, how does it change with time? How do we determine rt

Transcription

1 . TME-EVOLUTON OPERATOR Dyamical processes i quaum mechaics are described by a Hamiloia ha depeds o ime. Naurally he quesio arises how do we deal wih a ime-depede Hamiloia? priciple, he ime-depede Schrödiger equaio ca be direcly iegraed choosig a basis se ha spas he space of ieres. Usig a poeial eergy surface, oe ca propagae he sysem forward i small ime-seps ad follow he evoluio of he complex ampliudes i he basis saes. pracice eve his is impossible for more ha a hadful of aoms, whe you rea all degrees of freedom quaum mechaically. However, he mahemaical complexiy of solvig he ime-depede Schrödiger equaio for mos molecular sysems maes i impossible o obai exac aalyical soluios. We are hus forced o see umerical soluios based o perurbaio or approximaio mehods ha will reduce he complexiy. Amog hese mehods, ime-depede perurbaio heory is he mos widely used approach for calculaios i specroscopy, relaxaio, ad oher rae processes. his secio we will wor o classifyig approximaio mehods ad wor ou he deails of ime-depede perurbaio heory..1. Time-Evoluio Operaor Le s sar a he begiig by obaiig he equaio of moio ha describes he wavefucio ad is ime evoluio hrough he ime propagaor. We are seeig equaios of moio for quaum sysems ha are equivale o Newo s or more accuraely Hamilo s equaios for classical sysems. The quesio is, if we ow he wavefucio a ime r,, how does i chage wih ime? How do we deermie r, for some laer ime? We will use our iuiio here, based largely o correspodece o classical mechaics). To eep oaio o a miimum, i he followig discussio we will o explicily show he spaial depedece of wavefucio. We sar by assumig causaliy: precedes ad deermies, which is crucial for derivig a deermiisic equaio of moio. Also, as usual, we assume ime is a coiuous variable: lim (.1) Now defie a ime-displaceme operaor or propagaor ha acs o he wavefucio o he righ ad hereby propagaes he sysem forward i ime: U, (.) We also ow ha he operaor U cao be depede o he sae of he sysem. This is ecessary for coservaio of probabiliy, i.e., o reai ormalizaio for he sysem. f a a (.3) 1 1 Adrei Tomaoff, 11/8/14

2 - U, U, a1 1 U, a a a 1 1 (.4) This is a reflecio of he imporace of lieariy ad he priciple of superposiio i quaum mechaical sysems. While i a i, a a (.5) a ypically is o equal o This dicaes ha he differeial equaio of moio is liear i ime. Properies of U We ow mae some impora ad useful observaios regardig he properies of U. 1) Uiary. Noe ha for eq. (.5) o hold ad for probabiliy desiy o be coserved, U mus be uiary which holds if U 1 U. P U U (.6) ) Time coiuiy: The sae is uchaged whe he iiial ad fial ime-pois are he same U, 1. (.7) 3) Composiio propery. f we ae he sysem o be deermiisic, he i sads o reaso ha we should ge he same wavefucio wheher we evolve o a arge ime i oe sep or muliple seps. Therefore, we ca wrie 1,,, U U U (.8) 1 1 Noe, sice U acs o he righ, order maers: U, 1U1, U, 1 1 (.9) Equaio (.8) is already very suggesive of a expoeial form for U. Furhermore, sice ime is coiuous ad he operaor is liear i also suggess ha he ime propagaor is oly a depede o a ime ierval, U U (.1) 1 1 4) Time-reversal. The iverse of he ime-propagaor is he ime reversal operaor. From eq. (.8):

3 A equaio of moio for U -3 U, U, 1 (.11),,. (.1) 1 U U Le s fid a equaio of moio ha describes he ime-evoluio operaor usig he chage of he sysem for a ifiiesimal ime-sep, U,. Sice : U lim, 1 (.13) We expec ha for small eough, U will chage liearly wih. This is based o aalogy o hiig of deermiisic moio i classical sysems. Seig o, so ha U, U, we ca wrie U U i ˆ (.14) ˆ is a ime-depede Hermiia operaor, which is required for U o be uiary. We ca ow U,, he equaio of moio for U: wrie a differeial equaio for he ime-developme of du U U d lim (.15) So from (.14) we have: U, iˆ U, (.16) You ca ow see ha he operaor eeded a complex argume, because oherwise probabiliy desiy would o be coserved; i would rise or decay. Raher i oscillaes hrough differe saes of he sysem. We oe ha ˆ has uis of frequecy. Sice quaum mechaics fudameally associaes frequecy ad eergy as E, ad sice he Hamiloia is he operaor correspodig o he eergy, ad resposible for ime evoluio i Hamiloia mechaics, we wrie ˆ ˆ H (.17) Wih ha subsiuio we have a equaio of moio for U: i U, HU ˆ, (.18) Muliplyig from he righ by gives he TDSE:

4 -4 i Hˆ (.19) f you use he Hamiloia for a free paricle ( /m)( /x ), his loos lie a classical wave equaio, excep ha i is liear i ime. Raher, his loos lie a diffusio equaio wih imagiary diffusio cosa. We are also ieresed i he equaio of moio for U which describes he ime evoluio of he cojugae wavefucios. Followig he same approach ad U, acs o he lef: recogizig ha U, (.) we ge i U, U, Hˆ. (.1) Evaluaig he ime-evoluio operaor A firs glace i may seem sraighforward o iegrae eq. (.18). f H is a fucio of ime, he he iegraio of idu U Hd gives i U, exp Hd (.) Followig our earlier defiiio of he ime-propagaor, his expoeial would be cas as a series expasio? i 1 i U, 1 H d d d H H! (.3) This approach is dagerous, sice we are o properly reaig H as a operaor. Looig a he secod erm i eq. (.3), we see ha his expressio iegraes over boh possible imeorderigs of he wo Hamiloia operaios, which would oly be proper if he Hamiloias a,. Now, le s proceed a bi more carefully assumig ha he Hamiloias a differe imes differe imes commue: H H do o commue. egraig equaio (.18) direcly from o gives i U, 1 d H U, (.4) This is he soluio; however, i is o very pracical sice U, is a fucio of iself. Bu we ca mae a ieraive expasio by repeiive subsiuio of U io iself. The firs sep i his process is

5 -5 i i U d H d H U, 1 1, i i 1 dh d d H H U, (.5) Noe i he las erm of his equaio, ha he iegraio limis eforce a ime-orderig; ha is, he firs iegraio variable mus precede he secod. Picorially, he area of iegraio is The ex subsiuio sep gives i U, 1 dh i 3 i d dh H, d d dh H H U (.6) From his expasio, you should be aware ha here is a ime-orderig o he ieracios. For he hird erm, acs before, which acs before :. Wha does his expressio represe? magie you are sarig i sae ad you wa o describe how oe evolves oward a arge sae. The possible pahs by which oe ca shif ampliude ad evolve he phase, picured i erms of hese ime variables are: The firs erm i eq. (.6) represes all acios of he Hamiloia which ac o direcly couple ad. The secod erm described possible rasiios from o via a iermediae sae m. The expressio for U describes all possible pahs bewee iiial ad fial sae. Each

6 -6 of hese pahs ierferes i ways dicaed by he acquired phase of our eigesaes uder he imedepede Hamiloia. The soluio for U obaied from his ieraive subsiuio is ow as he posiive imeordered expoeial i U d H ˆ i T exp d H, exp i 1 1 d d d1h H 1 H 1 ( T ˆ is ow as he Dyso ime-orderig operaor.) his expressio he ime-orderig is (.7) (.8) So, his expressio ells you abou how a quaum sysem evolves over a give ime ierval, ad i allows for ay possible rajecory from a iiial sae o a fial sae hrough ay umber of iermediae saes. Each erm i he expasio accous for more possible rasiios bewee differe iermediae quaum saes durig his rajecory. Compare he ime-ordered expoeial wih he radiioal expasio of a expoeial: 1 i 1 1! d d1 H H 1 H 1 (.9) Here he ime-variables assume all values, ad herefore all orderigs for H i are calculaed. The areas are ormalized by he! facor. (There are! ime-orderigs of he imes.) (As commeed above hese pois eed some more clarificaio.) We are also ieresed i he Hermiia cojugae of U, moio i eq. (.1). f we repea he mehod above, rememberig ha, he we obai, which has he equaio of U acs o he lef, i U, 1 d U, H. (.3) Performig ieraive subsiuio leads o a egaive-ime-ordered expoeial: i, exp U d H i 1 1 d d d H H H (.31)

7 -7 Here he H i ac o he lef. Readigs 1. Cohe-Taoudji, C.; Diu, B.; Lalöe, F., Quaum Mechaics. Wiley-ersciece: Paris, 1977; p Merzbacher, E., Quaum Mechaics. 3rd ed.; Wiley: New Yor, 1998; Ch Muamel, S., Priciples of Noliear Opical Specroscopy. Oxford Uiversiy Press: New Yor, 1995; Ch.. 4. Saurai, J. J., Moder Quaum Mechaics, Revised Ediio. Addiso-Wesley: Readig, MA, 1994; Ch..

8 -8.. egraig he TDSE Direcly Oay, how do we evaluae he ime-propagaor ad obai a ime-depede rajecory for a quaum sysem? Expressios such as eq. (.7) are dauig, ad here are o simple ways i which o hadle his. Oe cao rucae he expoeial because usually his is o a rapidly covergig series. Also, he soluios oscillae rapidly as a resul of he phase acquired a he eergy of he saes ivolved, which leads o a formidable iegraio problem. Rapid oscillaios require small ime seps, whe i fac he ime scales. For isace i a molecular dyamics problem, he highes frequecy oscillaios may be as a resul of elecroically excied saes wih periods of less ha a femosecod, ad he uclear dyamics ha you hope o describe may occur o may picosecod ime scales. Raher ha geeral recipes, here exis a arseal of differe sraegies ha are suied o paricular ypes of problems. The choice of how o proceed is geerally dicaed by he deails of your problem, ad is ofe a ar-form. Cosiderable effor eeds o be made o formulae he problem, paricularly choosig a appropriae basis se for your problem. Here i is our goal o gai some isigh io he ypes of sraegies available, worig maily wih he priciples, raher ha he specifics of how i s implemeed. Le s begi by discussig he mos geeral approach. Wih adequae compuaioal resources, we ca choose he brue force approach of umerical iegraio. We sar by choosig a basis se ad defiig he iiial sae. The, we ca umerically evaluae he imedepedece of he wavefucio over a ime period by discreizig ime io small seps of widh =/ over which he chage of he sysem is small. A variey of sraegies ca be pursed i pracice. Oe possibiliy is o expad your wavefucio i he basis se of your choice () c () (.3) ad solve for he ime-depedece of he expasio coefficies. Subsiuig io he righ side of he TDSE, i Hˆ (.33) ad he acig from he lef by o boh sides leads o a equaio ha describes heir imedepedece: c () i H () c () (.34) or i marix form i c Hc. This represes a se of coupled firs-order differeial equaios i which ampliude flows bewee differe basis saes a raes deermied by he marix elemes

9 -9 of he ime-depede Hamiloia. Such equaios are sraighforward o iegrae umerically. We recogize ha we ca iegrae o a grid if he ime sep forward () is small eough ha he Hamiloia is esseially cosa. The eq. (.34) becomes i c () H () c () (.35) ad he sysem is propagaed as c ( ) c () c () (.36) The dowside of such a calculaio is he uusually small ime-seps ad sigifica compuaioal cos required. Similarly, we ca use a grid wih shor ime seps o simplify our ime-propagaor as ˆ i (, ) exp ˆ i U dh exp H ˆ ( ) (.37) Therefore he ime propagaor ca be wrie as a produc of propagaors over hese small iervals. lim 1 1 Uˆ Uˆ Uˆ Uˆ Uˆ lim 1 j Uˆ j (.38) Here he ime-propagaio over he j h small ime sep is ˆ i exp ˆ U j H j (.39) Hˆ Hˆ j j Noe ha he expressios i (.38) are operaors ime ordered from righ o lef, which we deoe wih he + subscrip. Alhough (.38) is exac i he limi (or ), we ca choose a fiie umber such ha H() does o chage much over he ime. his limi he ime propagaor does o chage much ad ca be approximaed as a expasio

10 -1 Uˆ j i 1 Hˆ (.4). j a geeral sese his approach is o very pracical. The firs reaso is ha he ime sep is deermied by / H which is ypically very small i compariso o he dyamics of ieres. The secod complicaio arises whe he poeial ad ieic eergy operaors i he Hamiloia do commue. Taig he Hamiloia o be Hˆ Tˆ Vˆ, e ihˆ / i( Tˆ Vˆ ) / e e itˆ / ivˆ / e (.41) The secod lie maes he Spli Operaor approximaio, wha saes ha he ime propagaor over a shor eough period ca be approximaed as a produc of idepede propagaors evolvig he sysem over he ieic ad poeial eergy. The validiy of his approximaio depeds o how well hese operaors commue ad he ime sep, wih he error scalig lie 1 ˆ ˆ [ T ( ), V ( )]( / ), meaig ha we should use a ime sep, such ha ˆ ˆ 1/ { / [ T( ), V( )]}. 1 This approximaio ca be improved by symmerizig he spli operaor as Here he error scales as ˆ / ˆ ihˆ / iv ˆ / iv / it e e e e (.4) ( / ) {[ T ˆ,[ T ˆ, V ˆ ]] [ V ˆ,[ V ˆ, T ˆ ]]}. There is o sigifica icrease i compuaioal effor sice half of he operaios ca be combied as o give ivˆ j1 ivˆ j / ˆ ivj / e e e (.43) U e e e e ˆ / ˆ iv ˆ ˆ iv / ivj / itj / j1. (.44) Readigs 1. Taor, D. J., roducio o Quaum Mechaics: A Time-Depede Perspecive. Uiversiy Sciece Boos: Sausilio, CA, 7.

11 Trasiios duced by a Time-Depede Poeial For may ime-depede problems, mos oably i specroscopy, we ca ofe pariio he problem so ha he ime-depede Hamiloia coais a ime-idepede par H ha we ca describe exacly, ad a ime-depede poeial V : H H V (.45) The remaiig degrees of freedom are discarded, ad he oly eer i he sese ha hey give rise o he ieracio poeial wih H. This is effecive if you have reaso o believe ha he exeral Hamiloia ca be reaed classically, or if he ifluece of H o he oher degrees of freedom is egligible. From (.45), here is a sraighforward approach o describig he imeevolvig wavefucio for he sysem i erms of he eigesaes ad eergy eigevalues of H. Hamiloia To begi, we ow he complee se of eigesaes ad eigevalues for he sysem H E (.46) The sae of he sysem ca he be expressed as a superposiio of hese eigesaes: c (.47) The TDSE ca be used o fid a equaio of moio for he eigesae coefficies c (.48) Sarig wih ad from (.5) i H (.49) c i H i Hc (.5) (.51) Already we see ha he ime evoluio amous o solvig a se of coupled liear ordiary differeial equaios. These are rae equaios wih complex rae cosas, which describe he feedig of oe sae io aoher. Subsiuig eq. (.45) we have: c i i H V c E V c (.5)

12 -1 or, c i i Ec V c. (.53) Nex, we defie ad subsiue ie m c m e bm (.54) which implies a defiiio for he wavefucio as ie b e (.55) This defies a slighly differe complex ampliude, ha allows us o simplify higs cosiderably. Noice ha b c. Also, b c. pracice wha we are doig is pullig ou he rivial par of he ime evoluio, he ime-evolvig phase facor, which ypically oscillaes much faser ha he chages o he ampliude of b or c. We will come bac o his sraegy which we discuss he ieracio picure. Now eq. (.53) becomes e ie b i e ie V b i or i V e b (.56) b (.57) This equaio is a exac soluio. is a se of coupled differeial equaios ha describe how probabiliy ampliude moves hrough eigesaes due o a ime-depede poeial. Excep i simple cases, hese equaios cao be solved aalyically, bu i is ofe sraighforward o iegrae umerically. Whe ca we use he approach described here? Cosider pariioig he full Hamiloia io wo compoes, oe ha we wa o sudy H ad he remaiig degrees of freedom H1. For each par, we have owledge of he complee eigesaes ad eigevalues of he Hamiloia: Hi i, Ei, i,. These subsysems will ierac wih oe aoher hrough Hi. f we are careful o pariio his i such a way ha Hi is small compared H ad H1, he i

13 -13 should be possible o properly describe he sae of he full sysem as produc saes i he subsysems: 1. Furher, we ca wrie a ime-depede Schrödiger equaio for he moio of each subsysem as: 1 i H 1 1 (.58) Wihi hese assumpios, we ca wrie he complee ime-depede Schrödiger equaio i erms of he wo sub-saes: 1 i i 1 H1 1 1 H Hi 1 (.59) The lef operaig by 1 ad maig use of eq. (.58), we ca wrie i H 1 Hi 1 (.6) This is equivale o he TDSE for a Hamiloia of form (.45) where he exeral ieracio V 1 Hi 1 comes from iegraig he 1- ieracio over he sub-space of 1. So his represes a ime-depede mea field mehod. Readigs 1. Cohe-Taoudji, C.; Diu, B.; Lalöe, F., Quaum Mechaics. Wiley-ersciece: Paris, 1977; p Merzbacher, E., Quaum Mechaics. 3rd ed.; Wiley: New Yor, 1998; Ch Niza, A., Chemical Dyamics i Codesed Phases. Oxford Uiversiy Press: New Yor, 6; Sec Saurai, J. J., Moder Quaum Mechaics, Revised Ediio. Addiso-Wesley: Readig, MA, 1994; Ch..

14 Resoa Drivig of Two-level Sysem Le s describe wha happes whe you drive a wo-level sysem wih a oscillaig poeial. V V cos (.61) V V cos (.6) Noe, his is he form you would expec for a elecromageic field ieracig wih charged paricles, i.e. dipole rasiios. a simple sese, he elecric field is E E cos ad he ieracio poeial ca be wrie as V will loo a he form of his ieracio a bi more carefully laer. We ow couple wo saes a ad b wih he oscillaig field. Here he eergy of he saes is ordered so ha b > a. Le s as if he sysem sars i a wha is he probabiliy of fidig i i b a ime? E, where represes he dipole operaor. We The sysem of differeial equaios ha describe his problem is: i b b V e a, b a, b i i 1 i i e e b V e Where wroe cos i is complex form. Wriig his explicily i 1 i ba ba 1 ib bv e e b V ib b a ba b bb b V 1 a a aa i i i 1 ab i ab e e bv b ab e e or i ba i ba e e e e (.63) i i (.64) (.65) Here he expressios have bee wrie i erms of he frequecy ba. Two of hese erms are dropped, sice (for our case) he diagoal marix elemes Vii. We also mae he secular approximaio (or roaig wave approximaio) i which he oresoa erms are dropped. Whe ba, erms lie i e or i ba e oscillae very rapidly (relaive o ba 1 V ) ad so do o coribue much o chage of c. (Remember, we ae he frequecies ba ad o be posiive). So ow we have: i i ba b b ba Vba e (.66)

15 -15 i i ba b a bb Vab e (.67) Noe ha he coefficies are oscillaig a he same frequecy bu phase shifed o oe aoher. Now if we differeiae eq. (.66): i ba b b b a Vba iba ba Vba e i i ba e (.68) Rewrie eq. (.66): b a i i ba b b e (.69) V ba ad subsiue (.69) ad (.67) io (.68), we ge liear secod order equaio for b b. Vba b ba b b b i b b 4 (.7) This is jus he secod order differeial equaio for a damped harmoic oscillaor: ax bx cx (.71) b a cos si x e A B 1 4ac b a Wih a lile more maipulaio, ad rememberig he iiial codiios we fid b ad b (.7) 1, Vba b b si R Vba ba P b (.73) Where he Rabi Frequecy 1 R Vba ba (.74) Also, P 1 P (.75) a The ampliude oscillaes bac ad forh bewee he wo saes a a frequecy dicaed by he couplig bewee hem. [Noe a resul we will reur o laer: elecric fields couple quaum saes, creaig cohereces!] A impora observaio is he imporace of resoace bewee he drivig poeial ad he eergy spliig bewee saes. To ge rasfer of probabiliy desiy you eed he b

16 -16 drivig field o be a he same frequecy as he eergy spliig. O resoace, you always drive probabiliy ampliude eirely from oe sae o aoher. The efficiecy of drivig bewee a ad b saes drops off wih deuig. Here ploig he maximum value of P b as a fucio of frequecy:

17 Schrödiger ad Heiseberg Represeaios The mahemaical formulaio of quaum dyamics ha has bee preseed is o uique. So far, we have described he dyamics by propagaig he wavefucio, which ecodes probabiliy desiies. Ulimaely, sice we cao measure a wavefucio, we are ieresed i observables, which are probabiliy ampliudes associaed wih Hermiia operaors, wih imedepedece ha ca be ierpreed differely. Cosider he expecaio value: U Aˆ U A ˆ Aˆ UAU ˆ U AU ˆ (.76) The las wo expressios are wrie o emphasize alerae picures of he dyamics. The firs, ow as he Schrödiger picure, refers o everyhig we have doe so far. Here we propagae he wavefucio or eigevecors i ime as U. Operaors are uchaged because hey carry o ime-depedece. Aleraively, we ca wor i he Heiseberg picure. This uses he uiary propery of U o ime-propagae he operaors as A ˆ UAU ˆ, bu he wavefucio is ow saioary. The Heiseberg picure has a appealig physical picure behid i, because paricles move. Tha is, here is a ime-depedece o posiio ad momeum. Schrödiger Picure he Schrödiger picure, he ime-developme of is govered by he TDSE or equivalely, he ime propagaor: i H ad U, (.77) he Schrödiger picure, operaors are ypically idepede of ime, A. Wha abou observables? For expecaio values of operaors A () A : ˆ ˆ() ˆ ˆ A i A i A A AH ˆ H Aˆ AH ˆ, (.78) f  is idepede of ime (as we expec i he Schrödiger picure), ad if i commues wih H, i is referred o as a cosa of moio.

18 -18 Heiseberg Picure From eq. (.76), we ca disiguish he Schrödiger picure from Heiseberg operaors: A ˆ() Aˆ U AU ˆ Aˆ (.79) S S H where he operaor is defied as Aˆ ˆ H U, AU S, (.8) Aˆ ˆ H AS Noe, he picures have he same wavefucio a he referece poi. Sice he wavefucio should be ime-idepede, H, we ca relae he Schrödiger ad Heiseberg wavefucios as S U, So, U, (.81) H S S H (.8) As expeced for a uiary rasformaio, i eiher picure he eigevalues are preserved: Aˆ i ai S i S ˆ U AUU i au S i i S Aˆ a H i H i i H The ime evoluio of he operaors i he Heiseberg picure is: Aˆ ˆ H ˆ U ˆ ˆ U AS U ASU ASU U AS U U i ˆ ˆ i ˆ A U H ASU U AS HU i ˆ i H A Aˆ H i AH ˆ, H H H H H H (.83) (.84) The resul i Aˆ H Aˆ, H (.85) H is ow as he Heiseberg equaio of moio. Here have wrie he odd looig HH U HU. This is maily o remid oe abou he ime-depedece of H. Geerally speaig, for a ime-idepede Hamiloia a ime-depede Hamiloia, U ad H eed o commue. U e ih/, U ad H commue, ad HH H. For

19 -19 Classical equivalece for paricle i a poeial The Heiseberg equaio is commoly applied o a paricle i a arbirary poeial. Cosider a paricle wih a arbirary oe-dimesioal poeial H p V( x) (.86) m For his Hamiloia, he Heiseberg equaio gives he ime-depedece of he momeum ad posiio as Here, have made use of V p x (.87) p x m (.88) ˆ ˆ ˆ 1 x, p i x (.89) ˆ ˆ ˆ 1 x, p i p (.9) Curiously, he facors of have vaished i equaios (.87) ad (.88), ad quaum mechaics does o seem o be prese. sead, hese equaios idicae ha he posiio ad momeum operaors follow he same equaios of moio as Hamilo s equaios for he classical variables. f we iegrae eq. (.88) over a ime period we fid ha he expecaio value for he posiio of he paricle follows he classical moio. x p x (.91) m We ca also use he ime derivaive of eq. (.88) o obai a equaio ha mirrors Newo s secod law of moio, F=ma: x m V (.9) These observaios uderlie Ehrefes s Theorem, a saeme of he classical correspodece of quaum mechaics, which saes ha he expecaio values for he posiio ad momeum operaors will follow he classical equaios of moio. Readigs 1. Cohe-Taoudji, C.; Diu, B.; Lalöe, F., Quaum Mechaics. Wiley-ersciece: Paris, 1977; p Muamel, S., Priciples of Noliear Opical Specroscopy. Oxford Uiversiy Press: New Yor, Niza, A., Chemical Dyamics i Codesed Phases. Oxford Uiversiy Press: New Yor, 6; Ch. 4.

20 - 4. Saurai, J. J., Moder Quaum Mechaics, Revised Ediio. Addiso-Wesley: Readig, MA, 1994; Ch..

21 -1.6. eracio Picure The ieracio picure is a hybrid represeaio ha is useful i solvig problems wih imedepede Hamiloias i which we ca pariio he Hamiloia as H H V (.93) H is a Hamiloia for he degrees of freedom we are ieresed i, which we rea exacly, ad ca be (alhough for us usually will o be) a fucio of ime. V is a ime-depede poeial which ca be complicaed. he ieracio picure, we will rea each par of he Hamiloia i a differe represeaio. We will use he eigesaes of H as a basis se o describe he dyamics iduced by V(), assumig ha V() is small eough ha eigesaes of H are a useful basis. f H is o a fucio of ime, he here is a simple ime-depedece o his par of he Hamiloia ha we may be able o accou for easily. Seig V o zero, we ca see ha he ime evoluio of he exac par of he Hamiloia H is described by i U, H U, (.94) i where, U, exp d H or, for a ime-idepede H,, U e ih We defie a wavefucio i he ieracio picure wavefucio hrough: S U, (.95) (.96) i erms of he Schrödiger (.97) or U (.98) S Effecively he ieracio represeaio defies wavefucios i such a way ha he phase ih accumulaed uder e is removed. For small V, hese are ypically high frequecy oscillaios relaive o he slower ampliude chages iduced by V. Now we eed a equaio of moio ha describes he ime evoluio of he ieracio picure wavefucios. We begi by subsiuig eq. (.97) io he TDSE: i S H S (.99)

22 U i H U - i U, H U, i U H V U, ` (.1) i U H V U i V (.11) V U, V U, (.1) where saisfies he Schrödiger equaio wih a ew Hamiloia i eq. (.1): he ieracio picure Hamiloia, V. We have performed a uiary rasformaio of V io he frame i l of referece of H, usig U. Noe: Marix elemes i V V l e Vl where ad l are eigesaes of H. We ca ow defie a ime-evoluio operaor i he ieracio picure: U, (.13) where U, exp d V Now we see ha S i U, U, U, U, U, S,,, (.14) (.15) U U U (.16) Also, he ime evoluio of cojugae wavefucio i he ieracio picure ca be wrie i i U, U, U, exp d V exp d H (.17) For he las wo expressios, he order of hese operaors ceraily maers. So wha chages abou he ime-propagaio i he ieracio represeaio? Le s sar by wriig ou he ime-ordered expoeial for U i eq. (.16) usig eq. (.14):

23 -3 i U, U, d U, V U, U i d d d U V U U V U,,, Here have used he composiio propery of U,,, (.18). The same posiive ime-orderig applies. Noe ha he ieracios V(i) are o i he ieracio represeaio here. Raher we used he defiiio i eq. (.1) ad colleced erms. Now cosider how U describes he imedepedece if iiiae he sysem i a eigesae of H l ad observe he ampliude i a arge eigesae. The sysem evolves i eigesaes of H durig he differe ime periods, wih he ime-depede ieracios V drivig he rasiios bewee hese saes. The firs-order erm describes direc rasiios bewee l ad iduced by V, iegraed over he full ime period. Before he ieracio phase is acquired as ie e /, whereas afer he ieracio phase is ie / acquired as e. Higher-order erms i he ime-ordered expoeial accous for all possible iermediae pahways. We ow ow how he ieracio picure wavefucios evolve i ime. Wha abou he operaors? Firs of all, from examiig he expecaio value of a operaor we see ˆ A ˆ A U, AU ˆ, U U AU ˆ U Aˆ (.19) where A U A U. (.11) S So he operaors i he ieracio picure also evolve i ime, bu uder H. This ca be expressed as a Heiseberg equaio by differeiaig A ˆ i A ˆ H, A (.111) Also, we ow i V (.11)

24 -4 Noice ha he ieracio represeaio is a pariio bewee he Schrödiger ad Heiseberg represeaios. Wavefucios evolve uder V, while operaors evolve uder H. Aˆ i For H, V H ; S H S Schrödiger Aˆ i For H H, V H, Aˆ ; Heiseberg (.113) The relaioship bewee U ad b Earlier we described how ime-depede problems wih Hamiloias of he form H H V could be solved i erms of he ime-evolvig ampliudes i he eigesaes of H. We ca describe he sae of he sysem as a superposiio where he expasio coefficies c c (.114) are give by, c U e UU ie/ U (.115) Now, comparig equaios (.115) ad (.54) allows us o recogize ha our earlier modified expasio coefficies b were expasio coefficies for ieracio picure wavefucios b U (.116) Readigs 1. Muamel, S., Priciples of Noliear Opical Specroscopy. Oxford Uiversiy Press: New Yor, Niza, A., Chemical Dyamics i Codesed Phases. Oxford Uiversiy Press: New Yor, 6; Ch. 4.

25 -5.7. Time-Depede Perurbaio Theory Perurbaio heory refers o calculaig he ime-depedece of a sysem by rucaig he expasio of he ieracio picure ime-evoluio operaor afer a cerai erm. pracice, rucaig he full ime-propagaor U is o effecive, ad oly wors well for imes shor compared o he iverse of he eergy spliig bewee coupled saes of your Hamiloia. The ieracio picure applies o Hamiloias ha ca be cas as H H V ad, allows us o focus o he ifluece of he couplig. We ca he rea he ime evoluio uder H exacly, bu rucae he ifluece of V. This wors well for wea perurbaios. Le s loo more closely a his. We ow he eigesaes for H : H E, ad we ca calculae he evoluio of he wavefucio ha resuls from For a give sae, we calculae b V : b (.117) as:, b U (.118) i U V d where, exp (.119) Now we ca rucae he expasio afer a few erms. This wors well for small chages i ampliude of he quaum saes wih small couplig marix elemes relaive o he eergy b b ; V E E spliigs ivolved ( ) As we will see, he resuls we obai from perurbaio heory are widely used for specroscopy, codesed phase dyamics, ad relaxaio. Le s ae he specific case where we have a sysem prepared i, ad we wa o ow he probabiliy of observig he sysem i a ime due o Expadig b exp dv i b d V i i V : P b d 1 1 d V V i Now, usig V U V U V we obai: e. (.1) (.11) (.1)

26 -6 i b d V i 1 e 1 1 firs-order (.13) i m i m 1 d d1e Vm Vm 1 i m e secod order (.14) The firs-order erm allows oly direc rasiios bewee ad, as allowed by he marix eleme i V, whereas he secod-order erm accous for rasiios occurrig hrough all possible iermediae saes m. For perurbaio heory, he ime-ordered iegral is rucaed a he appropriae order. cludig oly he firs iegral is firs-order perurbaio heory. The order of perurbaio heory ha oe would exed a calculaio should be evaluaed iiially by which allowed pahways bewee ad you eed o accou for ad which oes are allowed by he marix elemes. For firs-order perurbaio heory, he expressio i eq. (.13) is he soluio o he differeial equaio ha you ge for direc couplig bewee ad : i i b e V b (.15) This idicaes ha he soluio does o allow for he feedbac bewee ad ha accous for chagig populaios. This is he reaso we say ha validiy dicaes b b 1. f he iiial sae of he sysem is o a eigesae of H, we ca express i as a superposiio of eigesaes, wih b b b U (.16) Aoher observaio applies o firs-order perurbaio heory. f he sysem is iiially prepared i a sae, ad a ime-depede perurbaio is ured o ad he ured off over he ime ierval o, he he complex ampliude i he arge sae is jus relaed o he Fourier rasform of V evaluaed a he eergy gap. i i b d e V (.17) f he Fourier rasform pair is defied i he followig maer: V F V dv expi, (.18) 1 1 V F V dv expi (.19)

27 -7 The we ca wrie he probabiliy of rasfer o sae as P V. (.13)

28 -8 Example: Firs-order Perurbaio Theory Le s cosider a simple model for vibraioal exciaio iduced by he compressio of harmoic oscillaor. We will subjec a harmoic oscillaor iiially i is groud sae o a Gaussia compressio pulse, which icreases is force cosa. Firs, wrie he complee ime-depede Hamiloia: p 1 H T V x m (.131) Now, pariio i accordig o H H V i such a maer ha we ca wrie H as a harmoic oscillaor Hamiloia. This ivolves pariioig he ime-depede force cosa io wo pars: m exp (.13) p 1 1 H x x exp m H V () (.133) Here is he magiude of he iduced chage i he force cosa, ad is he ime-widh of he Gaussia perurbaio. So, we ow he eigesaes of H: H E H a a 1 E 1 (.134) Now we as, if he sysem is i before applyig he perurbaio, wha is he probabiliy of fidig i i sae afer he perurbaio? For : i i b d V e (.135)

29 E E, ad recogizig ha we ca se he limis o ad =, Usig i i b x d e e -9 (.136) i This leads o / b x e (.137) Here we made use of a impora ideiy for Gaussia iegrals: expax bx cdx exp c (.138) a a 1 b 4 a b 4a expax ibxdx exp (.139) Wha abou he marix eleme? x m a a m aa a a aa a a (.14) From hese we see ha firs-order perurbaio heory will o allow rasiios o 1, oly ad. Geerally his would o be realisic, because you would ceraily expec exciaio o =1 would domiae over exciaio o =. A real sysem would also be aharmoic, i which case, he leadig erm i he expasio of he poeial V(x), ha is liear i x, would o vaish as i does for a harmoic oscillaor, ad his would lead o marix elemes ha raise ad lower he exciaio by oe quaum. However for he prese case, x (.141) m So, b i m e (.14) ad we ca wrie he probabiliy of occupyig he = sae as m b P 4 e (.143) From he expoeial argume, sigifica rasfer of ampliude occurs whe he compressio pulse widh is small compared o he vibraioal period. 1 (.144)

30 -3 his regime, he poeial is chagig faser ha he aoms ca respod o he perurbaio. pracice, whe cosiderig a solid-sae problem, wih frequecies machig hose of acousic phoos ad ui cell dimesios, we eed perurbaios ha move faser ha he speed of soud, i.e., a shoc wave. The opposie limi, 1, is he adiabaic limi. his case, he perurbaio is so slow ha he sysem always remais eirely i =, eve while i is compressed. Now, le s cosider he validiy of his firs-order reame. Perurbaio heory does o allow for b o chage much from is iiial value. Firs we re-wrie eq. (.143) as P 4 e (.145) Now for chages ha do differ much from he iiial value, P 1 1 Geerally, he magiude of he perurbaio mus be small compared o. (.146) Oe sep furher The precedig example was simple, bu i racs he geeral approach o seig up problems ha you rea wih ime-depede perurbaio heory. The approach relies o wriig a Hamiloia ha ca be cas io a Hamiloia ha you ca rea exacly H, ad ime-depede perurbaios ha shif ampliudes bewee is eigesaes. For his scheme o wor well, we eed he magiude of perurbaio o be small, which immediaely suggess worig wih a Taylor series expasio of he poeial. For isace, ae a oe-dimesioal poeial for a boud paricle, V(x), which is depede o he form of a exeral variable y. We ca expad he poeial i x abou is miimum x = as 3 1 V 1 V 1 V V x x xy xyz! x! xy 3! x x yz, xyz x 1 () (3) 3 (3) (3) x V xy V3 x V x y V1 xy (.147) The firs erm is he harmoic force cosa for x, ad he secod erm is a bi-liear couplig whose magiude V () idicaes how much a chage i he variable y iflueces he variable x. (3) The remaiig erms are cubic expasio erms. V3 is he cubic aharmoiciy of V(x), ad he remaiig wo erms are cubic coupligs ha describe he depedece of x o y. roducig a ime-depede poeial is equivale o iroducig a ime-depedece o he operaor y, where he form ad sregh of he ieracio is subsumed io he ampliude V. he case of he

31 -31 (3) previous example, our formulaio of he problem was equivale o selecig oly he V erm, (3) so ha V, ad givig he value of y a ime-depedece described by he Gaussia waveform. Readigs 1. Cohe-Taoudji, C.; Diu, B.; Lalöe, F., Quaum Mechaics. Wiley-ersciece: Paris, 1977; p Niza, A., Chemical Dyamics i Codesed Phases. Oxford Uiversiy Press: New Yor, 6; Ch Saurai, J. J., Moder Quaum Mechaics, Revised Ediio. Addiso-Wesley: Readig, MA, 1994; Ch..

32 p Fermi s Golde Rule A umber of impora relaioships i quaum mechaics ha describe rae processes come from firs-order perurbaio heory. These expressios begi wih wo model problems ha we wa o wor hrough: (1) ime evoluio afer applyig a sep perurbaio, ad () ime evoluio afer applyig a harmoic perurbaio. As before, we will as: if we prepare he sysem i he sae, wha is he probabiliy of observig he sysem i sae followig he perurbaio? Cosa perurbaio (or sep perurbaio) The sysem is prepared such ha applied a : V V V. A cosa perurbaio of ampliude V is (.148) Here is he Heaviside sep respose fucio, which is for < ad 1 for. Now, urig o firsorder perurbaio heory, he ampliude i, we have: i i b d e V (.149) Here V is idepede of ime. Seig =, i i b V d e V exp i E E iv E i / e E 1 si / i i he las expressio, used he ideiy e 1 ie si 4 V E E P b si. Now (.15) (.151) f we wrie his usig he eergy spliig variable we used earlier: E E V P si / Compare his wih he exac resul we have for he wo-level problem:, (.15)

33 p. -33 P V V si V / (.153) As expeced, he perurbaio heory resul wors well for V <<. Le s examie he ime-depedece o P, ad compare he perurbaio heory (solid lies) o he exac resul (dashed lies) for differe values of. The wors correspodece is for = for which he behavior appears quadraic ad he probabiliy quicly exceeds uiy. is ceraily urealisic, bu we do o expec ha he expressio will hold for he srog couplig case: V. Oe begis o have quaiaive accuracy i for he regime P()P () <.1 or < 4V. Now le s loo a he depedece o. We ca wrie he firs-order resul eq. (.15) as where sic si V P sic / (.154) x x x. f we plo he probabiliy of rasfer from o as a fucio of he eergy level spliig E E we have:

34 p. -34 The probabiliy of rasfer is sharply peaed where eergy of he iiial sae maches ha of he fial sae, ad he widh of he eergy mismach arrows wih ime. Sice lim sicx 1, we x see ha he shor ime behavior is a quadraic growh i P lim P V (.155) The iegraed area grows liearly wih ime. Sice he eergy spread of saes o which rasfer is efficie scales approximaely as E E, his observaio is someimes referred o as a uceraiy relaio wih E. However, remember ha his is really jus a observaio of he priciples of Fourier rasforms. A frequecy ca oly be deermied as accuraely as he legh of he ime over which you observe oscillaios. Sice ime is o a operaor, i is o a rue uceraily relaio lie px. he log ime limi, he sic (x) fucio arrows o a dela fucio: si ax lim x ax (.156) lim P V E E (.157) The dela fucio eforces eergy coservaio, sayig ha he eergies of he iiial ad arge sae mus be he same i he log ime limi. Wha is ieresig i eq. (.157) is ha we see a probabiliy growig liearly i ime. This suggess a rasfer rae ha is idepede of ime, as expeced for simple firs-order ieics: P V w E E (.158) This is oe saeme of Fermi s Golde Rule he sae-o-sae form which describes relaxaio raes from firs-order perurbaio heory. We will show ha his rae properly describes log ime expoeial relaxaio raes ha you would expec from he soluio o dp d wp.

35 p. -35 Harmoic perurbaio The secod model calculaio is he ieracio of a sysem wih a oscillaig perurbaio ured o a ime. The resuls will be used o describe how a ligh field iduces rasiios i a sysem hrough dipole ieracios. Agai, we are looig o calculae he rasiio probabiliy bewee saes ad : V V cos (.159) V Vcos V i e e i (.16) Seig, firs-order perurbaio heory leads o i i Usig e 1 ie si b i i d V e iv i i d e e i i iv e 1 e 1 as before: i / i / V e si / e si / b (.161) (.16) Noice ha hese erms are oly sigifica whe. The codiio for efficie rasfer is resoace, a machig of he frequecy of he harmoic ieracio wih he eergy spliig bewee quaum saes. Cosider he resoace codiios ha will maximize each of hese: Firs Term max a : E Secod Term E E E E E E E Absorpio (resoa erm) Simulaed Emissio (ai-resoa erm)

36 p. -36 f we cosider oly absorpio, E E, we have: V 1 si P b (.163) We ca compare his wih he exac expressio: V 1 si V P b V. (.164) Agai, we see ha he firs-order expressio is valid for coupligs he deuig. The maximum probabiliy for rasfer is o resoace V ha are small relaive o Similar o our descripio of he cosa perurbaio, he log ime limi for his expressio leads o a dela fucio. his log ime limi, we ca eglec ierfereces bewee he resoa ad airesoa erms. The raes of rasiios bewee ad saes deermied from w P / becomes w V. (.165) We ca examie he limiaios of his formula. Whe we loo for he behavior o resoace, expadig he si(x) shows us ha P rises quadraically for shor imes: V lim P (.166) 4 This clearly will o describe log-ime behavior, bu i will hold for small P, so we require (.167) V A he same ime, we cao observe he sysem o oo shor a ime scale. We eed he field o mae several oscillaios for his o be cosidered a harmoic perurbaio. 1 1 (.168) These relaioships imply ha we require V. (.169)

37 p. -37 Readigs 1. Cohe-Taoudji, C.; Diu, B.; Lalöe, F., Quaum Mechaics. Wiley-ersciece: Paris, 1977; p McHale, J. L., Molecular Specroscopy. 1s ed.; Preice Hall: Upper Saddle River, NJ, 1999; Ch Saurai, J. J., Moder Quaum Mechaics, Revised Ediio. Addiso-Wesley: Readig, MA, 1994.