Pro. O. B. Wrgh, Auum Quaum Mechacs II Lecure Tme-depede perurbao heory Tme-depede perurbao heory (degeerae or o-degeerae sarg sae) Cosder a sgle parcle whch, s uperurbed codo wh Hamloa H, ca exs a superposo o saes. So, H E or he complee orhoormal se o waveucos. Ths mples ha he marx or Hamloa H s dagoal, ad bass. s he eergy Suppose ha H s depede o me, bu ha a = a me-depede perurbao s appled o he sysem, so ha H H W (), where s a parameer we ca vary. Prevously we used H sead o W, bu hs s jus a coveo. Here we use W sead. Sorry abou ha. Ad suppose ha he sysem s ally he sae, a egesae o H wh egevalue E. We choose he leer because sads or al sae. ca be a degeerae or a o-degeerae sae. We are eresed how he waveuco chages me, ha s, wha s ()? We ca also calculae he probably o he sysem beg aoher egesae o H a me. The leer sads or al sae. Ths s a commo problem ha s useul o solve. Ths s a approxmao, because o course he presece o he perurbao he egesaes o H are o he real egesaes o he ew sysem. We could also d he ew eerges o he sysem as a uco o me, bu you ca do ha usg he me-depede perurbao heory o Lecures 9 or or each me. The ew eerges ad ew saoary egesaes o he sysem do o deped o how he perurbao s swched o. I coras, a me he waveuco, ad he probables or beg each egesae o he bass, do deped o how (e.g. how as) he perurbao s swched o.
Pro. O. B. Wrgh, Auum Dervao o he resuls o me-depede perurbao heory Sce () s he ew waveuco, (). Le c () be he compoe o () he bass: ( ) c ( ) () where c ( ) ( ). (Ths s really he same hg as we used beore Lecure 9, N( ) C ( ), bu wh slghly dere oao. We wll roduce a expaso powers o a b laer o. Noce ha our prese problem, we smply he oao by droppg he subscrp o ().) Dee W W() o smply he oao eve more. The Schrödger equao gves d ( ) H W ( ) ( ) d Tae he er produc wh, ad use Eq. (): d c E c W c d ( ) ( ) ( ) ( ). Dee E E / ad le E / c ( ) a ( ) e, () sce we ow ha ampludes vary as E/ e geeral. Ths s jus o smply he oao a b, ad o remove he ow E/ e depedece.
Pro. O. B. Wrgh, Auum da () E / E E / E / E / e a ( ) e Ea ( ) e e W ( ) a ( ) d Thereore, da () EE / e W ( ) a ( ) d or da () e W ( ) a ( ) (3) d Le a a a a (4) () () ( ) ( ) ( ) ( )... We shall loo or a rs-order soluo. Soluo or o perurbao Frs cosder he suao,, whe here s o perurbao. Ad rs cosder a geeral al codo where we do o ow he al sae. I ha case, (jus rom he Schrödger equao ad Eq. ()) E / ( ) c ( ) c () e a e (rom Eqs. () ad (4)) E / E / ( ) a ( ) e Ths s jus he sadard evoluo o a sae me wh o perurbao. Noe ha eve whou a perurbao, he sae () evolves me, bu he probably c () o he sysem beg each sae does o chage. From a comparso o he erms he above equao, or : me, ad a ( ) c () as well. a ( ) c () or all Now cosder our parcular al codo ha he sysem s sae a =. 3
Pro. O. B. Wrgh, Auum Sce c ( ) ( ), so c () (because () or =) Thereore, a ( ) c (). Noe ha a () does o deped o me! We ca hereore wre a () a we wa. I would be more sesble! We shall use he relao a a b laer. Noe ha sce he sysem sars o he sae E / e, sce c () () whe ormalzed. he, or, he soluo s jus Soluo wh perurbao o rs-order For, we use Eqs. (3) ad (4) or level : d a a ( ) e W ( ) a a ( ) d () () or d a ( ) e W ( ) a ( ) d () () Mach coeces o : da () () e W ( ) e W ( ) d Iegrae wh respec o me: () a ( ) e W ( ) d (5) Ths s bascally all we eed o ow! 4
Pro. O. B. Wrgh, Auum The probably P o he sysem ha was ally sae beg oud s a sae s P P ( ) ( ) c ( ) rom Eq. (). We say ha P s he raso probably bewee he saes ad. Noe ha sce E / c ( ) a ( ) e, he a ( ) c ( ). To rs order, a a a. () ( ) ( ) Sce we are cosderg a raso bewee wo dere saoary saes o he uperurbed Hamloa, we ow ha a, so () ( ) ( ) ( ) ( ) where E E P c a e W d /. Ths equao s vald provded a P ( ). We ca chec he us o P () : E o dmesos. OK! New waveuco o rs order To d he ew waveuco (), all we have o do s use Eqs. (), (), (4) ad (5): ( ) c ( ) a ( ) e () E / E / () E / () e a e 5
Pro. O. B. Wrgh, Auum So, E / E / ( ) e e W ( ) d e or E / E / ( ) e e W ( ) d e Sce he secod erm hs equao s very small (he par caused by he perurbao), hs waveuco s ormalzed o rs order. We had a smlar suao or rs-order perurbao heory Lecure 9. The resul rom Lecure 9 loos le hs: H E E Ca you see he smlary? The egral over me he expresso or () gves a acor wh he dmesos o me. Togeher wh he / acor hs plays he role o (eergy) -, jus le he erm / ( E E) he above equao. So you see ha he expressos are que smlar. Example A D smple harmoc oscllaor s s groud sae a me. Is mass s m ad s aural agular requecy s. Cosder he perurbao W () / eexe. Please do o couse E wh eergy. Here s a elecrc eld. e s he charge. 6
Pro. O. B. Wrgh, Auum () Wha s he probably ha he oscllaor s he sae a? For smplcy, less us assume ha ( s also possble o ae ee ec.) We ow rom Eq. (5) ha () ( ) / a ee x e e d, where we have used. Use he ladder operaors: x a a a. where a ad m The oly o-zero value o a () () occurs whe =, sce oly x m m s o-zero (usg a ). To rs-order () ee / a( ) a ( ) e e d m ee / /4 e, m where we have used he resul x / b c x / b c /4 e e dx b e. e E e E P a e e m m / 3 / Thereore ( ) Ths resul s vald provded ha P. We ca see ha he resul depeds o he value o. I example., P, or 7
Pro. O. B. Wrgh, Auum () Fd a expresso or he waveuco a me o rs order. To d he waveuco a me use E / () E / ( ) e a ( ) e. Thereore, / / 3 / ( ) e ee x e e d e / / 3 / e ee e e d e m Sudde or slow perurbaos? The heory we have derved s geeral, ad apples o a boh sudde or slow perurbaos. Bu wha do we mea by ha? Sudde perurbao For a sudde perurbao, how do we dee sudde? d H d I we apply a chage over he me erval, he / / ( / ) ( / ) H( ) ( ) d o order I he. So a saaeous chage H produces o chage (o cludg dela ucos). Ths s rue H chages o a me scale small compared o he aural me scale T o he sysem. See he resul o he prevous example: we d ha P or small, so ha problem s he aural me scale o he sysem. 8
Pro. O. B. Wrgh, Auum Slow perurbao We call a slow perurbao a adabac perurbao. Cosder whe H() chages very slowly rom H() o H() a me. Assume we sar a = sae () o H(). The adabac heorem says: The sysem wll ed he correspodg sae () o H(). (We wll o prove hs heorem here.) Example: cosder a parcle ally a box o legh L(), ha expads slowly o he legh L() a =. How slow s slow eough or he sysem o rema he h sae? Mehod : sem-classcal mehod p, so he aural me scale T ca be assumed o be he me or oe oscllao L L ml ml he box T v p, where p v s he parcle velocy. m L So he chage s slow L dl ml d ml dl Tha s L d per cycle. We ca rewre hs v v walls parcle, sce v walls dl ad vparcle d p v. m Mehod : quaum mehod wh perurbao heory Use aural me scale T ~, where m s he smalles o he raso m requeces bewee he al sae ad ay possble al sae. Ths s because we are eresed he slowes respose me o he sysem o d s aural me scale. So m s he smalles o all he E E. 9
Pro. O. B. Wrgh, Auum I he case o our D box, E rom las wee. ml So E E E ~ m ml ad T ml ~ ~ E /. We oba he same resul as he sem-classcal mehod or he aural me scale T. These resuls suggess ha a degeerae sae (where E ) wll o be adabacally sable because T. Perodc Perurbao Cosder he case whe H( ) H W ( ) H Ve, ad he perurbao s swched o a =. We ow ha he amplude or he raso rom sae o sae a me ca, o rs order, be obaed rom () ( ) ( ) ( ) a e W d e V d V ( ) e ( ) Thereore V e ( ) / ( ) s ( ) / ( ) s 4 P c () V. ( ) / Ths s a covee way o wre or he depedece. Aoher way, we wa o see he depedece s ( ) P c ( ) V s. ( ) 6
Pro. O. B. Wrgh, Auum Depedece o s jus a s depedece. Bu, he rs orm o he equao s more useul o see he depedece ( ha case). s x x Ths graph shows he depedece Noe x= meas. - x The ma o-zero probably les he rego x The edges o hs rego (he pea he plo) le a or E E. ( ) E E I me s small, E, ad here s o preerece or up or dow E rasos. Bu whe (or / ), here s a preerece or E E. So he sysem eeds o go hrough a ew cycles o he perurbao able o selec he al sae. Ve order o be I ac a () () s he orm o a emporal Fourer rasorm o he perurbao over a e erval o me. So whe he me erval ges loger he wdh o he Fourer
Pro. O. B. Wrgh, Auum rasorm ges smaller. (A geeral resul o Fourer rasorms.) Wha happes aer a log me (.e. log compared o he aural me scale)? Cosder he case ha he susodal perurbao lass rom me T/ o T/, ad le T. I ha case T / () a Lm V e d T T / a sce e d ( a) = V 4 P V T / T / ( ) Lm e d Lm d T T Lm. T T T/ T/ Average raso rae R P V T R s he probably o a raso rom o over a u me erval Sce ( ax) ( x) a, he ( ) ( E E ), where E E E. Thereore P R V E E. T
Pro. O. B. Wrgh, Auum Noe ha he us o he dela uco are J -, so he us o R are ~ s -. Noe ha ( E E) de, so he us o ( E E) are J -. Trasos o a couum Ferm s Golde Rule Suppose he al sae or he raso s a colleco o levels ha orm a couum. A couum occurs whe some soluos o he Schrodger equao are plae waves raher ha boud saes. Ay eergy s allowed ad he saes are couous. A example s he case o a e poeal well or or a H aom oo (or E>). Le us derve a equao or hs case. Our equao or raso rae s P R V E E. T Sce we cao ppo a parcular al level, cosder a rage o eerges or he al level E E/: R E E E V E E ( / ) where he sum s over a oal o N saes labelled by he rage cosdered, ad he eergy E rages rom E E/ o E E/. Here E s deed as he ceral eergy. 3
Pro. O. B. Wrgh, Auum A couum meas ha here are ( E) de levels he eergy rage de aroud E. The quay ( E) s called he desy o saes. I ca be a oal desy o saes J -, or per u volume J - m -3. I ( E) de,.e. or a raso o oe sae, meas ha E E de/ E E de/ R de V (because he egral o a dela uco gves ). Noe E s a shorhad or E. Bu ( E) de N,.e. or N saes, meas ha E E de/ E E de/ R de V N V ( E) de R de (he al relao holds because de s very small) Thereore, R V ( E) Ferm s Golde Rule I he desy o saes ( E) s us o per u eergy per u volume, he he raso rae wll be measured us o per u volume per u me (m -3 s - ). Noe ha hs equao E E, he al sae eergy. 4