Part B: Many-Particle Angular Momentum Operators.

Transcription

1 Part B: Man-Partice Anguar Moentu Operators. The coutation reations deterine the properties of the anguar oentu and spin operators. The are copete anaogous: L, L = i L, etc. L = L ± il ± L = L L L L = L L L L S, S = i S, etc. S = S ± is ± S = S S S S = S S S S The one-eectron eigenfunctions for the L, L operators are the spherica haronics ˆ = ( ) h LY Y LY ˆ = hy ˆ LY ~ Y = h ( ) ( ) Y C (, Y ) ˆ LY ~ Y = h ( ) ( ) Y C (, Y ) Here I incuded the precise proportionait constants. For ater convenience the are abbreviated as C± (, ). The one-eectron spin eigenfunctions are denoted as Y / / / = α; Y = β. Epicit the various equations read / S ( ) 3 α = α = α 4 S, S, S α = α α = 0 α = β S ( ) 3 β = β = β 4 S, S, S β = β β = α β = 0 A singe eectron (a so-caed spin / partice) is awas described b the spin-functions α and β. Higher than spin / functions show up in an-eectron wave functions. Nucear spin operators (indicated Î ) aso satisf precise the sae coutation reations Iˆ, ˆ ˆ I = ih I and ccic perutations. Nucei on the other hand can have higher spins (the consist of protons and neutrons that individua are spin / partices).

2 This eans that the nucear spin functions ight for eape be a tripet (I=), or a quartet (I=3/). The atheatics undering a of these different phsica phenoena is precise the sae, and we wi focus on the case of anguar oentu for an ato consisting of a certain nuber of eectrons. The an-eectron operators are defined in an anaogous fashion L tota = L ( ) L ( )..., etc. S tota = S ( ) S ( )..., etc. The operators, tota tota tota S S S S tota S tota S tota S tota =, (and siiar L, tota ) are copicated (two-eectron) operators that contain ied ters ike S ( ) S ( ). We can awas epress things in ter of products of tota S S tota etc, so we do not need to use S direct. We can work with the su-operators on a product of functions. As eapes consider tota S ( α( ) α( )) = [ S ( ) S ( )]( α ( ) α ( )) = [( S ( ) α ( )) α ( ) α ( ) S ( ) α ( )] = α( ) α( ) α( ) α( ) = α( ) α( ) and tota S ( α( ) α( )) = [ S ( ) S ( )]( α ( ) α ( )) = [( S ( ) α ( )) α ( ) α ( ) S ( ) α ( )] = β( ) α( ) α( ) β( ) hence we can think of acting with the one-eectron operator on each one-eectron function separate and suing the resut. The procedure works in the sae wa if we use the L tota operator on a product of 0 spherica haronics. Let us take the p functions and abbreviate p = Y ; p = Y ; p = Y. Then tota L ( p ( ) p ( )) = ( p ( ) p ( ) ( p ( ) p ( )) tota L ( p ( ) p ( )) = ( 0 ) p ( ) p ( ) = p ( ) p ( ) For p-functions ( = ) the factor 0 ( ) ( ) = ( =, = 0, ). We aso wish to eaine what happens if we act on an antisetric Sater deterinant of spin-orbitas. Aso here we can just act with the one-eectron operator on each of the

3 product functions in the deterinant and su the resut. The reason is that the su operator is setric and coutes with an perutation of eectron abes, e.g. [ S ( ) S ( )][ P (, ) α ( ) β ( )] = [ S ( ) S ( )] α ( ) β ( ) = P(, )[( S ( ) S ( )) α ( ) β ( )] = [ S ( ) S ( )] α ( ) β ( ) This arguent is cear genera and so Sˆ... ˆtota ϕϕ = S ( ϕϕ...) tota a b a b Let us consider the spin orbitas in a p-anifod: p, p, p, p, p, p 0 0. In this notation p = p α; p = p β, etc. Beow I wi give a set of eapes of operations of spin and anguar oentu operations. You can verif the resuts, and see that the basic rues are not ver difficut. L p p = 0, L p p = p p S p p = p p = 0 ( antisetr!), S p p = 0 The above reations suffice to show that p p is the = coponent of the D utipet. You woud need to use the for ˆ ˆ ˆ ˆ ˆ L = L L L h L. A other functions in the utipet can be generated b acting successive b L. For eape L p p = ( p0 p p p0 ) L ( p p p p ) = ( p p p p ) Iportant, this eigenstate with eigenvaue = h is not a deterinant but a inear cobination of deterinants. In this state ou cannot sa that these orbitas are occupied and the rest ept. The wave function is ore copicated. You can act with the ˆL operator on this new function to show that the eigenvaue has not changed: () h is the answer ou shoud find. But it is ore work to show this epicit (give it a tr). You can keep apping the L operator unti ou find the state p p ( = h), and acting with L once ore ieds ero. 3

4 As another eape L p p = p p = 0, L p p = p p S p p = 0, S p p = p p L p p = ( p p p p ) = p p S p p = ( p p p p ) The first two equations estabish that p p 0 is the =, s = coponent of the 3 P utipet. Pease verif. A other 9 states can be obtained b successive appication of L and S as iustrated b the ast two eapes given. We wi do soe eercises with the anguar oentu operators eaborating on soe eapes as iustrated above. There is a vast iterature on the topic and the above is a ver brief suar. Let e re-ephasie that anguar oentu theor underies NMR spectra (nucear spin functions). Eectron spin resonance can be treated anaogous. This wi be iustrated in cass. To treat these phenoena we act on product of nucear spin functions. The principes are siiar, and spin functions are in fact a bit easier because the proportionait constants for spin partices are unit rather than. Let e aso ention that we can use a siiar treatent to find eigenfunctions of the tota anguar oentu operator Jˆ = Lˆ Sˆ, which is particuar reevant for atoic ter sbos. We wi go through soe eapes in cass and in the probe set. III. Genera decoposition of a product basis of anguar oentu eigenfunctions into eigenfunctions of the tota anguar oentu operators. In this section we consider the construction of eigenfunctions of the anguar oentu operators for a coposite partice (in fact it appies to an product function, for eape spin-orbitas pα ). The structure is ver genera. We wi consider the products for two partices, but fro this ou can construct products for an arbitrar nuber of partices 4

5 using the sae genera principes. You have seen eapes in NMR before we get to this part of the ecture notes. Consider the individua utipets,, =... and,, =... which are eigenfunctions of the anguar oentu operators ˆ ˆ L (), L () and ˆ (), ˆ () L L respective. Here we are using the Dirac notation for states:, indicates an eigenfunction characteried b the quantu nubers in the so-caed ket. The fu set of product functions,, spans a space of diension ( )( ). We want to decopose this product basis into a set of new basis functions that are eigenfunctions of the coposite anguar oentu operators Lˆ, L ˆ, where Lˆ = Lˆ() Lˆ(). Let us tota, tota indicate these eigenfunctions as LM,, M= L... L. Such a set of functions with a given vaue of L is caed a utipet, and the diension of the utipet is L. We wi see beow that the possibe eigenvaues L range fro L=,,...,, or, assuing that, we can aso sa that the utipets that are occurring are L= k, k = 0,..,. The tota diension of the space spanned b these utipets is {( k) } = ( )(( ) ) k k= 0, k= 0, = ( )( ) ( ) ( ( )) = ( )( ) This is consistent with the tota nuber of product functions, as shoud be the case. tota Let us rationaie this resut further b an epicit construction of the eigenfunctions. Each product function itsef is an eigenfunction of L ˆ, tot : Lˆ,, = h ( ),, tot,. Hence we can easi arrange the product functions in a tabe, such that aong a row, the a have the sae vaue of M. 5

6 Tabe: Arrangeent of product functions according to M-vaues (eigenvaue of L ˆ, tot ) M= M= M=,,,,,,,,,,,, 3.. M= M= M=,,,,,, 3,,,,,, # functions ( ) ( ) ( ) The overa utipet structure can be discerned now. In the second coun, top row, we find the function with the aiu vaue M = L=. This function is aso an eigenfunction of L. This foows because L ˆ,,, tota =0. You can copete the ˆtota arguent: wh is it an eigenfunction of L ˆtota generated b acting with L ˆ ˆ ˆ, tota = L () L (), hence? The other functions in this utipet are Lˆ,, = ( C (, ),, C (, ),, ), tota and so forth. It is seen that in genera we find a inear cobination of products states, ecept for the highest and owest possibe M-vaue in the space. This process continues as ou adder down and decrease the vaue of M, unti M = L=. The nuber of functions is precise the nuber of functions in the first coun, ( ). The net utipet is constructed b starting fro the eigenfunction that has M =, and is orthogona to the inear cobination function aread found. Aternative we ight tr to find that inear cobination such that acting with L ˆ, tota on this cobination ieds 0. Then it wi be an eigenfunction of both L and L, ˆtota ˆ tota. For eape 6

7 Lˆ, tota ( C (, ),, C (, ),, ). = C (, ) C (, )[,,,, ] = 0 This eans it wi be an eigenfunction of L with L=. ˆtota Even though we require a inear cobination of the product states to construct the true eigenfunctions, the nuber of functions in this second utipet is precise the sae as in the second coun of the tabe, nae ( ). The schee now repeats itsef. To start constructing the net utipet find the reaining highest M-state that is orthogona to the states aread found and adder down. Aternative, and this is usua easier, find that inear cobination of functions of specific M-vaues such that acting with L ˆ, tota ieds precise 0. The above schee is copete genera, and appies to a variet of probes in phsics. For eape we can construct eigenfunctions of the spin-operators in this fashion, and in this wa we find the proper cobinations of singet, doubet, tripet spin functions. The sae strateg appies to nucear spin functions. These probes are ver instructive: The give eact resuts because the basis set is copete, whie the agebra is often not so tedious. We can aso coupe different anguar oentu operators for eape to get eigenfunctions of the tota anguar oentu Jˆ = Lˆ Sˆ. These probes are awas copete anaogous if ou work on constructing an-partice functions. Essentia this is true because the anguar oentu operators for different partices coute. It is aso possibe to use anguar oentu theor to construct for eape integras and orbitas of higher -vaue. As an eape ou can evauate integras ike p ˆ d i j. The seection rues are easi evauated (eaning ou can know when an integra is ero). The precise evauation of non-ero integras requires ore work. To hep ou digest this ateria there are soe eercises! 7