( ) = c n ( t) n ( ) (2.4) ( t) = k! t

Transcription

1 Adrei Tokmakoff, MIT Deparme of Chemisry, /13/ TIME-DEPENDENT HAMILTONIAN Mixig of eigesaes by a ime-depede poeial For may ime-depede problems, mos oably i specroscopy, we ofe ca pariio he ime-depede Hamiloia io a ime-idepede par ha we ca describe exacly ad a ime-depede par H = H 0 + V.1) Here H 0 is ime-idepede ad V is a ime-depede poeial, ofe a exeral field. Niza, Sec..3., offers a ice explaaio of he circumsaces ha allow us o use his approach. I arises from pariioig he sysem io ieral degrees of freedom i H 0 ad exeral degrees of freedom acig o H 0. If you have reaso o believe ha he exeral Hamiloia ca be reaed classically, he eq..1) follows i a sraighforward maer. The here is a sraighforward approach o describig he ime-evolvig wavefucio for he sysem i erms of he eigesaes ad eergy eigevalues of H 0. We kow H 0 = E..) The sae of he sysem ca be expressed as a superposiio of hese eigesaes: = c.3) The TDSE ca be used o fid a equaio of moio for he expasio coefficies c k = k.4) Sarig wih = i H.5) c k = i k H ).6)

2 - iserig = 1 = i k H c subsiuig eq..1) we have: or, If we make a subsiuio c k ) = i k H 0 + V c = i % E k + V k ' c c k + i E c k k = i.7) c.8) V k..9) c m = e ie m b m,.10) which defies a slighly differe expasio coefficie, we ca simplify cosiderably. Noice ha b k = c k. Also, b k 0) = c k 0). I pracice wha we are doig is pullig ou he rivial par of he ime-evoluio, he ime-evolvig phase facor for sae m. The reasos will become clear laer whe we discuss he ieracio picure. I is easy o calculae b k ad he add i he exra oscillaory erm a he ed. Now eq..9) becomes e ie k b k e ie b = i V k.11) or i b k e i k b = V k.1) This equaio is a exac soluio. I 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 ca be solved aalyically, bu i s ofe sraighforward o iegrae umerically.

3 -3 Resoa Drivig of Two-level Sysem As a example of he use of hese equaios, le s describe wha happes whe you drive a wolevel sysem wih a oscillaig poeial. V = V cos = Vf.13) Noe: This is wha you expec for a elecromageic field ieracig wih charged paricles, i.e. dipole rasiios. I a simple sese, he elecric field is For a paricle wih charge q i a field E, he force o he paricle is which is he gradie of he poeial E = E 0 cos.14) F = q E.15) F x = V x = qe x V = qe x x.16) qx is jus he x compoe of he dipole mome µ. So marix elemes i V look like: More geerally, We ll look a his a bi more carefully laer. k V ) = qe x k x cos.17) V = E µ..18) So, = V cos = E 0 µ cos = V k cos = E 0 µ k cos..19) V V k We will ow couple our wo saes k ad wih he oscillaig field. Le s ask if he sysem sars i wha is he probabiliy of fidig i i k a ime?

4 -4 The sysem of differeial equaios ha describe his siuaio are: i b k = b = b V k e i k V k e i k % 1 ei + e +i.0) Or more explicily i b k = 1 b V e i k ) i k + e k + ) % ' + 1 b V k kk i b = 1 b V e i + e i ' + 1 b V e i k ) i k k + e k + % or % e i k + e i + e i ' ' + e i k '.1) Two of hese erms ca be dropped sice for our case) he diagoal marix elemesv ii = 0. We also make he secular approximaio roaig wave approximaio) i which he oresoa erms are dropped. Whe k, erms like e ±i or e i k + ) oscillae very rapidly relaive o V k 1 ) ad so do coribue much o chage of c. Remember ha k is posiive). So we have: b k = i b V k ei k ) b = i b V k k ei k ).).3) Noe ha he coefficies are oscillaig ou of phase wih oe aoher. Now if we differeiae eq..): b k = i % b V k e i k ) + i k )b V k e i k ) '.4) Rewrie eq..):

5 -5 b = i bk e i k ) V k.5) ad subsiue.5) ad.3) io.4), we ge liear secod order equaio for b k. b k + V k b k i k 4 b k = 0.6) This is jus he secod order differeial equaio for a damped harmoic oscillaor: x = e b a ax + bx + cx = 0.7) Acos µ + Bsi µ) µ = 1 1.8) 4ac b a % Wih a lile more work, ad rememberig he iiial codiios b k 0) = 0 ad b 0) = 1, we fid P k = b k = V k V k + k si r.9) Where he Rabi Frequecy R = 1 V k + k % 1 '.30) ) Also, P = 1 b k.31) The ampliude oscillaes back 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 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.

6 -6 The efficiecy of drivig bewee ad k saes drops off wih deuig. Here ploig he maximum value of P k as a fucio of frequecy: Readigs This lecure draws from 1. C. Cohe-Taoudji, B. Diu, ad F. Lalöe, Quaum Mechaics, Vol.. Wiley- Iersciece, Paris, 1977). J. J. Sakurai, Moder Quaum Mechaics. Addiso-Wesley, Readig, MA, 1994).

7 Adrei Tokmakoff, MIT Deparme of Chemisry, /13/ QUANTUM DYNAMICS The moio of a paricle is described by a complex wavefucio r, probabiliy ampliude of fidig a paricle a poi r a ime. If we kow r, 0 i chage wih ime? ha gives he, how does r, 0 )? r,) > 0.3) We will use our iuiio here largely based o correspodece o classical mechaics). We are seekig a equaio of moio for quaum sysems ha is equivale o Newo s or more accuraely Hamilo s) equaios for classical sysems. We sar by assumig causaliy: 0 derivig a deermiisic equaio of moio for r, parameer: precedes ad deermies. So will be. Also, we assume ime is a coiuous lim 0 = 0 ).33) Defie a operaor ha gives ime-evoluio of sysem. = U, 0 ) 0 ).34) This ime-displaceme operaor or propagaor is similar o he space-displaceme operaor which moves a wavefucio i space. r) = e ik r r 0 ) r 0 ).35)

8 -8 We also say ha U does o deped o he paricular sae of he sysem. This is ecessary for coservaio of probabiliy, i.e. o reai ormalizaio for he sysem. If he 0 ) = a ) + a 0 ).36) = U, 0 ) 0 ) = U, 0 )a ) + U, 0 = a a a 0.37) This is a reflecio of he imporace of lieariy i quaum sysems. While a i ypically o equal o a i 0), a = a 0.38) Properies of U, 0 ) 1) Uiary. Noe ha for eq..38) o hold ad for probabiliy desiy o be coserved, U mus be uiary P = = 0 ) U U 0 ).39) which holds oly if U = U 1. I fac, his is he reaso ha equaes uiary operaors wih probabiliy coservaio. ) Time coiuiy: U,) = 1..40) 3) Composiio propery. If we ake 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 0 ) or muliple seps 0 1 ):

9 -9 Noe, sice U acs o he righ, order maers: U, 0 ) = U, 1 )U 1, 0 ).41) = U, 1 )U 1, 0 ) 0 ) = U, 1 ) 1 ).4) Equaio.41) is already very suggesive of a expoeial form. Furhermore, sice ime is coiuous ad he operaor is liear i also suggess wha we will see ha he ime propagaor is oly depede o a ime ierval ad U 1, 0 ) = U 1 0 ).43) U 0 ) = U 1 )U 1 0 ).43 4) Time-reversal. The iverse of he ime-propagaor is he ime reversal operaor. From eq..41): U, 0 )U 0,) = 1.3) U 1, 0 ) = U 0,)..33) Fidig a equaio of moio for 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 0 +, 0 lim U 0 0 +, 0 )= 1.34) We expec ha for small, he differece bewee U 0, 0 ) ad U 0 +, 0 ) will be liear i This is based o aalogy o how we hik of deermiisic moio i classical sysems)

10 -10 U 0 +, 0 ) = U 0, 0 ) i ˆ 0 ).35) We ake ˆ o be a ime-depede Hermeia operaor. We ll see laer why he secod erm mus be imagiary. So, ow we ca wrie a differeial equaio for U. We kow ha U +, 0 ) = U +,)U, 0 )..36) Kowig he chage of U durig he period allows us o wrie a differeial equaio for he ime-developme ofu, 0. The equaio of moio for U is d U, 0 ) = lim d U, 0 ) U +, 0 0 U +, = lim % 0 1 ' U, 0.37) Where I have subsiued eq..35) i he secod sep. So we have: U, 0 ) = i ˆU, 0 ).38) 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 1) quaum mechaics says E = ad ) i classical mechaics he Hamiloia geeraes ime-evoluio, we wrie ˆ = Ĥ.39) Where ˆ ca be a fucio of ime. The Muliplyig from righ by 0 ) gives he TDSE i U, 0 ) = ĤU, 0 ).40)

11 -11 i = Ĥ.41) 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 recogizig ha U, 0 he lef: we ge acs o = 0 ) U, 0 ),.4) i U, 0 ) = U, 0 ) Ĥ.43) Evaluaig U, 0 ): Time-idepede Hamiloia Direc iegraio of eq..40) suggess ha U ca be expressed as: = exp i H 0 ) U, 0 Sice H is a operaor, we will defie his operaor hrough he expasio: % '.44) exp ih % 0 )' = 1+ ih 0 ) + i + ) *, - H 0 % +.45) Noe H commues a all. You ca cofirm he expasio saisfies he equaio of moio foru. To evaluae U for he ime-idepede Hamiloia, we expad i a se of eigekes: So we have H = E = 1.46)

12 -1 ad = exp ih 0 ) / U, 0 = exp ie 0 = U, 0 ) 0 ) ) = 0 ) = c c 0 % / % i exp % E % 0 ) '.47).48) Expecaio values of operaors are give by = A = 0 ) U, 0 ) AU, 0 ) 0 ) A.49) For a iiial sae 0 = c, we showed i eq ha * A = c m m m e +i m 0 ) m A e i 0 ) c,m = c * m c A m e i m 0 ),m = c m *,m c A m.50) which is Trρ)A). The correlaio ampliude was give i eq

13 -13 Evaluaig he ime-evoluio operaor: Time-depede Hamiloia A firs glace i may seem sraighforward o deal wih. If H is a fucio of ime, he he formal iegraio of i U = HU gives = exp i U, 0 % 0 ' H )d ).51) We would defie his expoeial as a expasio i a series, ad subsiue io he equaio of moio o cofirm i: U, 0 ) = 1 i H )d % i ' ) d d H The if we kow he eigesaes of H, we could use eq..46) o express U as = exp i U, H ) +.5) E ' * j )d % 0 ).53) However, his is dagerous; we are o reaig H as a operaor. We are assumig ha he Hamiloias a differe imes commue H ), H ) % = 0. I is oly he case for special Hamiloias wih a high degree of symmery, i which he eigesaes have he same symmery a all imes. This holds for isace i he case of a degeerae sysem, i.e., spi ½ sysem, wih a ime-depede couplig. Geerally speakig his is o he case. Now, le s proceed a bi more carefully assumig ha he Hamiloia a differe imes does o commue. Iegrae To give: U, 0 ) = 1 i U, 0 ) = i H )U, 0 ).54) d H )U, 0 ).55) 0 This is he soluio; however, i s o very pracical sice U, 0 ) is a fucio of iself. Bu we ca solve by ieraively subsiuig U io iself. Firs Sep:

14 -14 U, 0 ) = 1 i % d H 0 ' 1 i + = 1+ i., - / 0 d H 0 + i d H 0 U, 0 )* ) +., - / 0 0 d d H 0 H U, 0.56) Noe i he las erm of his equaio, he iegraio variable preceeds. Picorally, he area of iegraio is Nex Sep: = 1+ i U, 0 + i % ' + i % ' % ' ) d H 0 3 ) d d * H 0 0 H *) ) d d *) d ** H 0 0 * 0 H *) H ** U **, 0.57) 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 : 0. Imagie you are sarig i sae 0 = ad you are workig oward a arge sae = k. The possible pahs ad associaed ime variables are:

15 -15 The expressio for U describes all possible pahs bewee iiial ad fial sae. Each of hese pahs ierfere i ways dicaed by he acquired phase of our eigesaes uder he imedepede Hamiloia. The soluio for U obaied from his ieraive subsiuio is kow as he posiive) ime-ordered expoeial exp + i U, 0 % ' d H 0 * ) T ˆ % exp i ' d H 0 * ) 1 + i. = 1+, - / 0 d d d H 0 =1 H 1 H 1.58) ˆ T is kow as he Tyso ime-orderig operaor.) I his expressio he ime-orderig is: ) 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+ + ' ) d ) d 1 H ) H 1 ) H 1 ).60) 0 0 =1

16 -16 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.) We are also ieresed i he Hermeia cojugae ofu, 0 ), which has he equaio of moio i eq..43) U, 0 ) = +i U, 0 ) H.61) If we repea he mehod above, rememberig ha U, 0 ) acs o he lef: he from = 0 ) U, 0 ).6) U, 0 ) = U 0, 0 ) + i we obai a egaive-ime-ordered expoeial: d U, 0 H ).63) = exp i U, 0 = ' d H 0 ) % =1 * i - +,. / 0 d d 1 d 1 H H H.64) Here he H i ) ac o he lef. Readigs This lecure draws from he followig: 1. Merzbacher, E. Quaum Mechaics, 3rd ed. Wiley, New York, 1998), Ch Mukamel, S. Priciples of Noliear Opical Specroscopy Oxford Uiversiy Press, New York, 1995), Ch.. 3. Sakurai, J. J. Moder Quaum Mechaics, Revised Ediio Addiso-Wesley, Readig, MA, 1994), Ch..

17 Adrei Tokmakoff, MIT Deparme of Chemisry, // SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mahemaical formulaio of he dyamics of a quaum sysem is o uique. So far we have described he dyamics by propagaig he wavefucio, which ecodes probabiliy desiies. This is kow as he Schrödiger represeaio of quaum mechaics. Ulimaely, sice we ca measure a wavefucio, we are ieresed i observables probabiliy ampliudes associaed wih Hermeia operaors). Lookig a a ime-evolvig expecaio value suggess a alerae ierpreaio of he quaum observable: = Â ) = 0) U ÂU 0) = 0) U ) Â U 0 ) ) = 0) U ÂU Â 0.65) The las wo expressios here sugges alerae rasformaio ha ca describe he dyamics. These have differe physical ierpreaios: 1) Trasform he eigevecors: U. Leave operaors uchaged. ) Trasform he operaors: Â ) U ÂU. Leave eigevecors uchaged. 1) Schrödiger Picure: Everyhig we have doe so far. Operaors are saioary.. Eigevecors evolve uderu, 0 ) Heiseberg Picure: Use uiary propery of U o rasform operaors so hey evolve i ime. The wavefucio is saioary. This is a physically appealig picure, because paricles move here is a ime-depedece o posiio ad momeum. Le s look a ime-evoluio i hese wo picures: Schrödiger Picure We have alked abou he ime-developme of, which is govered by i = H.66)

18 -18 i differeial form, or aleraively = U, 0 ) 0 ) i a iegral form. I he Schrödiger picure, for operaors ypically A values of operaors Â) =  ) : = 0. Wha abou observables? For expecaio i Â) = i %  % +  +  ' =  H ) H Â.67) = Â, H ' = Â, H ' Aleraively, wrie for he desiy marix: i Â) = i Tr  = itr  % ' = Tr Â)* H, +, = Tr ) * Â, H +,.68) If  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. Heiseberg Picure From eq..65) we ca disiguish he Schrödiger picure from Heiseberg operaors: Â) =  ) = 0 S U ÂU 0 ) =  S H.69) where he operaor is defied as  H  H 0 = U, 0 = ÂS ÂS U, 0 ).70)

19 -19 Also, sice he wavefucio should be ime-idepede H = 0, we ca relae he Schrödiger ad Heiseberg wavefucios as S = U, 0 ) H.71) So, H = U, 0 ) S = S 0 ).7) I eiher picure he eigevalues are preserved:  i S = a i i S U ÂUU i S = a i U i S.73)  H i H = a i i H The ime-evoluio of he operaors i he Heiseberg picure is: ÂH = U  S U ) = U  S U + U  S U = i U H ÂS U i U  S H U +  % ' H + U ÂS U.74) = i H H  H i  H H H = i ) * Â, H +, H The resul: i  H = Â, H %.75) H is kow as he Heiseberg equaio of moio. Here I have wrie H H = U H U. Geerally speakig, for a ime-idepede Hamiloia U = e ih /, U ad H commue, ad H H = H. For a ime-depede Hamiloia, U ad H eed o commue. Paricle i a poeial

20 -0 Ofe we wa o describe he equaios of moio for paricles wih a arbirary poeial: For which he Heiseberg equaio gives: H = p + V x).76) m p = V x..77) x = p m.78) Here, I ve made use of ˆx, ˆp = iˆx%1.79) ˆx, ˆp = iˆp%1.80) These equaios idicae ha he posiio ad momeum operaors follow equaios of moio ideical o he classical variables i Hamilo s equaios. These do o ivolve facors of. Noe here ha if we iegrae eq..78) over a ime period we fid: x = p m + x 0 ).81) implyig ha he expecaio value for he posiio of he paricle follows he classical moio. These equaios also hold for he expecaio values for he posiio ad momeum operaors Ehrefes s Theorem) ad idicae he aure of he classical correspodece. I correspodece o Newo s equaio, we see m x = V.8)

21 -1 THE INTERACTION PICTURE 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 0 + V.83) H 0 is a Hamiloia for he degrees of freedom we are ieresed i, which we rea exacly, ad ca be alhough for us geerally wo be) a fucio of ime. V is a ime-depede poeial which ca be complicaed. I he ieracio picure we will rea each par of he Hamiloia i a differe represeaio. We will use he eigesaes of H 0 as a basis se o describe he dyamics iduced by V, assumig ha V is small eough ha eigesaes of H 0 are a useful basis o describe H. If H 0 is o a fucio of ime, he here is 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 0 is described by where, mos geerally, U 0, 0 U, 0 0 ) = i H 0 U 0, 0 ).84) i = exp + % 0 d H 0.85) ' bu for a ime-idepede H 0 U 0, 0 ) = e ih 0 0 ).86) We defie a wavefucio i he ieracio picure I hrough: S U 0, 0 ) I.87) or I = U 0 S.88) Effecively he ieracio represeaio defies wavefucios i such a way ha he phase accumulaed uder e ih 0 is removed. For small V, hese are ypically high frequecy oscillaios relaive o he slower ampliude chages i cohereces iduced by V.

22 - We are afer a equaio of moio ha describes he ime-evoluio of he ieracio picure wave-fucios. We begi by subsiuig eq..87) io he TDSE: i S = H S.89) U, 0 0 ) I = i H ) U 0, 0 ) I U 0 I + U 0 I ) U 0, 0` ) I = i H + V 0.90) i H U + U I 0 0 I 0 ) U 0 I = i H + V 0 i I = V I I.91) where V I = U 0, 0 ) V U 0, 0 ).9) I saisfies he Schrödiger equaio wih a ew Hamiloia: he ieracio picure Hamiloia, V I, which is he U 0 uiary rasformaio ofv. Noe: Marix elemes i V I = k V I l = e i lk V kl where k ad l are eigesaes of H 0. We ca ow defie a ime-evoluio operaor i he ieracio picure: where U I, 0 I = U I, 0 ) I 0 ).93) % = exp + i ' d V I 0 )..94) Now we see ha S = U 0, 0 ) I = U 0, 0 )U I, 0 ) I 0 ) = U 0, 0 )U I, 0 ) S 0 ).95)

23 -3 U, 0 ) = U 0, 0 )U I, 0 ).96) Usig he ime ordered expoeial i eq..94), U ca be wrie as U, 0 ) = U 0, 0 ) + ) =1 i % ' U 0 *,* V * 1 * * d* d* 1 d* 1 U 0,* )V * )U 0 *,* 1 ) + 0 U 0 * 1, ) where we have used he composiio propery ofu, 0 ). The same posiive ime-orderig applies. Noe ha he ieracios Vτ i ) are o i he ieracio represeaio here. Raher we used he defiiio i eq..9) ad colleced erms. For rasiios bewee wo eigesaes of H 0 l ad k): The sysem evolves i eigesaes of H 0 durig he differe ime periods, wih he ime-depede ieracios V drivig he rasiios bewee hese saes. The ime-ordered expoeial accous for all possible iermediae pahways. Also, he ime evoluio of cojugae wavefucio i he ieracio picure is expressed as % +i U, 0 ) = U I, 0 ) U 0, 0 ) = exp ' d V I 0 ) exp % or U 0 = e ih 0 ) whe H 0 is idepede of ime. The expecaio value of a operaor is: +i ' d H 0 0 ).98) = Â ) = 0 ) U, 0 ) ÂU, 0 ) 0 ) = 0 ) U I U 0 ÂU 0 U I 0 ) = I ÂI I Â.99)

24 -4 where A I U 0 A S U 0.100) Differeiaig A I gives: Â = i I H, 0 ÂI %.101) also, = i I V I I.10) Noice ha he ieracio represeaio is a pariio bewee he Schrödiger ad Heiseberg represeaios. Wavefucios evolve uderv I, while operaors evolve uder H 0. For H 0 = 0,V = H Â = 0; = i S H Schrödiger S For H 0 = H,V = 0 Â % ' = 0 Heiseberg = i H, Â ;.103)

25 -5 The relaioship bewee U I, 0 ) ad b ) For problems i which we pariio a ime-depede Hamiloia, H = H 0 + V.104) H 0 is he ime-idepede exac zero-order Hamiloia ad V is a ime-depede poeial. We kow he eigekes ad eigevalues of H 0 : H 0 = E.105) ad we ca describe he sae of he sysem as a superposiio of hese eigesaes: = c The expasio coefficies c k are give by c k = k.106) = k U, 0 ) 0 ).107) Aleraively we ca express he expasio coefficies i erms of he ieracio picure wavefucios b k = k I = k U I 0 ).108) This oaio follows Cohe-Taoudji.) Noice c k = k U 0 U I 0 = e i k k U I 0 = e i k b k.109) This is he same ideiy we used earlier o derive he coupled differeial equaios ha describe he chage i he ime-evolvig ampliude of he eigesaes: i b k e i k b = V k.110) So, b k is he expasio coefficie of he ieracio picure wavefucios. Remember b k = c k ad b k 0) = c k 0). If ecessary we ca calculae b k ad he add i he exra oscillaory erm a he ed.

26 Adrei Tokmakoff, MIT Deparme of Chemisry, 3/14/ PERTURBATION THEORY Give a Hamiloia H = H 0 + V where we kow he eigekes for H 0 : H 0 = E, we ca calculae he evoluio of he wavefucio ha resuls fromv : I = b.111) usig he coupled differeial equaios for he ampliudes of. For a complex imedepedece or a sysem wih may saes o be cosidered, solvig hese equaios is pracical. Aleraively, we ca choose o work direcly wihu I, 0, calculae b k as: b k = k U I, 0 ) 0 ).11) where U I, 0 % = exp + i 0 ' V I )d ).113) Now we ca rucae he expasio afer a few erms. This is perurbaio heory, where he dyamics uder H 0 are reaed exacly, bu he ifluece of V o b is rucaed. This works well for small chages i ampliude of he quaum saes wih small couplig marix elemes relaive o he eergy spliigs ivolved b k b k 0) ; V E k E ) As we ll see, he resuls we obai from perurbaio heory are widely used for specroscopy, codesed phase dyamics, ad relaxaio. Trasiio Probabiliy Le s ake he specific case where we have a sysem prepared i, ad we wa o kow he probabiliy of observig he sysem i k a ime, due ov. P k = b k b k = k U I, 0 ).114)

27 -7 b k = k exp + i ' d V I 0 ).115) % b k = k i d k V I 0 + i ' % ) d d 1 k V I )V I ) usig k V I = k U 0 V U 0 = e i k V k.117) So, b k = k i d 1 e i k % 1 V k 1 firs order.118) 0 + m i% ' e i* m ) V m ) 1 ) +.119) ) d) d) 1 e i* mk ) V km ) 0 0 secod order The firs-order erm allows oly direc rasiios bewee ad k, as allowed by he marix eleme i V, whereas he secod-order erm accous for rasiios occurig hrough all possible iermediae saes m. For perurbaio heory, he ime ordered iegral is rucaed a he appropriae order. Icludig 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 k you eed o accou for ad which oes are allowed by he marix elemes. For firs order perurbaio heory, he expressio i eq..118) is he soluio o he differeial equaio ha you ge for direc couplig bewee ad k : b = i k ei k V k b 0).10)

28 -8 This idicaes ha he soluio does allow for he feedback bewee ad k ha accous for chagig populaios. This is he reaso we say ha validiy dicaes b k b k 0. If 0 is o a eigesae, we oly eed o express i as a superposiio of eigesaes, 0 = b 0 ad b k = b 0) k U I..11) Now here may be ierferece effecs bewee he pahways iiiaig from differe saes: P k = c k = b k = k b.1) Also oe ha if 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 k is jus he Fourier rasform of V) evaluaed a he eergy gap k. b k = i + d e i k V % k ).13) If he Fourier rasform is defied as F V V % = 1 +) * d V exp i ) ',.14) he P k = V k )..15)

29 -9 Example: Firs-order Perurbaio Theory Vibraioal exciaio o compressio of harmoic oscillaor. Le s subjec a harmoic oscillaor o a Gaussia compressio pulse, which icreases is force cosa. Firs wrie he Hamiloia: Now pariio i accordig o H = H 0 + V : H = T + V = p m + 1 k ) x.16) k = k 0 + k k 0 = m k = k 0 exp 0 ) p H = m + 1 k 0 x + 1 k 0 x exp ' 0 ) % ) H 0 V ) % ' ) ).17).18) H 0 = E H 0 = a a + 1 % ' E = + 1 % '.19)

30 -30 If he sysem is i 0 a 0 =, wha is he probabiliy of fidig i i a =? For 0 : b = i Usig 0 = E E 0 ) = : b = i k x + 0 d 0 d V 0 e i 0.130) 0 % e i e '.131) So, b = i k x 0 e % / 0.13) Here we used: Wha abou he marix eleme? x = m a + ) a = m aa + a a + aa + a a ).133) Firs-order perurbaio heory wo allow rasiios o = 1, oly = 0 ad =. Geerally his would be realisic, because you would ceraily expec exciaio o v=1 would domiae over exciaio o v=. A real sysem would also be aharmoic, i which case, he leadig erm i he expasio of he poeial Vx), 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 0 = m.134) So, b = i k 0 m% e %.135)

31 -31 k ad P = b = 0 e %4 = m ' k 0 k 0 ) + e %4.136) * From he expoeial argume, sigifica rasfer of ampliude occurs whe he compressio pulse is shor compared o he vibraioal period. << 1.137) Validiy: Firs order perurbaio heory does allow for b o chage much from is iiial value. For P << 1 % ' k 0 k 0 * ) << 1.138) Geerally, he perurbaio δk) mus be small compared o k 0, i.e. H 0 >> V, bu i should also work well for he impulsive shock limi σω<<1).

32 -3 FIRST-ORDER PERTURBATION THEORY A umber of impora relaioships i quaum mechaics ha describe rae processes come from 1 s order perurbaio heory. For ha, here are a couple of model problems ha we wa o work hrough: Cosa Perurbaio Sep-Fucio Perurbaio) ) =. A cosa perurbaio of ampliude V is applied o 0. Wha is P k? V = 0 ) V = % 0 < 0 V 0 To firs order, we have: k U 0 V U 0 = V e i k 0 ) b k = k i d e i% k 0 ) Vk 0.13 Here V k is idepede of ime. Now, assumig k ad seig 0 = 0 we have b k = i V d k e i k.140) 0 = V k E k E 1 exp i k %.141) = iv k ei k / si E k E k / ).14) Where I usede i 1 = ie i si ). Now

33 -33 P k = b k = 4 V k Wriig his as we did i Lecure 1: E k E si 1 k.143) P k = V si /.144) where = E k E ). Compare his wih he exac resul we have for he wo-level problem: P k = Clearly he perurbaio heory resul works for V << Δ. We ca also wrie he firs-order resul as V V + si + V / ).145) P k = V where sic x) = si x) x. Sice lim sic x) = 1, x0 sic /.146) lim P 0 k = V.147) The probabiliy of rasfer from o k as a fucio of he eergy level spliig E k E ) : Area scales liearly wih ime. Sice he eergy spread of saes o which rasfer is efficie scales approximaely as E k E <, his observaio is someimes referred o as a uceraiy relaio

34 -34 wih E. However, remember ha his is really jus a observaio of he priciples of Fourier rasforms, ha frequecy ca oly be deermied by he legh of he ime period over which you observe oscillaios. Sice ime is o a operaor, i is o a rue uceraily relaio like p x. Now urig o he ime-depedece: The quadraic growh for Δ=0 is ceraily urealisic a leas for log imes), bu he expressio should hold for wha is a srog couplig case Δ=0. However, le s coiue lookig a his behavior. I he log ime limi, he sic x) fucio arrows rapidly wih ime givig a dela fucio: lim si ax ) lim = ax x ).148) P k = V k E k % E ).149) 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..149) 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 kieics:

35 -35 w k = P k = V k E k E ).150) 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.

36 -36 Slowly Applied Adiabaic) Perurbaio Our perurbaio was applied suddely a > 0 sep fucio) V = 0 )V This leads o uphysical cosequeces you geerally ca ur o a perurbaio fas eough o appear isaaeous. Sice firs-order P.T. says ha he rasiio ampliude is relaed o he Fourier Trasform of he perurbaio, his leads o addiioal Fourier compoes i he specral depedece of he perurbaio eve for a moochromaic perurbaio So, le s apply a perurbaio slowly... V = V e here η is a small posiive umber. 1 is he effecive ur-o ime of he perurbaio. The sysem is prepared i sae a =. Fid P k. b k = k U I = i d e i k % k V e b k = iv k exp ' + i k ) * + i k exp ' + i E k E ) / ) = V * k E k E + i V P k = b k = k exp ' ) * + k = V k exp ' ) * E k E ) + This is a Lorezia lieshape i k wih widh.

37 -37 Gradually Applied Perurbaio Sep Respose Perurbaio The gradually ured o perurbaio has a widh depede o he ur-o rae, ad is idepede of ime. The ampliude grows expoeially i ime.) Noice, here are o odes i P k. Now, le s calculae he rasiio rae: w kl = P k = V k e + k Look a he adiabaic limi;0. Seig e 1 ad usig lim 0 = % + k k w k = V k k ) = V k Ek E ) We ge Fermi s Golde Rule idepede of how perurbaio is iroduced

38 -38 Harmoic Perurbaio Ieracio of a sysem wih a oscillaig perurbaio ured o a ime 0 = 0. This describes how a ligh field moochromaic) iduces rasiios i a sysem hrough dipole ieracios. Agai, we are lookig o calculae he rasiio probabiliy bewee saes ad k: V = V cos = µe 0 cos.151) To firs order, we have: V k b k = k I = V k cos = V k ei + e i % = i d V k 0 e i% k.15) = iv k d e i % k +% ) e i % k % 0 ' ) seig ) = V k e i % k +% ) 1 % k + % + ei % k % ) 1 * + '* % k % ) + Now, usig e i 1 = ie i si ) as before: b k = iv k ' ' / si k ) / % k e i k / si k + ) / % % k + + ei k +.154) Noice ha hese erms are oly sigifica whe k. As we leared before, resoace is required o gai sigifica rasfer of ampliude.

39 -39 Firs Term Secod Term max a : = + k = k E k > E E k < E E k = E + E k = E Absorpio resoa erm) Simulaed Emissio ai-resoa erm) For he case where oly absorpio coribues, E k > E, we have: P k = b k = V k or si 1 k E 0 µ k si 1 k k )% k )%.155) We ca compare his wih he exac expressio: V P k = b k = k k 1 si % + V k V k + k.156) ' which pois ou ha his is valid for coupligs V k ha are small relaive o he deuig = k ). The maximum probabiliy for rasfer is o resoace k =

40 -40 Limiaios of his formula: By expadig si x = x x3 3 +, we see ha o resoace = k 0 lim 0 P k = V k 4.157) This clearly will o describe log-ime behavior. This is a resul of 1 s order perurbaio heory o reaig he depleio of. However, i will hold for small P k, so we require << V k.158) A he same ime, we ca observe he sysem o oo shor a ime scale. We eed he field o make several oscillaios for i o be a harmoic perurbaio. These relaioships imply ha > 1 1 k.159) V k << k.160)

41 -41 Adiabaic Harmoic Perurbaio Wha happes if we slowly ur o he harmoic ieracio? V = V e cos b k = i ' e d V % k e i + e i * i k + ), + = V ' k e i k + ) e k + )+ i + e i k ) ) k + i Agai, we have a resoa ad ai-resoa erm, which are ow broadeed by. If we oly cosider absorpio: V P k = b k = k 1 e 4 k + which is he Lorezia lieshape ceered a k = wih widh =. Agai, we ca calculae he adiabaic limi, seig 0. We will calculae he rae of rasiios k = P k /. Bu le s resric ourselves o log eough imes ha he harmoic perurbaio has cycled a few imes his allows us o eglec cross erms) resoaces sharpe. w k = V k + k + ) % k ' *, +,

42 Adrei Tokmakoff, MIT Deparme of Chemisry, 3/1/ FERMI S GOLDEN RULE The rasiio rae ad probabiliy of observig he sysem i a sae k afer applyig a perurbaio o from he cosa firs-order perurbaio does allow for he feedback bewee quaum saes, so i urs ou o be mos useful i cases where we are ieresed jus he rae of leavig a sae. This quesio shows up commoly whe we calculae he rasiio probabiliy o o a idividual eigesae, bu a disribuio of eigesaes. Ofe he se of eigesaes form a coiuum of accepig saes, for isace, vibraioal relaxaio or ioizaio. Trasfer o a se of coiuum or bah) saes forms he basis for a describig irreversible relaxaio. You ca hik of he maerial Hamiloia for our problem beig pariioed io wo porios, H = H S + H B + V SB, where you are ieresed i he loss of ampliude i he H S saes as i leaks io H B. Qualiaively, you expec deermiisic, oscillaory feedback bewee discree quaum saes. However, he ampliude of oe discree sae coupled o a coiuum will decay due o desrucive ierfereces bewee he oscillaig frequecies for each member of he coiuum. So, usig he same ideas as before, le s calculae he rasiio probabiliy from o a disribuio of fial saes: P k. P k = b k Probabiliy of observig ampliude i discree eigesae of H 0 E k ) : Desiy of saes uis i1 E k, describes disribuio of fial saes all eigesaes of H 0 If we sar i a sae, he oal rasiio probabiliy is a sum of probabiliies P k = P k..161) We are jus ieresed i he rae of leavig ad occupyig ay sae k or for a coiuous disribuio: k

43 -43 For a cosa perurbaio: Pk = Pk = de k E k ) P k.16) si de k E k ) E k E ) / 4 V k.163) E k E Now, le s make wo assumpios o evaluae his expressio: 1) E k ) varies slowly wih frequecy ad here is a coiuum of fial saes. By slow wha we are sayig is ha he observaio poi is relaively log). ) The marix eleme V k is ivaria across he fial saes. These assumpios allow hose variables o be facored ou of iegral P k = V k + de k 4 si E k E ) /.164) E k E Here, we have chose he limis + sice E k ) is broad relaive o P k. Usig he ideiy wih a = / we have + d si a = a%.165) P k = V k.166) The oal rasiio probabiliy is liearly proporioal o ime. For relaxaio processes, we will be cocered wih he rasiio rae, w k :

44 -44 w k = Pk w k = V k.167) Remember ha P k is ceered sharply a E k = E. So alhough ρ is a cosa, we usually wrie eq..167) i erms of E k = E ) or more commoly i erms of E k E ) : w k = E = E k ) V k.168) w k = V k w k = de k E k ) w k Ek E.169) This expressio is kow as Fermi s Golde Rule. Noe he raes are idepede of ime. As we will see goig forward, his firs-order perurbaio heory expressio ivolvig he marix eleme squared ad he desiy of saes is very commo i he calculaio of chemical rae processes. Rage of validiy For discree saes we saw ha he firs order expressio held for V k << k, ad for imes such ha P k ever varies from iiial values. P k = w k 0 ) << 1.170) w k However, rasiio probabiliy mus also be sharp compared o E k ), which implies >> / E k.171) So, his expressio is useful where

45 -45 E >> w k k >> w k..17)