Angular Momentum Theory.

Transcription

1 Angua Momentum Theoy. Mace Nooijen, Univesity of Wateoo In these ectue notes I wi discuss the opeato fom of angua momentum theoy. Angua momentum theoy is used in a age numbe of appications in chemica physics. Some eampes ae: atomic obita theoy, otationa specta, many eecton theoy of atoms, NMR and ESR spectoscopy, spin-obit couping. A of these topics can be teated in a unified and eegant way using the theoy of angua momentum. The topic of angua momentum theoy wi aso intoduce us to chemica systems that ae not easiy descibed using singe deteminant moecua obita theoy. We wi fist ook at an atenative soution to the soution of the time-independent Schödinge equation fo the hydogen atom. The hydogen atom is the simpest atom and can be soved eacty. It is a usefu mode fo a othe atoms. The S.E. fo the hydogen atom can be educed to a one-body pobem in thee dimensions. Even moe essentia the pobem has spheica symmety, and it wi theefoe be advantageous to use spheica coodinates to descibe the soutions and tacke the pobem. The angua pat of the pobem shows up in many guises in physica chemisty and is not esticted at a to finding atomic obitas. We wi use a vey powefu way of finding soutions to this pobem that can be used in pecisey the same way fo such divese pobems as finding the eigenfunctions of the igid oto, the desciption of spin eigenstates (singets, tipets etc.), o hypefine spitting in atomic absobtion specta. The method of soving the angua pobem invoves woking with opeatos and commutatos and this type of appoach is used vey often in the iteatue nowadays. It is usefu to know, and vey eegant too, I may add - I can't esist teing you about it fo the shee beauty of it. The second pat of the Hydogen atom pobem invoves the adia pat of the Schödinge equation. I wi aso discuss this in some detai, but it has fa ess genea appicabiity than the angua pat. In the second pat of these notes we wi conside angua

2 momentum theoy fo many-patice systems. The teatment wi be quite genea, and the focus is on the concepts athe than mathematica detai. Pat A: The hydogen atom. I. Intoduction. As aways in quantum theoy we anayse the cassica pobem fist in ode to deive the quantum hamitonian. The cassica enegy of a poton and an eecton woud consist of the kinetic enegy + Couomb inteaction pp pe e E = + m m 4πε (6.) p e e p whee the subscipt p indicates the poton whie e efes to the eecton. As discussed vey nicey in MS (pobems 5-5/5-6), any two-body pobem in which the potentia enegy depends ony on the distance between the two bodies can be sepaated in a cente of mass pobem and a one-body pobem that invoves the educed mass. Hence if we define R c m = m p p p + mee ; M = mp + me (6.a) + m e mm e p mp = e p; µ = = me me (6.b) m + m m + m the enegy becomes PC p e E = + ; = M µ 4πε e p e p (6.3) The cassica enegy can be immediatey tansated into the quantum mechanica Hamitonian by making the substitutions p ih, etc, and eaiing that p = p + p + p. This eads to the quantum mechanica Hamitonian y

3 $H tota = h C e h M µ 4πε (6.4) whee = + + y (it is ponounced "naba squaed"). We ae inteested in finding the eigenstates of this hamitonian. The cente of mass motion sepaates easiy if we ty the soution Φ( Rc ) Ψ( ), which eads to h CΦ( Rc) e + h Ψ = Etota MΦ( R ) Ψ( ) [ µ 4πε ] ( ) (6.5) c Each of the tems on the eft hand side must equa a constant. The tota enegy is given by the sum of the tansationa enegy of the cente of mass and the intena enegy of the hydogen atom. One possibe soution fo the cente of mass wave function is a pane wave Φ( R ) = e C ik R c with enegy E CM = h k. It is infinitey degeneate though (evey M diection of k with the same ength coesponds to the same enegy) and this tansationa enegy can take on any positive vaue. It is not quantied. Hencefoth we wi ony conside the intena wave function Ψ( ) which is associated with the Hamitonian $H = h e e h µ 4πε m 4πε e (6.6) The atte Hamitonian coesponds to a fied poton in the oigin. We coud have stated fom this fomuation and we woud have made ony a vey sma eo. Ou pobem becomes H$ Ψ( y,, ) = EΨ( y,, ) Because the potentia ony depends on the distance to the oigin it is convenient to us spheica coodinates. This wi aow us to sepeate vaiabes, and investigate vaious pobems sepaatey. Defining = sinθ cos ϕ ; y = sinθ sin ϕ ; = cos θ; (6.7) < ; θ π; ϕ π; with the coesponding invese tansfomation c h c h y = + y + / / ; ϕ = actan( ) ; θ = accos( + y + ) (6.8) we can wite the Schödinge equation 3

4 $ h e HΨ(, θϕ, ) = ( ) Ψ(, θϕ, ) = EΨ(, θϕ, ) µ 4πε (6.9) In ode to discuss this pobem futhe we need to obtain in spheica coodinates. This can be done by staightfowad but vey tedious manipuation using essentiay the chain ue. Fo eampe f (, θ, ϕ) = f + f θ + f ϕ * * * fo any f (, θ, ϕ ) (chain ue). θ ϕ + θ + ϕ θ ϕ = + + = + + = = y / y ( ) / ( ) * sinθcosϕ So the pocedue woud be: "Take deivatives, and epess (eventuay) eveything in spheica coodinates." Vey tedious! Let us wok out one eampe epicity, as it invoves a famous opeato: $ L i ( y ) i ( y = ) i ( y ) i ( y ) = y y + θ θ y + ϕ ϕ h h h h θ y ϕ and hence $ L = ih. ϕ y y y y = = θ y mess y y θ y = *( ) = ϕ y y y ϕ y = ( ) = + y + y h Simia manipuations yied fo : µ h h = + µ µ L$ ( ) µ whee $ $ L L L $ L $ = + + is pecisey the squae of the angua momentum opeato y (6.) $ =h [ (sin θ ) + ] (6.) sin θ θ θ sin θ ϕ L 4

5 Pease note that h is pat of the definition of the opeato L $. The opeato L $ was encounteed befoe when we discussed the igid oto. It shows up vey fequenty in quantum mechanics, and beow we wi discuss the eigenfunctions of this opeato, which ae functions of the angua coodinates ony. Fo now et us just assume that such eigenfunctions eist and et us denote them g a ( θ, ϕ ), whee $ Lg( θϕ, ) = h ag( θϕ, ), (6.) a such that the eigenvaue is h a. Assuming this, we ty soutions fo the Schödinge equation of the fom Ψ(, θ, ϕ ) = f ()* g a (, θ ϕ) (6.3) and substituting this fom in the Schödinge equation Hf $ ( ) g ( θϕ, ) = Ef ( ) g ( θϕ, ) (6.4) we obtain a a a h h + = µ ( f ) ( a e µ 4πε ) f( ) Ef( ) (6.5) This is a one-dimensiona diffeentia equation fo f ( ), that depends on the eigenvaue a of the angua pat of the wave function. We ae hence eft with two subpobems: opeato (the so- Pobem a: Find the pecise eigenvaues and eigenfunctions of the L $ caed angua pobem). Pobem b: Find soutions of the adia equation fo fied vaue of a. Intemeo: Woking with commutatos. In the foowing we wi make etensive use of commutation eations. The foowing ues come in vey handy:. ca$, B$ = A$, cb$ = c A$, B$ 5

6 In the above c is a numbe, but it coud aso be an opeato that commutes with both $ A and B! $ If an opeato commutes with eveything ese we can aways simpy put it in font of eveything.. poof: AB $ $, C$ = A$ B$, C$ + A$, C$ B$ ABC $ $ $ CAB $$ $ = ( ABC $ $ $ ACB $ $ $ ) + ( ACB $$$ CAB $$$ ) = A$ B$, C$ + A$, C$ B$ This ue aows us to epess unknown commutatos in tems of eementay commutatos and opeato poducts. It is often a vey quick way to simpify mattes. Eampes abound beow! 3. AB $, $ + C$ = AB $ ( $ + C$ ) ( B$ + C$ ) A$ = AB $, $ + AC $, $ II. Soution of angua pobem using opeato ageba. We wi show that the angua eigenvaue pobem can be soved using ony the commutation eations between the angua momentum opeatos. So eveything foows fom the foowing 'definitions' : L$, $ $ Ly = ihl L$, $ $ L = ihly L$, L$ i L$ y = h L$ = L$ + L$ + L$ y (A.) In fact you deived these anticommutation eations yousef (MS pobem 4-7) fom the definition of L $ = p, and the commutation eations, p = i hδ, whee k, abe the k k thee catesian diections in space (,y,). Using the above mentioned ues this is vey easy: 6

7 L$, L$ = yp $$ p $$, p $$ p $$ y y = yp $$ p$, $ y p$, p$ p$, p$ + p $$ $, p$ = ih( p $$ yp $$ ) = ihl$ y y y Pease note how we moved commuting opeatos in font fo immediate simpifications! We wi aso use the foowing auiay opeatos: L$ = L$ + il$ + L$ = L$ il$ Using the commutation eations you can deive: y y (A.) $, $ $ L L L, L$ L$ = =, L$ = (A.3) y $ $ $ $ L = L L+ + L + hl LL $ $ L$ = + hl + (A.4) $, $ $ L L = L, L$ = (A.5) + L$, L$ = h L$ (A.6) + + L$, L$ =h L$ (A.7) eampes: $, $ $, $ $ L L L L L, L$ L$ = + +, L$ y = + L$ L$, L$ + L$, L$ L$ + L$ L$, L$ + L$, L$ L$ y y y y = ih( L $ L $ + L $ L $ ) + ih( L $ L $ + L $ L $ ) = y y y y (A.8) L$, L$ = L$, L$ + il$ = L$, L$ + i L$, L$ + y y = ihl + i( ihl ) = h( L $ + il $ ) = hl$ y y + (A.9) Let us now continue and deive the eigenvaues of $ L eations. In fact we wi use that since $ L, L $ just by using the commutation = these opeatos must have common eigenfunctions. We coud have used just as easiy $ L o $ L y, but is is the standad 7

8 convention to use the pai L $ and L $. Let us ca these common eigenfucntions g ab, ( θ, ϕ ), whee a is the eigenvaue of $ L whie b is the eigenvaue of L. They ae abitay (ea) numbes at the moment. Howeve because we immediatey deduce that Lg $ Lg $ = ag (A.) ab, ab, = bg (A.) a, b a, b $L = L + L + L (A.) a b y (A.3) Net, if g ab, is a common eigenfunction of $ L, L $ then Lg $ + ab,, a new function, is aso an eigenfunction of both these opeatos. Poof: $ ( $ ) $ $ L L g = L L g = a( L $ g ) (A.5) + ab, + ab, + ab, L$ ( L$ g ) = L$ L$ g + hl$ g = ( b+ h )( L $ g ) (A.6) + ab, + ab, + ab, + ab, hence Lg $ + ab, is an eigenfunction with eigenvaues ( ab+, h ), o Lg $ + ab, is eo... It is seen that acting with L + keeps the eigenvaue of L $ the same but it inceases the eigenvaue of $L by the amount h. This ceay indicates that the eigenvaue a of L $ is degeneate: in genea thee is moe than one eigenfunction with the same eigenvaue! Simiay $ ( $ ) $ $ L L g = L L g = a( L $ g ) (A.7) ab, ab, ab, L$ ( L$ g ) = L$ L$ g hl$ g = ( bh )( L $ g ) (A.8) $ is a common eigenfunction of $ L, L $ Lg ab, ab, ab, ab, ab, with eigenvaues ( ab, h ) (o it is eo...). $L + and $ L ae caed adde opeatos. They define eigenfunctions having adjacent eigenvaue of $ L (shift b by ±h) but they eave the eigenvaue of $ L unchanged. 8

9 Howeve the adde opeatos cannot act indefinitey, since b a. Let us ca the maimum eigenvaue bma = h. This means that the net highe function geneated by L $ + has to vanish! $ + h = (A.9) Lg a, In the seque I wi suppess h in the subscibt: g a, h ga,. It wi tun out that the natua unit of angua momentum is h, and the fomuas take a simpe fom if we wite b ma = h. Since is abitay, this is not a imitation, just a convenience. Inteestingy enough we can immediatey find the eigenvaue of L $ if bma = h. We use the specific fom fo L $ that acts with L $ + fist (see Eqn. A.4), because we know Lg $ + a, =. $ ( $ $ $ $ Lg = LL + + L + hl) g = ( h + h ) g = h ( +) g (A.) a, a, a, a, Simiay acting by L $ the addeing down pocess must end. Let us ca kh the minimum vaue, such that Lg $ a, k =. The coesponding eigenvaue of L $ is given by (now we use the fom of L $ in which L $ acts fist (eqn. A.4)): Lg ˆ = ( LL ˆ ˆ + Lˆ hlˆ ) g = ( h k + h k) g = h k( k+ ) g (A.) a, k + a, k a, a, k Since the eigenvaue a is the same we must have k =. We can summaie the above esut by saying we get the compete set of eigenvaues: a = ( + ) h, eigenvaues of L $, (A.) Coesponding vaues of L $ : mh, m=, +,...,, (A.3) And we can use as the defining equations fo the (unnomaied) eigenfunctions: o Lg $ = ; g = Lg $ (unnomaied, addeing down) (A.4) +, m, m, Lg $ = ; g = Lg $ (unnomaied, addeing up) (A.5), m, + + m, 9

10 What ae aowed vaues of, (which up to now was ony esticted to be )? By aising by each time we must end up at + and this means that thee ae ony two types of possibiities A. is intege B. is haf intege. The fist possibiity can descibe functions g m, ( θ, ϕ ) (see beow). With the second type of eigenvaue we cannot associate a we defined eigenfunction in the angua coodinates howeve. Sti, they tun out to have a physica meaning. They tun up when we descibe the spin of patices! Each vaue of descibes a set of eigenfunctions gm,, m =, +,...,,. They ae said to fom a mutipet of dimension +. In the tabe beow I have isted the owest types of mutipets. With each vaue of we have + vaues of m, that ange fom, +,...,,, as shown L $ ( h ) L$ ( h ) degeneacy L $ spatia spin ( + ) m "name" "name" = s singet = -,, 3 p tipet = 6 -,-,,, 5 d quintet = 3-3,-,-,,,,3 7 f septet = 3 4, - doubet = , 3,, 4 - quatet = ,, 3 5,,, 6 - setet

11 At this point we have shown the genea stuctue of the soutions, which we deived using ony the commutation eations between the opeatos. The fist fou eations in this section is the ony thing we needed, and a of the est foows o can be deived. Pesenty, in quantum mechanics the commutation eations ae taken as the definition of angua momentum. This is fo eampe why spin is consideed angua momentum: the spin opeatos simpy satisfy the same commutation eations! If we assume the standad definition fo angua momentum we can do a itte moe and aso deive the coesponding eigenfunctions in spheica coodinates. The opeatos L$, L$, L$ + can be epessed in spheica coodinates, just ike we did fo L. They woud take the fom: L$ = ih ϕ $ iϕ cos L = e [ i ] + θ + h (A.6) θ sinθ ϕ $ iϕ cos L = e [ i ] θ h θ sinθ ϕ It is easy to find soutions that ae eigenfuctions of L $ i f = m f f = e im ϕ h ( ϕ) h ( ϕ) ( ϕ) (A.7) ϕ Bounday condition: f ( ϕ + π ) = f ( ϕ) m is intege (ony intege vaues aowed fo m!) This is the eason that the haf-intege (spin) functions cannot be epessed in θ, ϕ coodinates. They woud not be singe-vaued functions in 3d-space! Net we can sove fo the θ -pat. In spheica coodinates the eigenfunctions g m, ae conventionay denoted as Y m ( θ, ϕ ). They ae caed the spheica hamonics. The ϕ - dependent pat of these functions is detemined above, and the Y m ( θ, ϕ ) can be witten as m m imϕ Y ( θ, ϕ) = P ( θ ) e, (A.8)

12 whee the P m ( θ ) ae so-caed associated Legende poynomias. They can be easiy geneated using the adde opeatos. If we take the function with m = it has to satisfy LY + (, ) = m iϕ θ ϕ ; Y ( θ, ϕ) = Pbg θ e (A.9) And using the spheica coodinate fom fo the L $ + opeato we find iϕ iϕ cosθ e e [ P θ θ sin θ ] ( ) = (A.3) It is easiy veified that the soution is P ( θ ) = (sin θ ) = sin θ as cosθ sin θcosθ sin sin θ = θ The highest m-vaued function in a mutipet, Y bθ, ϕg hence has the simpe fom i Y ( θ, ϕ) = sin θe ϕ. (A.3) A of the othe functions in the mutipet can be found by acting with L $. Thee is one futhe simpification in that we ony need to geneate functions up to m =. One can show that m m Y ( θϕ, ) = P ( θ) e Hence the θ -pat is the same fo +m and m. This foows fom the fom of $ L and $ m LP ( θϕ, ) e im ϕ and $ m LP ( θ ) e $ imϕ = (sin θ ) + sinθ θ θ sin θ ϕ L im ϕ yieds the same diffeentia equation fo P m ( θ ) $ m L ( (sin ) ) P m ( ) ( ) P m = θ θ = + ( θ ) sin θ θ θ sin θ Let us ook at the non-tivia eampe of = (d-functions). Y e i (, ) sin θ ϕ = θ ϕ (genea fomua) L$ Y (, )~sin cos e i θϕ θ θ ϕ

13 L$ Y (, ) ~ ( sin + cos + cos ) = θ ϕ θ θ θ 3cos θ $ L Y (, )~sin cos e i θϕ θ θ ϕ $ iϕ L Y ( θϕ, )~( sin θ+ cos θcos θ) e ~sin θe iϕ L $ cos ( sin cos θ sin sin ) = θ θ θ θ It is seen that the fom of the functions is geneated quite easiy. The nomaiation factos ae ess impotant, athough even they can be obtained vey geneay fom the commutation eations! To sove the Schödinge equation fo the Hydogen atom we actuay ony equie angua eigenfunctions of L $, not of L $. We deveop the above fomaism because of its eegance and geneaity. In pactice it is easie to think of ea eigenfunctions. We can theefoe make inea combinations of degeneate eigenfunctions of L $. In paticua we can combine P m ( θ ) e im ϕ m and P ( θ ) e imϕ into P m ( θ )cos( mϕ ) and P m ( θ)sin( mϕ ). Fo the above d-functions this eads to the famiia catesian foms of the d-obitas sin θcosϕ = sin θ(cos ϕ sin ϕ) ~ y sin θsin ϕ = sin θsinϕcos ϕ ~ sinθcosθcos ϕ ~ y sinθcosθsin ϕ ~ 3 3 cos ~ y 3

14 A angua functions can easiy be geneated this way: Stat fom sin θe iϕ, act with L $ sequentiay, and combine e ± imϕ into cos( mϕ ), sin( mϕ ). Voia! III. The adia equation fo the Hydogen atom and its soutions. Above we discussed the angua pat of the equations that detemine the atomic obitas fo the Hydogen atom in geat detai. Hee we wi discuss how the fu set of soutions can be obtained. The Hamitonian in spheica coodinates was given by $ h $ H ( ) L e = + µ µ 4πε m And we ty the function Ψ(, θ, ϕ) = f() Y (, θ ϕ) [ ( ) ( (( ) h + h + ) ( )] ( θϕ, ) = ( ) ( θϕ, ) µ f e µ 4πε f Y m Ef Y m Let us simpify the notation somewhat and mutipy though with µ, and define h µ E = ε. Let us aso use µ e h 4πε h (.) (.) me e =, whee a 4 is the Boh adius. This πε h a yieds the adia equation + + = ( f f ) ( ) ( ) a f ( ) ε f ( ) (.3) o bette yet, f f ( + ) + f( ) = a f ( ) ε f ( ) (.4) One moe substitution to make. Ty f( ) = p( ) e α, whee p( ) wi be a poynomia in, p ( ) = a+ b+ c +..., hence f = p e α p e α α ( ) f = p p α + [ α p ( )] e e α is mutipied thoughout, so we can cance this. We end up with an equation fo p( ) α 4

15 dp α = d p d p dp ( ) ( ) α α p ( ) p ( ) d d a p ( ) ε p ( ) (.5) Let us fist eamine some of the owe degee equations befoe discussing the genea soution. Remembe that = fo s-obitas, = woud yied p obitas, and so foth. Let me just ist some soutions: dp d p =, p( ) = : = =, substitute in (.5) d d α α = ε a ε = α ; α = ; a and theefoe a / h f( ) = e ; E = µ a Anothe eampe: dp d p =, p( ) = c =, = d d (.6) (.7) α + ( c) + α ( = a c ) α ( c ) ε( c ) (.8) in such an equation the tems must match fo each powe in, hence ode : α = ε and we obtain the soutions: o α ode unity: + α / a = ode : αc+ c/ a = (.9) ε = α, α = / a, c= a (.) / a h h f( ) = ( a ) e ; E = = µ ( a ) 4 µ a (.) 5

16 Let us take the simpest eampe of a p function p dp d p =, ( ) =, =, = : d d Genea chaacteistics: + + α α + α = ε a / a h α = / a, ε =/ 4a f( ) = e, E = 4µ a Suppose highest powe in poynomia is m. Then, substituting in (.5): Tems of ode m: α = ε (aways!) Tems of ode m : α + mα = ( m + ) α = a a α =, = = ( m+ ) a E h µ ( + ), m a m,,,... Independent of! Usuay we put n= m+ and ca it the pincipe quantum numbe: n / na h e, E = (n : highest powe in ) ma n e s Ca s the smaest eponent in p( ), p ( ) = ( + a+ b+...). In equation (.5) we wi then obtain tems stating fom s : ss( s ) + ( + ) = s= The fist adia soution coesponding to Y m θ, ϕ Depends on quantum numbe. b g (no adia nodes) stats with! 6

17 Summay of genea soutions: (,, ) p () e Y m, (, θ ϕ); na / Ψ nm,, θ ϕ = n, h En ( ) = µ an whee p, ( ) = a + b n n n is a poynomia in having n adia nodes. Possibe enegy eves and thei degeneacies: n = 3,,,... = 3,,,,..., n m=, +,...,, n Degeneacy E n : ( + ) = n+ = n+ n( n ) = n = n = We note that fo the hydogen atom the enegy ony depends on the pincipa quantum numbe n. Hence the s, p obitas ae degeneate as ae 3s, 3p, 3d and so foth. This is ony tue fo one-eecton atoms (H, Ne 7+, etc), but not fo many-eecton atoms in genea. 7

18 Pat B: Many-Patice Angua Momentum Opeatos. The commutation eations detemine the popeties of the angua momentum and spin opeatos. They ae competey anaogous: L$, L$ = ihl$, etc. y L$ = L$ ± il$ ± $ $ $ $ L = L $ + L + L hl = LL $ $ + L$ + hl$ + y S$, S$ = ihs$, etc. y S$ = S$ ± is$ ± $ $ $ $ S = S $ + S + S hs = SS $ $ + S$ + hs$ + y The one-eecton eigenfunctions fo the $ L, L $ opeatos ae the spheica hamonics ˆ m m ( θϕ, ) = ( + ) h ( θϕ, ) LY Y LY ˆ ( θϕ, ) = mhy ( θϕ, ) m m ˆ m m+ m+ m+ LY + ( θ, ϕ)~ Y ( θ, ϕ) = h ( + ) mm ( + ) Y ( θ, ϕ) C+ (, my ) ( θ, ϕ) ˆ m m m m LY ( θ, ϕ)~ Y ( θ, ϕ) = h ( + ) mm ( ) Y ( θ, ϕ) C(, my ) ( θ, ϕ) Hee I incuded the pecise popotionaity constants. Fo ate convenience they ae abbeviated as C± (, m). The one-eecton spin eigenfunctions ae denoted as Y / / / = α; Y = β. Epicity the vaious equations ead / $ S ( ) 3 α = + h α = h α $ S ( ) 4 3 β = + h β = h β 4 S$, S$, S $ α = hα + α = α = hβ S$, S$, S $ β = hβ + β = hα β = A singe eecton (a so-caed spin / patice) is aways descibed by the spin-functions α and β. Highe than spin / functions show up in many-eecton wave functions. Nucea spin opeatos (indicated Î ) aso satisfy pecisey the same commutation eations Iˆ, ˆ ˆ I y = ih I and cycic pemutations. Nucei on the othe hand can have highe spins (they consist of potons and neutons that individuay ae spin / patices). 8

19 This means that the nucea spin functions might fo eampe be a tipet (I=), o a quatet (I=3/). The mathematics undeying a of these diffeent physica phenomena is pecisey the same, and we wi focus on the case of angua momentum fo an atom consisting of a cetain numbe of eectons. The many-eecton opeatos ae defined in an anaogous fashion L $ tota = L $ () + L $ ( ) +..., etc. S $ tota = S $ () + S $ ( ) +..., etc. The opeatos $, tota $ tota $ tota S S S S$ tota S$ tota S$ tota S$ tota = + +, (and simiay L$, tota ) ae y y compicated (two-eecton) opeatos that contain mied tems ike S $ () S $ ( ). We can aways epess things in tem of poducts of $ tota S S $ tota + etc, so we do not need to use S $ diecty. We can wok with the sum-opeatos on a poduct of functions. As eampes conside and $ tota S ( α( ) α( )) = [ S $ () + S$ ( )]( α ( ) α ( )) = [( S$ () α ()) α ( ) + α () S$ ( ) α ( )] h h = α( ) α( ) + α( ) α( ) = hα( ) α( ) $ tota S ( α( ) α( )) = [ S $ () + S$ ( )]( α ( ) α ( )) = [( S$ () α ()) α ( ) + α () S$ ( ) α ( )] = hβ() α( ) + hα() β( ) hence we can think of acting with the one-eecton opeato on each one-eecton function sepaatey and summing the esut. The pocedue woks in the same way if we use the L $ tota opeato on a poduct of spheica hamonics. Let us take the p functions and abbeviate p = Y; p = Y ; p = Y. Then $ tota L ( p ( ) p ( )) = h ( p( ) p ( ) + ( p ( ) p ( )) + $ tota L ( p () p ( )) = ( ) hp () p ( ) = h p () p ( ) Fo p-functions ( = ) the facto h ( + ) mm ( + ) = h ( =, m=, ). We aso wish to eamine what happens if we act on an antisymmetic Sate deteminant of spin-obitas. Aso hee we can just act with the one-eecton opeato on each of the 9

20 poduct functions in the deteminant and sum the esut. The eason is that the sum opeato is symmetic and commutes with any pemutation of eecton abes, e.g. [ S $ () + S$ ( )][ P (, ) α ( ) β ( )] = [ S$ () + S$ ( )] α ( ) β ( ) = P(, )[( S $ () + S$ ( )) α ( ) β ( )] = [ S$ ( ) + S$ ()] α ( ) β () This agument is ceay genea and so S ˆ... ˆ tota ϕϕ = S ( ϕϕ...) tota a b a b Let us conside the spin obitas in a p-manifod: p, p, p, p, p, p. In this notation p = p α; p = p β, etc. Beow I wi give a set of eampes of opeations of spin and angua momentum opeations. You can veify the esuts, and see that the basic ues ae not vey difficut. L$ pp =, L$ pp = h pp + S$ p p = p p = ( antisymmety!), S$ p p = The above eations suffice to show that pp is the m = component of the D mutipet. You woud need to use the fom ˆ ˆ ˆ ˆ ˆ + L = L L + L +h L. A othe functions in the mutipet can be geneated by acting successivey by $ L. Fo eampe L$ pp = h ( pp + pp ) L$ ( p p + p p ) = h ( p p + p p ) Impotanty, this eigenstate with eigenvaue m = h is not a deteminant but a inea combination of deteminants. In this state you cannot say that these obitas ae occupied and the est empty. The wave function is moe compicated. You can act with the ˆL opeato on this new function to show that the eigenvaue has not changed: (+) h is the answe you shoud find. But it is moe wok to show this epicity (give it a ty). You can keep appying the L $ opeato unti you find the state p p ( m =h ), and acting with $ L once moe yieds eo.

21 As anothe eampe L$ pp = h pp =, L$ pp = h pp + S$ pp =, S$ pp = h pp + L$ p p = h ( p p + p p ) = h p p S$ p p = h( p p + p p ) The fist two equations estabish that pp is the m =, ms = component of the 3 P mutipet. Pease veify. A othe 9 states can be obtained by successive appication of L $ and S $ as iustated by the ast two eampes given. We wi do some eecises with the angua momentum opeatos eaboating on some eampes as iustated above. Thee is a vast iteatue on the topic and the above is a vey bief summay. Let me e-emphasie that angua momentum theoy undeies NMR specta (nucea spin functions). Eecton spin esonance can be teated anaogousy. This wi be iustated in cass. To teat these phenomena we act on poduct of nucea spin functions. The pincipes ae simia, and spin functions ae in fact a bit easie because the popotionaity constants fo spin patices ae unity athe than. Let me aso mention that we can use a simia teatment to find eigenfunctions of the tota angua momentum opeato Jˆ = Lˆ+ Sˆ, which is paticuay eevant fo atomic tem symbos. We wi go though some eampes in cass and in the pobem set. III. Genea decomposition of a poduct basis of angua momentum eigenfunctions into eigenfunctions of the tota angua momentum opeatos. In this section we conside the constuction of eigenfunctions of the angua momentum opeatos fo a composite patice (in fact it appies to any poduct function, fo eampe spin-obitas pα ). The stuctue is vey genea. We wi conside the poducts fo two patices, but fom this you can constuct poducts fo an abitay numbe of patices

22 using the same genea pincipes. You have seen eampes in NMR befoe we get to this pat of the ectue notes. Conside the individua mutipets, m, m =... and, m, m =... which ae eigenfunctions of the angua momentum opeatos Lˆ (), L ˆ () and Lˆ (), L ˆ () espectivey. Hee we ae using the Diac notation fo states:, m indicates an eigenfunction chaacteied by the quantum numbes in the so-caed ket. Moe on this notation shoty. The fu set of poduct functions, m, m spans a space of dimension (+ )( + ). We want to decompose this poduct basis into a set of new basis functions that ae eigenfunctions of the composite angua momentum opeatos Lˆ, L ˆ, whee Lˆ = Lˆ() + Lˆ(). Let us indicate these eigenfunctions as tota, tota tota L, M, M = L... L. Such a set of functions with a given vaue of L is caed a mutipet, and the dimension of the mutipet is L+. We wi see beow that the possibe eigenvaues L ange fom L= +, +,...,, o, assuming that, we can aso say that the mutipets that ae occuing ae L= + k, k =,..,. The tota dimension of the space spanned by these mutipets is {( + k) + } = ( + )(( + ) + ) k k=, k=, = ( + )(+ ) + (+ ) ( ( + )) = ( + )( + ) This is consistent with the tota numbe of poduct functions, as shoud be the case. Let us ationaie this esut futhe by an epicit constuction of the eigenfunctions. Each poduct function itsef is an eigenfunction of L ˆ, tot : Lˆ, m, m = h ( m + m ), m, m tot,. Hence we can easiy aange the poduct functions in a tabe, such that aong a ow, they a have the same vaue of M.

23 Tabe: Aangement of poduct functions accoding to M-vaues (eigenvaue of L ˆ, tot ) + M= M= + M= +,,,,,,,,,,,, 3.. M= +, +,, +, +,, + 3 M= + M=, +,,, +,, # functions ( + ) + ( + ) + ( + ) + The ovea mutipet stuctue can be discened now. In the second coumn, top ow, we find the function with the maimum vaue M = L= +. This function is aso an eigenfunction of L ˆtota. This foows because L ˆ,, +, tota =. You can compete the agument: why is it an eigenfunction of L ˆtota geneated by acting with L ˆ ˆ ˆ, tota = L() + L(), hence? The othe functions in this mutipet ae Lˆ,, = ( C (, ),, + C (, ),, ), tota and so foth. It is seen that in genea we find a inea combination of poducts states, ecept fo the highest and owest possibe M-vaue in the space. This pocess continues as you adde down and decease the vaue of M, unti M = L=. The numbe of functions is pecisey the numbe of functions in the fist coumn, ( + ) +. The net mutipet is constucted by stating fom the eigenfunction that has M = +, and is othogona to the inea combination function aeady found. Atenativey we might ty to find that inea combination such that acting with L ˆ +, tota on this combination yieds. Then it wi be an eigenfunction of both L ˆtota and L, ˆ tota. Fo eampe 3

24 Lˆ +, tota ( C+ (, ),, C+ (, ),, ). = C (, ) C (, )[,,,, ] = + + This means it wi be an eigenfunction of L with L= +. ˆtota Even though we equie a inea combination of the poduct states to constuct the tue eigenfunctions, the numbe of functions in this second mutipet is pecisey the same as in the second coumn of the tabe, namey ( + ) +. The scheme now epeats itsef. To stat constucting the net mutipet find the emaining highest M-state that is othogona to the states aeady found and adde down. Atenativey, and this is usuay easie, find that inea combination of functions of specific M-vaues such that acting with L ˆ +, tota yieds pecisey. The above scheme is competey genea, and appies to a vaiety of pobems in physics. Fo eampe we can constuct eigenfunctions of the spin-opeatos in this fashion, and in this way we find the pope combinations of singet, doubet, tipet spin functions. The same stategy appies to nucea spin functions. These pobems ae vey instuctive: They give eact esuts because the basis set is compete, whie the ageba is often not so tedious. We can aso coupe diffeent angua momentum opeatos fo eampe to get eigenfunctions of the tota angua momentum Jˆ = Lˆ+ Sˆ. These pobems ae aways competey anaogous if you wok on constucting many-patice functions. Essentiay this is tue because the angua momentum opeatos fo diffeent patices commute. It is aso possibe to use angua momentum theoy to constuct fo eampe integas and obitas of highe -vaue. As an eampe you can evauate integas ike p d ˆ i j. The seection ues ae easiy evauated (meaning you can know when an intega is eo). The pecise evauation of non-eo integas equies moe wok. To hep you digest this mateia thee ae some eecises! 4