, j_, m_, m_??

17 ' / ) * ( + 4-3, = ; 7 5? < 9 : 6 8 H F E D I A G C L J K Combned J sae of boh parcles J j, m j j j, m j, m m j, m J j, m j j j, m j, m m j, m Here ndex () and () refer o parcles Now, for he oal angular momenum we have : J J j,m j j j,m j,m m j,m We wll fnd he relaon beween j, m, j, m and j, m Les denoe he sae n whch one parcle has j_,m_ and oher j_, m_ by j, m j, m Les check wha are he values of J, J_z j, m j, m m j, m j, m m m m j, m j, m m m m j, m j, m

18 N M P O m and j of combned sae Thus z projecon of he combned momenum s jus he sum of z projecons for ndvdual parcles Wha abou j? m m m m_= -j_, -j_ +,,, j_-, j_ m_= -j_, -j_ +,,, j_-, j_ Snce m_max = j_+j_ and m_mn= -j_ -j_ we mus have j max j j Noe ha a correspondng number of m saes s (j_+j_)+ j, m j, m Noe ha we have ( j_ +)(j_+) of such saes Thus much more Thus here mus be oher values of j possble Wha are hey?

19 W R U V S Q T ` _ \ [ X ] Z ^ Y d b f c h g e a k l j n m p s q o r w v x u { y ƒ ~ } z Fndng j values, rsng and lowerng operaors J + J y J y and J y J y J y [ J y, ] J J + J y J y J y [ J y, ] J j,m J + j,m [ j j [ j j ] j,m- ] j,m+ Les make use of hs operaors o ry o fnd values of j : = m m J j, m j, m m m j, m j, m J + j, m j, m = J J J J j, m j, m

20 ˆ Š Ž Œ œ š Ÿ ž ª «³ ² ± º µ ¹ ¾ ½» À ¼ Are egensaes of J,JZ, J, JZ egensaes of J,Jz? I s easer o use hs form of J^ J J J J J J J J + = J J J J = J J J J J J = J j,m j,m j j j j m m m m m m m j j j j m j j m m m m j j j,m j,m + j,m- j,m+ + j,m+ j,m-

21 Å Á Æ Ä Â É È Î Ê Ï Í Ë Ò Ñ Ø Ö ÔÕ Ó Ú Ù ÝÞ ÛÜ connued We see ha n general j, m j, m s no an egensae of combned J^ (even f s an egensae of combned J_z) unless : j j j j Ths happens n wo cases : Ã m m Ì m m j j j j Ç m m Ð m m and m j or m j and m j or m j I should be clear ha wo cases: m j and m j correspond o j=j_+j_ Les check : m j and m j

22 ì ë ê é æ å è ç ó ò ñ ï î í ð â á à ß ä ã Combned saes J j,m j,m = j j j j j j j j,m j,m j j j j j j j j j j,m j,m = j,m j,m THUS : wo cases: m j and m j correspond o j j j Wha are abou oher saes of j whch mus exs? Wha s he mnmal value of j? We can fgure ha ou by checkng norms :

23 ú ø ý û ù õ ô ü ö þ ÿ Mnmal value of j: J + j,m <jm J + j,m <jm J jm [ j j ] = [ j j m m m m ] Les assume ha jj and nser m=j and m=-j Then J + j,m [ j j j j j j ] j j j If j<j we nser m=-j and m=j and consderng boh of hese cases we conclude: j j j We also conclude ha snce m and m can change by, also combned j value changes n seps of

24 ! # $ " % ( ' & ), / - * : G F D C B = < ; H E Glebsch-Gordan coeffcens We hus have : j j j j j, m j, m j j j j j, m j j, m j j, m j j, m j LH Bu n general: j, m j j m j m j j m, m m j C m, m j, m j, m Glebsch -Gordan coeffcens Les check he number of saes n he rgh hand sde and lef hand sde: j j j j j j j j j j j j j j j = = j j j j j j j j j j 4j j j RH

25 O M K N L J Q P I Z Y U T S W X V R [ ^ ] \ _ b a ` d e c g f n k h m j u p o r w q v s Glebsch Gordan coeffcens How o fnd all he coeffcens? General formula s no avalable, bu one can fnd hem consrucng he j,m saes For example, s easy o buld all saes wh j=j+j and all values of m by applyng m-lowerng operaor J- : j J j j j, m j j j j j,m- J j, m j j, m j J J j, m j j, m j = j j m m j, m j, m + j j m m j, m j, m = j j, j j, j + j j, j j, j j j, j j lj j, j j, j j, j j, j j j j j j j, j j j j j j, j j, j j j j j, j j, j

26 ~ y x { } z Œ Š ˆ Ž ƒ ž œ š Ÿ J=/ and j=/ Le's ry he las formula for wo spns j=/ and j=/ We expec: j=j+j= and j=j-j = We should have he followng four saes: j= m=,,- and j=, m= From wha we know already we can consruc wo saes :, = /,/ /,/ and,- = /,-/ /,-/ Usng he formula: j j j, j j j, j j, j j, j j, j j j j j We ge :,,,,,,,,,, Wha s he j=,m= sae? I mus be orhogonal, normalzed and have Jz= j

27 ª «µ ² ± ¹ Â ¾ ½ º ¼» Ä Á Ã À j=/,j=/ The only possbly lef for j=,m= sae s :,,,,,, = /,/ /,/,- = /,-/ /,-/,,, ³ How o check f j= s rue? We can use,, J J J J J J J Check as a par of homeworks Ths noes wll connue