Time-Dependent Perturbation Theory

Transcription

1 Tme-Depede Perurbao Theory Mchael Fowler 7/6/7 Iroduco: Geeral Formalsm We look a a Hamloa H H + V( ), wh V( ) some me-depede perurbao, so ow he wave fuco wll have perurbao-duced me depedece Our sarg po s he se of egesaes of he uperurbed Hamloa H E, oce we are o labelg wh a zero, o E, because wh a me-depede Hamloa, eergy wll o be coserved, so s poless o look for eergy correcos Wha happes sead, provded he perurbao s o oo large, s ha he sysem makes rasos bewee he egesaes of H Of course, eve for V, he wave fucos have he usual me depedece, ψ ( ) E / ce wh he c s cosa Wha happes o roducg ( ) me depedece, ( ) ( ) ψ c e E / V s ha he c s hemselves acqure ad hs me depedece s deermed by Schrödger s equao wh H H + V : () so E c() e ( H + V () ) c() e / E / E () ( ) ( ) c e V c e / E / Takg he er produc wh he bra me Em / Em E, ad roducg ωm, m m ( ) ω ω cm m V ce Vme c Ths s a marx dffereal equao for he c s :

2 ω c V Ve c ω c Ve V c c 3 V33 c 3 ad solvg hs se of coupled equaos wll gve us he c () s, ad hece he probably of dg he sysem ay parcular sae a ay laer me If he sysem s al sae a, he probably amplude for beg sae f a me s o leadg order he perurbao ω c () δ V ( ) e d f The probably ha he sysem s fac sae f a me s herefore f () ( ) ω c V e d Obvously, hs s oly gog o be a good approxmao f predcs ha he probably of raso s small oherwse we eed o go o hgher order, usg he Ieraco Represeao (or a exac soluo lke ha he ex seco) Example: kckg a oscllaor Suppose a smple harmoc oscllaor s s groud sae a I s perurbed by a / τ small me-depede poeal V () eexe Wha s he probably of dg he rs exced sae a +? Here ( ) / τ, ad x /mω ( a+ a ) V ee x e evaluaed I s ( ) ( ) / ee / /mωπτe ωτ, from whch he probably ca be I s worh hkg hrough he physcal erpreaos for very log ad for very shor mes, ad explag he sgcace of he me for whch he probably s a maxmum

3 3 The Two-Sae Sysem: a Exac Soluo For he parcular case of a wo-sae sysem perurbed by a perodc exeral eld, he marx equao above ca be solved exacly Of course, real physcal sysems have more ha wo saes, bu fac for some mpora cases wo of he saes may be oly weakly coupled o oher degrees of freedom ad he aalyss he becomes releva A famous example, he ammoa maser, s dscussed a he ed of he seco For a wo-sae sysem, he, he mos geeral wave fuco s ad he dffereal equao for he c () s s: E () ( ) + ( ) ψ c e c e / E / ω ω c Ve e c ω ω c Ve e c Wrg ω + ω α for coveece, he coupled equaos are: c Ve c α c Ve c α These wo rs-order equaos ca be rasformed o a sgle secod-order equao by dffereag he secod oe, he subsug c from he rs oe ad c from he secod oe o gve V c αc c Ths s a sadard secod-order dffereal equao, solved by pug a ral soluo c() c( ) e Ω α α V Ths sases he equao f Ω ± +, so, reverg o he 4 orgal ω + ω α, he geeral soluo s: c () e Ae + Be Takg he al sae o be c ( ) c ( ) ( ω ω) ω ω V ω ω V + +, gves A -B To x he overall cosa, oe ha a,

4 4 c V V ( ) c ( ) Therefore V ω ω V c () s + ω ω V + () Noe parcular he resul f ω ω : V c s Assumg E > E, ad he wo-sae sysem o be ally he groud sae, hs meas ha afer a me h/4v he sysem wll ceraly be sae, ad wll oscllae back ad forh bewee he wo saes wh perod h/v Tha s o say, a precsely med perod spe a oscllag eld ca drve a colleco of molecules all he groud sae o be all a exced sae The ammoa maser works by sedg a sream of ammoa molecules, ravelg a kow velocy, dow a ube havg a oscllag eld for a dee legh, so he molecules emergg a he oher ed are all (or almos all, depedg o he precso of gog velocy, ec) he rs exced sae Applcao of a small amou of elecromagec radao of he same frequecy o he ougog molecules wll cause some o decay, geerag ese radao ad herefore a much shorer perod for all o decay, emg cohere radao A Sudde Perurbao A sudde perurbao s deed here as a sudde swch from oe me-depede Hamloa H o aoher oe H, he me of swchg beg much shorer ha ay aural perod of he sysem I hs case, perurbao heory s rreleva: f he sysem s ally a egesae of H, oe smply has o wre as a sum over he egesaes of H, The orval par of he problem s esablshg ha he chage s sudde eough, by esmag he acual me ake for he Hamloa o chage, ad he perods of moo assocaed wh he sae ad wh s rasos o eghborg saes (We dscussed oe example las semeser a elecro he groud sae a oe-dmesoal box ha suddely doubles sze Oher favore examples clude a aom wh sp-orb couplg a magec eld ha suddely reverses (Messah p 743), ad he reaco of orbg elecros o uclear α - or β -decay)

5 5 Harmoc Perurbaos: Ferm s Golde Rule Le us cosder a sysem a al sae perurbed by a perodc poeal ( ) V Ve ω swched o a For example, hs could be a aom perurbed by a exeral oscllag elecrc eld, such as a cde lgh wave Wha s he probably ha a a laer me he sysem be sae f? Recall he marx dffereal equao for he c s : ω c V Ve c ω c Ve V c c 3 V33 c 3 Sce he sysem s deely sae a, he ke vecor o he rgh s ally c, cj The rs-order approxmao o keep he vecor c, o he rgh, ha s, o solve he equaos cj c V e ω f() Iegrag hs equao, he probably amplude for a aom al sae o be sae f afer me s, o rs order: ( ω cf () f V e d f V The probably of raso s herefore P f () cf f V ( ω e ( ω (( ω ) ( ω / s /

6 6 ad we re eresed he large lm s α Wrg α ( ω ω) /, our fuco has he form Ths fuco has a peak a α, α wh maxmum value, ad wdh of order /, so a oal wegh of order The fuco has more peaks a α ( + /) π These are bouded by he deomaor a /α For large her corbuo comes from a rage of order / also, ad as he fuco eds o a δ fuco a he org, bu mulpled by Ths dvergece s ellg us ha here s a e probably rae for he raso, so he lkelhood of raso s proporoal o me elapsed Therefore, we should dvde by o ge he raso rae To ge he quaave resul, we eed o evaluae he wegh of he δ fuco erm We use he sξ sadard resul dξ π ξ o d sα dα π, ad herefore α sα lm πδ ( α) α Now, he raso rae s he probably of raso dvded by he large lm, ha s, ( ) () ( ω lm ( ω P f R f () lm f V ( ( )) f V πδ ω ω π f V δ ( ω s / / Ths las le s Ferm s Golde Rule: we shall be usg a lo You mgh worry ha he log me lm we have ake he probably of raso s fac dvergg, so how ca we use rs order perurbao heory? The po s ha for a raso wh ω ω, log me meas ( ω, hs ca sll be a very shor me compared wh he mea raso me, whch depeds o he marx eleme I fac, Ferm s Rule agrees exremely well wh experme whe appled o aomc sysems Aoher Dervao of he Golde Rule Acually, whe lgh falls o a aom, he full perodc poeal s o suddely swched o, o a aomc me scale, bu bulds up over may cycles (of he aom ad of he lgh) Baym rederves he Golde Rule assumg he lm of a very slow swch o,

7 7 ( ) e Ve V ε ω wh ε very small, so V swched o very gradually he pas, ad we are lookg a mes much smaller ha /ε We ca he ake he al me o be, ha s, ( ω ω ε) ( ω ω ε) e cf () f V e d f V ω ω ε so ad he me rae of chage e cf () f V ω ω + ε ε ( ) d d I he lm ε, he fuco ε e cf () f V ω ω + ε ε ( ) gvg he Golde Rule aga ε ( ) ( ω) πδ ω ω ω + ε Harmoc Perurbaos: Secod-Order Trasos Somemes he rs order marx eleme f V s decally zero (pary, Wger Eckar, ec) bu oher marx elemes are ozero ad he raso ca be accomplshed by a drec roue I he oes o he eraco represeao, we derved he probably amplude for he secod-order process, (wrg E / ( ) ω f ( ) ω( ) ω () < S( ) > < S( ) > c d d e f V e V e ω, ec) Takg he gradually swched-o harmoc perurbao ( ), as above, ε ω V e Ve, ad he al me ( ) ω f ( ωf ω ω ε) ( ω ω ω ε) c () < f V >< V > e d d e e The egrals are sraghforward, ad yeld S

8 8 ε ( ) ( ω ωf ) e < f V >< V > c () e ω ω ω ε ω ω ω ε f Exacly as he seco above o he rs-order Golde Rule, we ca d he raso rae: d π < f V >< V > c d ω ω ω ε ( ) () 4 δ ( ωf ω ω) 4 (The he deomaor goes o o replacg he frequeces ω wh eerges E, boh he deomaor ad he dela fuco, remember ha f E ω, δ ( ω) δ ( E) ) Ths s a raso whch he sysem gas eergy ω from he beam, oher words wo phoos are absorbed, he rs akg he sysem o he ermedae eergy ω, whch s shorlved ad herefore o well deed eergy here s o eergy coservao requreme o hs sae, oly bewee al ad al saes Of course, f a aom a arbrary sae s exposed o moochromac lgh, oher secod order processes whch wo phoos are emed, or oe s absorbed ad oe emed ( eher order) are also possble cc