Wintr 3 Chm 356: Introductory Quantum Mchanics Chaptr 7b Elctron Spin and Spin- Orbit Coupling... 96 H- atom in a Magntic Fild: Elctron Spin... 96 Total Angular Momntum... 3 Chaptr 7b Elctron Spin and Spin- Orbit Coupling H- atom in a Magntic Fild: Elctron Spin If lctron in orbital has angular momntum L!, on has a magntic momnt! m = q!! L L m m This magntic momnt can intract with a magntic fild and th intraction nrgy is givn by V = m! B!! B is masurd in Tsla T = Nwton/(ampr mtr) If w tak! B to b in dirction thn And th total Hamiltonian would b V = B m = Ĥ = Ĥ + m B L m B L Th wavfunctions w obtaind ψ n,l,m = f n,l (r)y l m (θ,ϕ) including th magntic fild E n,l,m = E () n,l + B(m!) m ar also ignfunctions of Ĥ ach lvl nl, splits in (l + ) sublvls This is what would b xpctd, from classical considrations. This is not what is obsrvd! 96
Wintr 3 Chm 356: Introductory Quantum Mchanics Strn and Grlach passd silvr atoms (with lctron in s- orbital) through inhomognous magntic fild. 9! Thy found this splits th bam into two. Classically on would xpct (l + ) lins for particl with angular momntum l. Hr l + = l =!! This was th first vidnc that lctrons nd anothr quantum numbr: half- intgr angular momntum. Jumping to what w know now, w introduc an intrinsic angular momntum oprator Ŝ x,ŝ y,ŝ Ŝ = Ŝx + Ŝ + y Ŝ This oprator is postulatd to hav xactly th sam commutation rlations as Lˆ, Lˆ, L ˆ Hnc Ŝ x,ŝ y = i!ŝ Ŝ,Ŝx = i!ŝ y Ŝ,Ŝ y = i!ŝx From out gnral discussion w know th possibl ignstats as Ŝ s,m s =! s(s +) s,m s x y Ŝ s,m s = m s! s,m s Whr I indicat th stats as s,m s : Dirac brackt notation (pag 59. MQ) Th splitting of silvr atom bam indicats ignstats s =, ms = = α s =, ms = = β Chaptr 7b Elctron Spin and Spin- Orbit Coupling 97
Wintr 3 Chm 356: Introductory Quantum Mchanics Evrything w discussd on oprators and angular momntum applis to spin, also th stats α, β ar orthonormal α α = α *(σ )α(σ )dσ = β β = β *(σ )β(σ )dσ = α β = α *(σ )β(σ )dσ = What ar th coordinats that w intgrat ovr? W do not hav a clu!! But, also, w do not nd it: W know α, β ar orthogonal bcaus thy ar ignstats of a Hrmitian oprator L ˆ with diffrnt ignvalus ±!. Hnc thy must b orthogonal! (S chaptr 4). W now hav sts of angular momntum oprators Lˆ, Lˆ, L ˆ : orbital angular momntum x y Sˆ, Sˆ, S ˆ : spin angular momntum x y L ˆ, S ˆ = α β Th oprators Lˆ, Sˆ, Lˆ, S ˆ all commut, and thy also commut with th Hamiltonian H = T + V, for th H- atom Thn w can charactri ach ignstat through nlm,,, Sm, 5 quantum numbrs Lt us facilitat th notation somwhat l s Not: Latr on this will rquir modification, as w ar nglcting rlativistic ffcts, which introduc so- calld spin- orbit coupling. Chaptr 7b Elctron Spin and Spin- Orbit Coupling 98
Wintr 3 Chm 356: Introductory Quantum Mchanics W can indicat th l - quantum numbr as s, p, d and gt functions P, P, P m l =,, W could thn writ, including spin Pα, Pα, P α Pβ, P β, P β In what follows w will focus on on particular n quantum numbr, which w can supprss. Morovr I can indicat th β - spin function by an ovrbar. Thn w gt th 6 p- functions Or th d- functions p, p, p, p, p, p d, d, d, d, d, d, d, d, d, d This indicats th lm,, Sm, quantum numbrs s s In this way w can labl th xact ignstats of th (non- rlativistic) Hamiltonian. Th splitting of th lins in a magntic fild would thn b dtrmind by th Hamiltonian Ĥ = Ĥ + B m ˆL + g B m S whr g... This factor g dtrmining th ratio btwn spin and orbital intractions with th magntic fild, can b calculatd using rlativistic quantum fild thory. (Schwingr, Tomanaga, Fynman). Far byond our aim. It agrs to about digits with th xprimntal valu! (that is lik masuring th distanc from hr to Nw York up to a millimtr!) Th p, p functions ar ignfunctions of this magntic Hamiltonian, and w can asily calculat th nrgy splitting. Unfortunatly, this dos not giv corrct rsults!! Th splitting du to th magntic fild is vry small. Thr ar othr corrctions to th nrgy lvls in Hydrogn atom du to rlativity (think Einstin). Thy ar of at last comparabl importanc, and cannot b nglctd whn discussing magntic ffcts. Th rlativistic Hydrogn atom is dscribd by th Dirac quation. This is far mor complicatd than w wish to discuss. Chaptr 7b Elctron Spin and Spin- Orbit Coupling 99
Wintr 3 Chm 356: Introductory Quantum Mchanics On can approximat ffcts by including so- calld spin- orbit intraction in th Hamiltonian. ( Ĥ R) = T ˆ + V ˆ + ξ(r) ˆL Ŝ ˆL Ŝ = ˆL x Ŝ x + ˆL y Ŝ y + ˆL S It is calld L S coupling or spin- orbit coupling Thn th magntic intraction can b includd as ˆ ( R) ˆ g H = H ˆ + L + S m m Lt us first xamin th nrgy lvls for th non- rlativistic Hamiltonian, i.. w nglct th spin- orbit or L S coupling trm. ˆ ˆ ˆ H = H ˆ + BL + g BS m m Lt us dnot! B m = γ and us g =, thn Ĥ = H + γ! ˆL + γ! Ŝ Lt us considr th allowd p+ s mission lins: Normal Zman ffct: H, only includ L includ S Chaptr 7b Elctron Spin and Spin- Orbit Coupling
Wintr 3 Chm 356: Introductory Quantum Mchanics Du to S ˆ all α - lvls go up by γ all β - lvls go down by γ. Sinc transitions cannot chang spin, Δ =, I gt sam transitions as without spin!! m s Our conclusion thus far. If on dos not considr spin lvls split in a magntic fild using p 3 qual spacd lvls d 5 qual spacd lvls m L ˆ If w includ spin, for singl lctron stats thn all α - stats shift up by unit of γ!, all β - lvls shift down by unit of γ!, and th transition nrgis np multipls ar split by th sam amount α n' s ar not affctd by spin. Morovr all α m!b. W would not s th ffcts of spin. This is what was originally obsrvd in arly xprimnts. It is calld th normal Zman ffct and it was xplaind by Lornt (two Dutch physicists). It appars as if only th L trm is prsnt. Howvr w do obsrv th ffcts of spin in mission spctra! Th story is mor complicatd. Th complications occur alrady for th H- atom without a magntic fild. Thr is a substantial corrction du to what is calld spin- orbit intraction. A good way to think about this is as follows: W usually think of th lctron as wiing around th nuclus. From th point of viw of th lctron w can ust as asily think that th nuclus is wiing about th lctron. This moving nuclus, with its angular momntum gnrats a magntic fild. This magntic fild intracts with th spin of th lctron. Compar th lctron with your position on th spinning arth. Th sun riss and sts from our standing still point of viw, and movs with incrdibl vlocitis, in this fram. For a chargd particl th magntic forc would b larg. It is calld spin- orbit coupling Chaptr 7b Elctron Spin and Spin- Orbit Coupling
Wintr 3 Chm 356: Introductory Quantum Mchanics H so = Lˆ S ˆ m r 3 Spin- orbit coupling is usually said to b a rlativistic ffct. This is bcaus it ariss in a natural way from th fully rlativistic Dirac quation. So dos spin; it ariss naturally. And so do particls and antiparticls, which also aris from th Dirac quation. Lt m say somthing mor about spin. Th spin oprators ar bst rprsntd by matrics. σ x =!, σ =! i y i, σ =! Ths matrics satisfy th commutation rlations of angular momntum Morovr σ x,σ y = i!σ S =! S =! Hnc α = β = =! =! * matrics hav only ignvctors. This is why w hav only α, β Th Dirac quation is a 4*4 matrix quation and w gt ( spin * mass) solutions. Th splitting du to th magntic fild is vry small. Thr ar othr corrctions to th nrgy lvls in Hydrogn atom du to rlativity. Thy ar of at last comparabl importanc, and cannot b nglctd whn discussing magntic ffcts. Th rlativistic Hydrogn atom is dscribd by th Dirac quation. This is far mor complicatd than w wish to discuss. On can approximat th ffcts by including spin- orbit intraction in th Hamiltonian. Hˆ = Tˆ+ Vˆ + ξ() r Lˆ Sˆ ( R) Lˆ Sˆ = Lˆ Sˆ + Lˆ Sˆ + Lˆ Sˆ x x y y It is calld L S coupling or spin- orbit coupling Thn th magntic intraction can b includd as Chaptr 7b Elctron Spin and Spin- Orbit Coupling
Wintr 3 Chm 356: Introductory Quantum Mchanics ˆ ( R) ˆ H = H ˆ + L + g S m m Our original spin- orbitals p, p, p, p, p, p ar howvr not ignstats of th coupling ˆ ( R ) H including L S W can classify th ignstats of ˆ ( R ) H by doing a littl mor angular momntum thory. Lt m sktch th rsult, as this is, finally, an accurat dscription. Total Angular Momntum W hav Lˆ, Lˆ, L ˆ and Sˆ, Sˆ, S ˆ Morovr This is bcaus ˆL and x y x y ˆLα,Ŝβ = αβ=, xy,, Ŝ act on diffrnt coordinats Now w dfin total angular momntum Jˆ x Jˆ = Lˆ + Sˆ ; Jˆ = Lˆ + Sˆ ; Jˆ = Lˆ + Sˆ x x x Ĵ = J x + J y + J, J ˆ and J ˆ satisfy th usual commutation rlations: y y y y Jˆ, ˆ ˆ ˆ, ˆ ˆ x J y Lx Sx Ly S = + + y = Lˆ, ˆ ˆ, ˆ ˆ, ˆ ˆ, ˆ x L y Lx S y Sx L y Sx S + + + y = i! ˆL + + + i!ŝ Morovr And Likwis = i!ĵ Ĵ = J x + J y + J ˆ ˆ = L + S + L S L S = L S + L S + L S x x y y = J L S ( ) J L L L S L x, = x, + x, = Chaptr 7b Elctron Spin and Spin- Orbit Coupling 3
Wintr 3 Chm 356: Introductory Quantum Mchanics But thn also And J S ˆ L S S S ˆ ˆ,,, α = α + α = [ ] J Jˆ Sˆ Lˆ J L S α, = α, = J, L = J, S = Hnc w can driv without too much troubl that th oprators Morovr ths oprators commut with L S, and also with ξ () rlˆ Sˆ. It thn follows that th angular momntum oprators Hamiltonian ˆ ( R ) H. And w can classify th stats with quantum numbrs nls,,, m, ˆ ˆ ˆ J, L, S and J ˆ all commut. ˆ ˆ ˆ J, L, S, J ˆ commut with th rlativistic As for th non- rlativistic cas, th angular momntum problm, dfining ls,, m, from th radial quation, and can b solvd onc and for all. is indpndnt ˆ L l, s,, m = l(l +)! l,s,,m ˆ S l, s,, m = s(s +)! l,s,,m ˆ J l, s,, m = ( +)! l,s,,m Jˆ l, s,, m = m! l,s,,m Ths quations hold for th Hydrogn atom, but latr on w will s that thy ar vry similar for many- lctron atoms, which also hav sphrical symmtry. Can w dduc what th ignstats of Ĵ and J ˆ might b? First w not that ignfunctions of J ar asy. Ĵ l,s,m l,m = ( m l + m s )! l,s,m l,m Eg. Ĵ p = +!p Ĵ p =!p W also know that acting with Ĵ + on, should giv : th highst m valu in th multiplt. It is asy to find th highst function: ml = l, ms = s Chaptr 7b Elctron Spin and Spin- Orbit Coupling 4
Wintr 3 Chm 356: Introductory Quantum Mchanics Hnc in th p- manifold p = p α m = 3, 3 = is th highst m function. 5 In th d- manifold it is d = dα with m = + = Acting with Ĵ w can gt th othr function in th multiplt. W gt ( = ) in 3 + = 4 functions W ar lft with functions in th p- manifold (6 spin- orbital in total). This crats a,,, functions. = multiplt. How dos this work, for th d- functions? l = highst m l. 5 m = l + s = + = 5 3 3 5 m =,,,,, ( J + ) = 6 functions In total w hav functions othr multiplt has 4 functions 3 = l = = For th Hydrogn atom w can construct always = l + = 5 l+ + = l+ Chaptr 7b Elctron Spin and Spin- Orbit Coupling 5
Wintr 3 Chm 356: Introductory Quantum Mchanics ( l ) = l l + = l ( l ) + + l is th total numbr of spin- orbitals of angular momntum. W labl th final ignstats as S+ L J Th - multiplt hav slightly diffrnt nrgis du to th L S = J L S qual LS, is sn to dpnd on th J quantum numbr. ( ) coupling, which for Finally, w ar rady to discuss how ths rlativistic lvls split in a magntic fild; Within a multiplt J th lvls split in J + qually spacd lvls du to a magntic fild Th splitting dpnds on ( nl,, ) Allowd transitions: Δ l = ±, Δ s =, Δ =, ± m J Chaptr 7b Elctron Spin and Spin- Orbit Coupling 6
Wintr 3 Chm 356: Introductory Quantum Mchanics All diffrnt nrgis. Tiny splitting, but this is how w can xprimntally accss dgnracis of nrgy lvls. Th splitting of nrgy lvls in a magntic fild is a complicatd subct, bcaus rlativistic ffcts hav to b considrd at th sam tim. Magntic transitions using nucli as in NMR ar much simplr as w only nd to considr angular momntum thory itslf. Bttr pictur of transitions including spin- orbit: B = diffrnt transitions, diffrnt frquncis Slction ruls Δ l = ±, Δ s =, Δ =, ±, Δ =, ± No J = J = transitions m p 3 s (m = ) 3 transitions p s (m = ) transitions p 3 s (m = ) 3 transitions Δ =, ± m Chaptr 7b Elctron Spin and Spin- Orbit Coupling 7
Wintr 3 Chm 356: Introductory Quantum Mchanics p s (m = ) transitions transitions in total To good approximation on can valuat th shifts in magntic fild from ˆ L ˆ + S m m Nglcting spin- orbit coupling. Chaptr 7b Elctron Spin and Spin- Orbit Coupling 8