Intro Microelec Theory
Intro Microelec Theory ECE 6451
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Lecture 9 Nondegenerate amp Degenerate Time Independent and Time Dependent Perturbation Theory Reading Notes and Brennan Chapter 41 amp 42 Georgia Tech ECE 6451 Dr Alan Doolittle Nondegenerate Time Independent Perturbation Theory If the solution to an unperturbed system is known including Eigenstates Pn0 and Eigen energies Bum 0 0 0 HOLPn En LIIn then we seek to nd the approximate solution for the same system under a slight perturbation most commonly manifest as a change in the potential of the system H Pn 2 En LP To do this we expand the Hamiltonian into modi ed form H 2 H0 ng Where g is a dimensionless parameter meant to keep track of the degree of smallness we will eventually set gl but for now we keep it H0 gHP LPn where H is the perturbation term in the Hamiltonian As g gtO then H gtH0 Pn gt Pn0 and En gtEn0 Georgia Tech ECE 6451 Dr Alan Doolittle Nondegenerate Time Independent Perturbation Theory If the perturbation is small enough it is reasonable to write the wavefunction as LP kiln gulf gZLPnQ and the energy as En 2 E50 gag gZEg Thus we can write the Schrodinger equation as HT 2 ET H0 ngX l jo g l jl gz l f E20 gEil ngf W50 g l jl gz l f or HP E Pn 0 g HOW E0gtw0gt 1 lt1 0 0 lt1 1 0 g HOLIII I HpLIII l En LIIn En LPquot For any Choice of g2 HOW H t 35 Ef gy thpgm g each term in S O 3 parenthesis must g m be equal to 0 g4 gt 0 Georgia Tech ECE 6451 Dr Alan Doolittle Nondegenerate Time Independent Perturbation Theory Given H l E P 0 n O 3 g How E0gtw0gt g1H HPL1150gt E0 le HEio i g3 g4 For any choice of g each term in parenthesis must be equal to 0 1st order perturbation theory seeks values 0 and 2nd order perturbation theory will seek values of and A k g2Enm gEn l Enw Georgia Tech gzl g k ECE 6451 Dr Alan Doolittle 1st Order Perturbation Theory Given the term that is 1st order in g gH0 l 1 H350 Egon why 2 0 and using the Fundamental Expansion Postulate for Tn using the basis vectors I j0 s Georgia Tech 1 o Pn Zaj 1 j J 1 o o 1 1 0 HOT Hp Pn En Pn En Pn 0 0 0 0 1 0 H0Zaj 1 j Hp 1 n En 2611312 En Pn J J But since 0 0 HOWn EOTn then 2 aim 111119 E 0Zaj 1 0 35 j j ECE 6451 Dr Alan Doolittle 1st Order Perturbation Theory Cont d ZajEjLij P11711150 E gtZ 61130 E j 139 As we have done many times we will multiply by 11 and integrate over all space I ajEflPrgomPfo jdv I L1120 lellioklv IE2 aijr omPfo jdv I E 1gtL11 gtL1 gtdv j 139 Using the fact that 11 is orthogonal to LP unless m j amEg j L11 gtpr gtdv E gtam E51 j managde or M w Hptrg gtgt z Egmam E 1ltLPE1 mm Cons1der the case when mn a E0 LPN H LP0gt Ewe E1ltLP0 LP0gt m m m p 71 71 m 71 m 71 U M w HpLP gt z Eggtltxpnltogt mm w Georgia Tech ECE 6451 Dr Alan Doolittle 1st Order Perturbation Theory To nd the coef cient am s consider the case where min amEZ CPL We Esgtltwgt L119gt w prwgogtgt z Egm Efn am o 1 allLPimgt am Em Em which is valid for all m n except m n Georgia Tech ECE 6451 Dr Alan Doolittle 1st Order Perturbation Theory For the case mn we have to consider the normalization condition T50 gwy IP50 glrg 1 U Tim g2 61mg Tim g2 am Prgodv 1 But since all cross terms of the form I Pogt PEOdv or IT O PE10dv equal 6mm l gam ga gZZama 1 which is only true for all g when am 2 at a a 0 Thus the original equation I 2 910 g I O 112 w g2 hecom es ltle M Georgia Tech NOTE since am 2 0 for m n the sum is only performed over m 7 n ECE 6451 Dr Alan Doolittle In summary Georgia Tech 1st Order Perturbation Theory E E20 gas where W 1Hp 1 50gt EIEI ltLP 50 V1150 LP 2 L110 g P where 0 lt0 lt Pm 19an gt o Em E0 m LPG 2 min ECE 6451 Dr Alan Doolittle 1st Order Perturbation Theory Things to consider 1 To calculate the perturbed 11th state wavefunction all other unperturbed wavefunctions must be known 2 Since the denominator is the difference in the energy of the unperturbed nth energy and all other unperturbed energies only those energies close to the unperturbed nth energy signi cantly contribute to the 1st order correction to the wavefunction 3 g can be set equal to l for convenience or rigidly determined by the normalization condition on Pn Georgia Tech awn 9 where m CKWHJE9gt Equot lt I l 1 gt qge9gmw where ltw20gtIlewsogtgt LPG z LP0 quot 0 0 m min En Em ECE 6451 Dr Alan Doolittle 2Ild Order Perturbation Theory Given the term that is 2nd order in g g2H0w2gt my ngge nggv nggm 0 and using the Fundamental Expansion Postulate for THO using the basis vectors I j0 s 5 zbj l jo Ho l f HPLPf Egon E Ef gy 0 0 0 0 1 0 2 0 HOZ bj l jo Hp Z app E Z by E 251131 E Pn j j j 139 Again multiplying by Wit and integrating over all space l w0gtH0ZbJWjodv IWLOVHP Z aqujodv j j j EOgtZ bjW0 l 0dv j E5gtZ a ijmjdv IE2W0 1 Odv j 139 But since Horn E350 and j nytoy l jmdv 6m then mg Z a j LII0gtHPLP gtdv meg amEgu E95quot 139 Georgia Tech ECE 6451 Dr Alan Doolittle Cont d 2Ild Order Perturbation Theory mg Z at I 50prwa mg amE gt Eme or 139 mg Z 50011510 al ijgt mey amE gt Eme f To nd the 2nd order energy correction consider the case of mn Georgia Tech E52 2 2 5210119 le trig anEgD J39 or pulling out of the summation the m n term 2 0 0 0 0 1 En EMT lelrj gt Mirquot Hp 1 quot gt anEn jin Inserting the result for E51 from the lst order solutiorr E5 2 1101 le 1If anltlr gt Hp 11150 anltan0 jin E5 2 2511011 le 1If fin Inserting the result for a from the 1st order solution HP T5 TS H T20 E512 PEr 0 gtjltlprfvm IHPIT0gt Kw le H 2 E2Z quot i jin Eno Ej0 ECE 6451 Dr Alan Doolittle 2Ild Order Perturbation Theory To nd the coef cient bm s consider the case where min mg Z 50011510 all L11 mg amEgD Eme j n bm E50 E5 Z 55011510 Hpkrjogtgt amE gt j n P an Pim E0 bm ltEogt392Logts gt1Er2ogt 50 al L115 aJwyallw m E0gt E0gt Er Em J LIIlt0gtH L115 kuH 50 ijH 50gt it ammo ltEwtgmgtlt w gt1 n J n m b Em ESP Em ESP ltltwiogtIHprsmgtXWlewa WIHprs gtgtXltwsmIHPIW 5quot E0gt EOgtXE0gt EOgt E0gt E0gtXEOgt Egt Georgia Tech ECE 6451 Dr Alan Doolittle 2Ild Order Perturbation Theory We also need to nd the value of bIn for the case where mn To do this we again examine the normalization condition Wm gxylt1gt g2q1lt2 U 0 lt0 2 lt0 lt0 lt0 2 lt0 Pn gZaj l j g ij l j Pn gZaj l j g ij l j dv l J J J J But since all cross terms of the form I l l dv or I l joy PJlOWv equal 6m 1 2 1 K PQWH W50 Adding this to the result for bm with m i n and inserting this into the expression for l nm H b 2 IHP Pjogtgtltq1ogtalqJJltogt 510 IHP PVEOgtXlt 50 p 5 1239 ago IHP j n 50 gulf gz l izgt 1 2 2 L1150 EEO ENE E E Mm Es E E0gtgt2 Georgia Tech ECE 6451 Dr Alan Doolittle 2Ild Order Perturbation Theory Cont d Thus the original equation P 2 1110 gP1 g21P2 1 quot T50 g2 am P gZme PS HZ fin becomes WWWMWWLWWWWWMWJWWMW E50 w Egt E50 Es x125 E59 2 129 2 1110 H P0 7 wwwwwmwwlwwwwwmmw E x125 E gt Em x125 Em n Tquot g2 Zg Z fin l 239 jin me all T5 2 E50 E 02 Po m Georgia Tech ECE 6451 Dr Alan Doolittle 2Ild Order Perturbation Theory In Summary K I S Hp 1 0gt 2 0 0 0 2 En En gltLPn Hp LP gtg XL E0E0 jin n j 11 Ts g27 ylt 1 3fi if gtw g222ltltw gt IHP PS gtX P2 lam00 W IHPIT2 gtgtXltT20gt IHP 1 5 gt12 1 39HP39T39EOgt2 E50 E50 Egm ES Em Eff E gt ES 2 M Ego E50 2 n m fin Georgia Tech ECE 6451 Dr Alan Doolittle Perturbation Theory Consider an important and illustrative example Small electric eld applied to an in nite potential barrier quantum well What effect does this have on the ground state energy IPOn1 E0n1 E0n1 123 0 1za Unperturbed Well 2a 0 12a Perturbed Well When the electric eld is applied the energy bands bend resulting a redistribution of the electron in the well toward the right side Since the energy on this side of the well is lower the new energy of the ground state is expected to be smaller than the unperturbed ground state We previously solved this as an asymmetric solution OltXltL and had states that only depended on sin functions For reasons that will become obvious we rede ne our limits as symmetric 12 a ltXlt 12 a which will require wave function solutions of the form T20 J2 sin j for m even and T20 J2 cos mj for m odd a a a a m l27r2h2 0 Eml 2 ma Georgia Tech ECE 6451 Dr Alan Doolittle Perturbation Theory We rst attempt a 1st order perturbation solution of the form 0 0 E Elo gE0 where E ltan HP LPW gt n n n 7 ltP0 LIJOgt A k A k Wm WM 12a 0 12a 1 a 0 a Unperturbed W611 Pzerturbed Well Note that since our ground state wave function solution is normalized the denominator in the above equation is equal to 1 In case 0n an Hp 39qSOX Electric eldVoltsmeter times meters Volts 9 Energy by iq Thus for the ground state nl odd index lH q80x Pigf dx q8xcos2 n 21y dx 2 2 0 integrand is an odd mction Georgia Tech ECE 6451 Dr Alan Doolittle Perturbation Theory Since our 1st order perturbation correction to the ground state energy resulted in a zero correction factor that deviated from our physical understanding of the system ie we expect a lower ground state energy we now must consider the 2nd order correction 2 KW all W 0gtgt E20 EEO En Eff gltT 0gt Hptrgfogtgt gzz fin A k A k Wm E0nl 12a 0 12a 1 a 0 a Unperturbed W611 Pzerturbed Well Note that since our 1st order correction was equal to O the middle term in the above equation is equal to 0 We will examine odd and even indexes in the summation separately Georgia Tech ECE 6451 Dr Alan Doolittle Perturbation Theory Consider the Odd indexes in the summation lt P20Hpi 1 f gt2 K I SEH qeax 1 02 Consider for j odd T53 qeax1w0gtgt Elt2gt Z fin Elt2gt Z fin a x m qsax PJ0 dx W a ltw523lt qeaxiwogtgt Georgia Tech ECE 6451 Dr Alan Doolittle Perturbation Theory Consider the Even indexes in the summation 0 0 ler gt 2 7 y n 0 0 fin E23 EE ICES qeaxlw E52 2 2 0 0 n lEn1 EJ l Consider for j even qsaxj LP 2 2 2 N m N m L113 qeax 1 0 dx E cos n a1 x Wax sin17 xdx 0 Wl Ti 2 71er lo 31er So the Even indexes in the summation contribute nonzero values LESS L252 qsaxx TE 2 L252 Georgia Tech ECE 6451 Dr Alan Doolittle Perturbation Theory Thus the correction term for Even indexes is 2 K all Pfm E Z 7 0 0 jisodd E53 Ej lo5 1lesinl j l lleiTsm jl n jgld n l27r2iz2 j27r22iz2 2ma2 2ma2 2 Technically this is an in nite summation However since the denominator increases proportionally to j2 the relative weight of higher order j terms rapidly decreases and the solution converges after just a few terms Note Since the denominator is always negative and the numerator is always positive the 2nd order correction is always negative resulting in a lower energy than the unperturbed ground state as expected 16 K 2 Ti HM En1 E 2E 0 2 Ne ative Number Georgia Tech quot 1 quot 1 g g g ECE 6451 Dr Alan Doolittle Degenerate Perturbation Theory Thus far it was assumed that the energy values of each state are never the same If this is not the case the correction terms can blow up to in nity En 2 E50 gltT ogt Hptrg 0gt gzz jin To 0gt Tn prsogz mm TO 940 H P gtk 1 H 940 CW H P gtIZ P H W 2 J j m n g 2 Em EoEo Eo Em EltoElto Elto J m n j n m n m n m How do we handle the case where the energy of multiple states is degenerate jin 110 W2 E10E20E30E40 130 140 Georgia Tech ECE 6451 Dr Alan Doolittle Degenerate Perturbation Theory The solution to this problem is to transform the degenerate basis vectors into another set which results in zero numerator terms En Eff gltT ogt Hptrgfogtgt gzz 11 0 E 0 1P 1P g I quot71 En Em 2 W m WIle as lew gt 1 l villas g 2 Z Em E 0 Em E0 Z T m n J V m 0 0 V 0 0 2 m EP Ef EP Ef 21quot E gtE gt jin Simple Example Changing to an equivalent basis set in real space A convenient set of equivalent basis vectors can be selected for any given problem 130 pa 00c 1200c 4120 110 110 Georgia Tech ECE 6451 Dr Alan Doolittle Degenerate Perturbation Theory Lets assume that h various unperturbed states result in the same degenerate energies Any speci c one of these states say the mth state can be written as where mcan be any valuem 1 2 3h 11 still represents the state of the electron while m represents which of the h degenerate states are being considered Since each of the above unperturbed states is an Eigenfunction of HQ with h degenerate Eigen energies En0 any linear combination of these degenerate basis states is also an Eigenfunction of H0 see Brennan Chapter 1 Thus we can construct a new set of basis sets labeled 0L that are linear combinations of the degenerate Eigenstates h Info 531153 wherem 123h ml The task is simply to nd the appropriate values of the coef cients bm To do this we simply nd the set of bma s that force the numerator Tom H ppm m p n Prfl Z P0 i 0 0 i m min I PSWYHA IWU dx quot o m 1 Tquot Z iE0 E0i T to equal zero when the denominator is also zero thus removing the singularity Georgia Tech ECE 6451 Dr Alan Doolittle Degenerate Perturbation Theory The problem reduces to a diagonalization of the submatrix of index h by de ning new basis sets such that 11 H11 H1 Hln Trgl H H 3 1 11 11 I 3139 l 2 CH1 HP P 2 HM H M 3 T532mh 2 P522144 Hquot1 H n T50 H11 0 0 H1h1 39 39 39 Hln Trigifrtn 0 H11 0 0 T5923 2 HPT 0 0 Hhh S H h11 39 39 i 1 Hquot1 Hm T50 Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory In all previous cases nothing observable happens because the states are assumed static unchanging Most useful systems require transitions between states For example optical absorption and electronhole pair recombination require a change from one state to another This inherently requires TIME DEPENDENT Perturbation Theory A k A k l1quot0n1 130 1 12a 0 12a 2a 0 12a 2a 0 12a Time tlt0 Time t0 Time tgtgt0 Given wave functions are sluggish in nature waves do not change instantly when the perturbation changes consider a water wave as an analogy an instantaneous change in the perturbation results is a more gradual time change in the wave function and thus the distribution of particles Note the expectation value of the total potential energy is assumed to change instantly as the perturbation energy changes instantly but the expectation value of the kinetic energy changes gradually Note terms like gradually and sluggish are somewhat misleading in that these changes can often happen in fractions of a nanosecond However compared to the instantaneous perturbation these changes are considered slower Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory The starting point for time dependent perturbation theory is the time dependent Schrodinger equation see lecture 6 a my 20 we y wequot h Of Breaking this into a time independent HO part and a SMALL time dependent part H39 t Ho gH39tgt Considering the solution before H39 t begins E mt 1 n it Since 10 2 Time H 150 Em 150 Thus the general solution for the time dependent solution before the perturbation is 0 2 an IMP subject to the normalization condition 0 0gt which given orthonormality of the individual eigenfunctions IE0 reduces to 2 an an 1 Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory For the time dependent solution h 6 H gH39lttgt 1 3t We express the wave functions as a linear combination of the complete basis set of unperturbed wavefunctions 0 x f Z 0 0 X t NOTE The time evolution of n the wave function depends Substitution of this expression into the Schrodinger Equation yeilds 0 the me evolution of the weighing coefficients to be h a determined and the known 0 0 0 Ho 2 an x t gH39 an x t Z Ta an x 0 time evolution ofthe basis 391 n 1 391 unperturbed states see or previous slide 2a mm x t 2a ogH39 on x t zg an 0 x a Zan age x 0 But from the unperturbed solution the 1st and 4th terms cancel leaving h a The time dependent perturbation can be 2 an t gH39 t 0 xD t f 2 an t r 0 xD t 4 described as a time evolution of the coefficients n 1 n t of the basis set wave vectors As we have many times before we multiply by a specific wave function 110 x t and integrate over all space 39 gl ZanUWliO x tH39 W150 x tdv my x W150 x 0 a 0 l 0 a 0 l 0 game ganlttj ltxtH ma gtltxtdv or game ganlttlt 3ltxtlH t Km Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Important Observations Since the weighting coef cients are known before the perturbation knowing the time derivative rate of change of the coef cients implies full knowledge of the weighting coef cients after the perturbation occurs a 0 l 0 a 0 l 0 game giganmmxw t gtltxtdv or game galttlt ltxtlH t km Each individual state is connected through the perturbation Hamiltonian Ie state m is connected to every other state all 11 states Via the perturbation Hamiltonian If the Hamiltonian does not allow a transition from one state to another ie the matrix element is zero then the weighting coef cient for that state remains unchanged ie static in time The change in each individual coef cient in time depends on couplings between ALL other available states Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Consider an expanded form of the previous equation garquot r zan 0W x rH39 t lo x 0 U gama an0r oxrlH39t Lxr an1r 0xrlH39t 510am anjr 0xrlH39t oat gamlr anor 1xrH39t Lxr an1r 1xrH39t Eixr anjr 1xrH39t i xr h a 01m gz t nJ gt expr an0r jxrH39t 3Lxtgt an1r jxrH39t 510am anjr jxrH39t Considering the vastness of this system of equations a simpli cation similar to what we did for the unperturbed system is in order Since the wave function evolution in time is entirely determined by the coef cients we will expand the coef cients in orders of g dza J dg2 da 0 2 aj aj gd gjg 0r 0 39 2 aj aj gajg aj Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Using this expansion the previous system of equations becomes l Eag ga0 g2a0 gag ga0 g2a0 x tH t Lxtgtga1 ga1 gza1 xtH t xtgt gao ga g2a X 0xtlet fty 0 7Eallto gal gzal gag ga0 g2a0 glx tH t n x tgt ga1 ga1 g2al lt glxtH39t n 1xtgt ga0 ga g2a X rr1xtHlt rljxtgt 7ga o gaj gig ga3 ga0 g2a0 lt jxtH t 30 x tgtga1 ga1 g2a1 lt jxtH39t rf 1xtgt ga ga g2a X jxrH39t n xrgt For our purposes we will only consider 1st order time dependent perturbation theory THANK Gonz For this we will consider only the linear terms in g Again for any choice of g the terms on the left hand side must equate to the terms of equal magnitude on the right hand side This is done on the following page Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Considering equality of the linear terms 13 a gtlt 20ltx t H39 t 5531 x o a lt E 20ltx t H39 t 5531 x 0 a5 lt E 20ltx t H39 0 v559 x o l 3 a1 a E 21ltx t H39 0 552m o af gtlt 21ltxtgt H39 t 5521 x o a gtlt E 21ltxtgtH39lttgt 55 o l 334 a gtlt 2jltxtgt H39 t Egtoltxtgtgt af gtlt 2jltxtgt H39 0 rm a gtlt 2jltxtgt H39 t i ltxtgt i6t The importance of this equation is that it is an approximate 1st order solution that only depends on the station unperturbed coefficients and the unperturbed basis wave functions Thus it depends on the initial condition unperturbed coefficients and wave functions of the system Each equation in the above system of equations represents the growth or decay of an individual Eigenfunction s magnitude and thus represents the growth or decay of the probability of the state being occupied or empty All states are connected via the perturbation Hamiltonian H t Thus if a particular state say m3 grows or decays in amplitude it must come at the expense of some other state for example mll decaying or growing in amplitude The rate of growth or decay is set by the 3m1 and 11th equations NOTE this is just a simple example and in practice all states are coupled so growth and decay involves transitions from all states In some cases the perturbation Hamiltonian will prevent certain transitions resulting in a inability to transition between states 7 called forbidden transitions Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Example 1 Consider the case when the initial state of a system is known to be in only the j h state all zero coef cients except ajl and the perturbation is turned on at time t0 ie H tH ut where H is a constant and ut is a step function H tH ut The systems of equations A 333 ago 1X 20ltxtgtH39lttgt i ltxtgt H H t gai a 1 21ltxt H39 t E ltx1gtgt reduces to equations of the form a IX PIEO x lH39 xgteiEin0E 0Z for m 01 2 but m i j and H 0 W where we have used the fact that since 0gt T510 X h then 0 7i E nh EEOR h j H39 11130 Xe h J ltTxH1P30xgteimmjtj Where wmj Efno E 0 PEEgtlt6 Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Thus this general equation a39 41 H39W O W f 012 b 39 5am y lt m x l njxgte orm 7 j 7 mm can be directly integrated to result in a mt ft 1 JiltlrgtxlH39wg gtxgte K ut form0l2butm j a The integration constant can be evaluated by restricting a mt0 at t0 resulting in v 1 0 y 0 eiwmjt 39 1 amt lt Pm xH l Pnzjxgt ut form Ol2butm 7b a m Finally since the perturbation is de ned as small the jth coef cient is simply Since fort S O a j 2 ago then a O and H tH ut for t gt O a can be found through normalization of Ha the total wavefunction but in general a39j z 0 t0 t Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Important Observations a39 t ltWx H39W ixgt l ut form o 1 2butm 2 j m m l The matrix element in the above expression ltgt term is known as the transition matrix element and describes which transitions are allowed and disallowed and how strongly the mth and j h states are coupled 2 If the transition matrix element integrand is odd a very common occurrence the integral is 0 meaning that a transition between the mth and j h state is forbidden 3 In general this transition matrix element is responsible for a variety of selection rules in atomic nuclear and semiconductor bulkquantum well optical spectra emission or absorption resulting from electronhole transitions between states 4 Even though a speci c transition is forbidden from say the jth state to the mth state the mth state may still eventually become populated by indirect and thus slower transitions of the form j 9 k 9 m etc Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Example 2 Turning on a Harmonic Perturbation at time t0 This case is important as many excitations such as electromagnetic radiation ac electric and magnetic elds all can be periodic in nature This problem proceeds identical to the previous example except 2 5t 6 got jltwgtx Axei o e imo utkrfgtjxgtei wt for m o 1 2 but m 7 j t0 t din X H39 LEE Xgteiwmjt for m 01 2 but m i 5a 71jltLIJIn xAx111152xgtem jtei t e iwo for m o 1 2 but m 7 j can be directly integrated to result in a I Z L110gtxAx w 0gtxgtelwmot 1 840 1 ut form0l2butm j h wmj Do wmj 0 Where the integration constant was again evaluated by restricting a mt0 at t0 Finally since the perturbation is de ned as small the jth coef cient is simply av 0fort 0anddj zOfortgtO Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Important Observations 0 0 iagtmjcoot iamfcoot a I W XlXl Dupe 1 id 0339 um for m 2 0717 27 but m ij mm 030 1 Two resonances exist in the above coupling equation wmjoao where the coupling between the mth and j h state is extremely strong These resonances occur when the mth state is exactly hoa0 away from the initially occupied j h state 2 States not on resonance still can be coupled to the initially occupied j h state but with a lesser coupling strength than those directly on resonance 3 While this equation predicts a strong coupling at resonance a mt 9 00 our restriction of a small perturbation ie a mt small indicates that 1st order time dependent perturbation theory is insuf cient to accurately describe this case and thus it is expected that a mt will NOT actually9 00 and that higher order approximations will be needed to accurately describe this condition 4 Even though a speci c transition is forbidden from say the jth state to the mth state the mth state may still eventually become populated by indirect and thus slower transitions of the form j 9 k 9 m etc Georgia Tech ECE 6451 Dr Alan Doolittle Time w Perturbation Theory Fermi s Golden Rule Since the probability of state m being occupied is found by Wm l m and due to the normalization of the basis wave functions this reduces to a mta mt Thus we need to consider the value of a mta mt To simplify this procedure we note that only one term in the square braces need be considered for the two cases near resonance a I W 0xlAX rf0Jxgtezwmmot 1 exawoot 1 h wmj030 any 030 ill form20 1 2 39bmm j 1 2 s1n203mj motj ut for E N Ego h03O 0O2 4 lt P3XIAX PEX r12 03ml atat and 4 lt1PEEX AX IP2JXgt2 SmZG mJ otj 73912 mm may atat ut for E N Ego hmo The reason we make this simplification is that in deriving Fermi s Golden Rule the functional form of the above equations will be convenient for integration Before we proceed lets examine these equations Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory The probability of all states including far offresonant states becoming occupied increases with time L0ga mta mt l u u 2 sm2 comJ 7 can aim t aim t m 4 XAXPJX 2 ut for E m El hcou hz mm icon 2 J and ammo 2 2 l PX AOIP EBJX er 2 mm in Ina for E m ESE hmu 71 mm mu Georgia Tech pom m tSO Eff z Ego mo E30 0 lt0 Em E h030 ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory Instead of discrete state to state transitions it is often useful to consider a discrete state to band of states transition Examples of this are donor states to conduction band transitions acceptor states to valence band transitions or simply defectimpurity states to either conductionvalence band transitions These bands can be described by their density per unit energy see lecture 7 or since Ehoa the density number per frequency 0 centered around the transition frequency poamj Assuming that this density of states poa Inj does not change quickly with cam we can find the new probability density of state m by integrating the previous expression over pm For example 2 2 1 t K P 5 x Altxgt W102 xgt Sm 5 3m 0 quot12 ammj mf pamjdamj for E H E30 heO atat Georgia Tech ECE 6451 Dr Alan Doolittle Time Dependent Perturbation Theory 2 sin2oamj 0Ot mmj 490 Making some assumptions about this function 1 The transition matrix element is a slowly varying function of pm 2 The density of states poamj also does not change quickly with omj 3 Both of the above two conditions can be achieved by noting that since the sin2 function sharpens in in mm with increasing time we can always wait long enough in time to make this part of the integrand the most rapidly varying portion of the integrand 1n ij lltWESxgtAxgtl P 2Jltxgt atant I W 0 N 0 pamjdamj forEIn EJ h030 Georgia Tech ECE 6451 Dr Alan Doolittle