Lctur 18 1
Classical Magntic Dipol In gnral, a particl of mass m and charg q (not ncssarily a point charg), w hav q g L m whr g is calld th gyromagntic ratio, which accounts for th ffcts of non-point charg distribution. In th prsnc of a magntic fild, th intraction btwn th fild and th magntic dipol rsults in additional potntial nrgy L U B B m
Application to Quantum Mchanics Assuming that th classical rsult rmains valid in quantum mchanics, th nrgy of a (n,l) stat now is E E n E n m B m B B, L B E n h m h m m B Bohr magnton Th nrgy lvl is, thrfor, split into l+1 lvls. For instanc, an l=1 stat is split into thr, whil an l=0 stat is not split. In othr words, a transition from th formr to lattr stat can produc photons of thr distinct nrgis (i.., thr lins), rathr than on. 3
Zman Effct For instanc, in a magntic fild of strngth 4 T, th splitting is E 3.7 h m 10 B 3 (9.3 10 J.3 10 4 4 J / T )(4 T ) V Tchnical rquirmnts: A high rsolution spctromtr Strong magntic fild 4
In Non-Uniform Magntic Fild If th applid magntic fild is not uniform, thr will b a forc acting upon a magntic dipol, F U ( B) If th gradint of th fild is along th -axis, th forc is F ( B ) B L m A bam of particls with a mixtur of stats corrsponding to diffrnt quantum numbr m will b split into l+1 bams aftr it passs through a non-uniform fild. 5
Strn-Grlach Exprimnt Succss: it allows visualiation of th quantiation of angular momntum, on of th ky prdictions of quantum mchanics. Pul: som atom bams (.g., silvr) wr sn to split into two or anothr vn numbr of pics. How? 6
Additional Exprimntal Puls Whn th magntic fild is sufficintly strong, not all Zman splitting can b accountd for with th l quantum numbr. Anomalous Zman Effct Hyprfin Structur Intrinsic Angular Momntum of th Elctron In atoms with many lctrons, thr can b at most two lctrons in any givn (n,l,m) stat. 7
Spin of th Elctron Singl lctrons hav a kind of intrnal angular momntum calld spin, with a quantum numbr analogous to l, i.., Sˆ l s( s 1) h l whr l is an ignfunction of th spin oprator squard and s is th spin. For lctrons, s=1/. Thrfor, th -componnt of th spin oprator can only hav two possibl valus, Sˆ m h s, m s 1 8
Complt Wav Function of th Hydrogn Atom Taking into account th spin quantum numbr of th lctron, th wav function of th hydrogn atom is nlmm s ( r, ) Rnl ( r) Ylm (, ) which dtrmins th probability that th lctron is at a givn location and has on of th two possibl spin projctions. Not that c is also an ignfunction of S, i., Sˆ 1 1 1 h 3 4 h Doubly Dgnrat! 9
Dgnracy Rvisitd Rcall that th nrgy lvls of th hydrogn atom dpnd only on th principl quantum numbr n and that for a givn n, thr is a n-fold dgnracy in l, a (l+1)-fold dgnracy in m, and doubl dgnracy in s. Thrfor, th total dgnracy associatd with a stat of principl quantum numbr n is n 1 n 1 n( n 1) (l 1) 4 l n 4 n n l 0 l 0 In othr words, thr ar n possibl wav functions that all corrspond to th n th bound-stat nrgy. 10
Strn-Grlach Exprimnt Rvisitd Pul: Som atom bams (.g., silvr) wr sn to split into two or anothr vn numbr of pics. How? Answr: A complt wav function must includ th lctron spin. For instanc, thr ar now two wav functions for th ground stat of th hydrogn atom, 100+1/ and 100-1/, corrsponding to an atom with its lctron spin paralll or antiparalll to th dirction of th magntic fild applid. In gnral, th ground stat of a hydrogn atom is a linar suprposition of ths wav functions, C 1 C 100 1/ 100 1/ 11
Magntic Momnt of th Elctron Similar to th magntic momnt arising from th orbital angular momntum of th lctron, th lctron also has an intrinsic magntic momnt du to its intrinsic angular momntum (i.., spin). By analogy, th intrinsic magntic momnt can b writtn as g S m whr g is th gyromagntic ratio of th lctron. Th drivation of th gyromagntic ratio rquirs rlativistic quantum mchanics. To a vry high prcision, it is. Th dviation from ld to th dvlopmnt of quantum lctrodynamics (QED). 1
Anomalous Zman Effct Bcaus th lctron has intrinsic magntic momnt, th potntial nrgy du to th prsnc of a magntic fild is now U mag m ( g S tot B L) B ( B m ( L orb ) B g S ) Th chang in th nrgy of th hydrogn du to th xtrnal magntic fild is thn E mag Bh m ( m g m s ) 13
An Exampl Considr th n=, l=1 nrgy stats of hydrogn. A magntic fild of 1 T is applid. For l=1 stats, th allowd valus for m is 1, 0, +1, so th fild split th (n,l)=(,1) lvl into thr lvls, if th lctron spin is not takn into account (i.., normal Zman ffct). Th magnitud of splitting du to th spin is actually twic as larg, E 1.84 hb m 10 3 (1.6 J 10 1. 19 C)(1.05 10 9.1 4 10 V 31 10 kg 34 J s)(1 T ) 14
Anomalous Zman Effct m=1 m s =1/ m=0 l=1 m=-1 m s =-1/ m=1 m=0 m=-1 15
Total Angular Momntum Classically, th total angular momntum is givn by which can hav any valu btwn L+S and L-S. J L In quantum mchanics, th J oprator should bhav just lik othr angular momntum oprators, such as L and S, S J j( j 1) h, j l s or l s J m jh, m j j, j 1,, j 1, j 16
Addition of Angular Momnta In gnral, if J 1 is on angular momntum and J is is anothr, th rsulting total angular momntum J= J 1 + J has th valu j ( j 1) h for its magnitud, whr j can b any of th valus, j 1 +j, j 1 +j -1,, j 1 -j 17
Exampls 1) Two lctrons ach has ro orbital angular momntum. Th total angular momntum is thn j = s 1 + s, so j has two possibl valus, j = ½ + ½ = 1 and j = ½ - ½ = 0. ) An lctron in an atom has orbital angular momntum L 1, with quantum numbr l 1 =, and a scond lctron has orbital angular momntum L, with quantum numbr l = 3. What ar th possibl quantum numbrs for th total orbital angular momntum? L = L 1 + L -> th maximum valu allowd is l 1 + l = 5, and th minimum valu allowd is l 1 l =1. Thrfor, th possibl quantum numbrs ar 5, 4, 3,, and 1. 18