5.74 Introductory Quantum Mechanics II

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1 MIT OpeCourseWare hp://ocw.mi.edu 5.74 Iroducory Quaum Mechaics II Sprig 009 For iformaio aou ciig hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.

2 drei Tokmakoff, MIT Deparme of Chemisry, 3// IRREVERSIBLE RELXTION 1 I may o seem clear how irreversile ehavior arises from he deermiisic TDSE, alhough his is a hallmark of all chemical sysems. To show how his comes aou, we will descrie he relaxaio of a iiially prepared sae as a resul of couplig o a coiuum. We will show ha firs-order peruraio heory for rasfer o a coiuum leads o irreversile rasfer a expoeial decay whe you iclude he depleio of he iiial sae. The Golde Rule gives he proailiy of rasfer o a coiuum as: w P π V ρ (E k E ) P w k ( 0 ) (3.1) P 1 P The proailiy of eig oserved i k varies liearly i ime. This will clearly oly work for shor imes, which is o surprise sice we said for firs-order P.T. k ( ) k ( 0). Wha log-ime ehavior do we expec? ime-idepede rae is also expeced for expoeial relaxaio. I fac, for expoeial relaxaio ou of a sae, he shor ime ehavior looks jus like he firs order resul: ( ) (0exp ) ( w ) P P 1 w + " (3.) So we migh elieve ha w represes he age o he relaxaio ehavior a 0. P w (3.3) 0 The prolem we had previously was we do accou for depleio of iiial sae. From a exac soluio o he wo-level prolem, we saw ha proailiy oscillaes siusoidally ewee he wo saes wih a frequecy give y he couplig:

3 3- Ω R Δ +V Bu we do have a wo-sae sysem. Raher, we are relaxig o a coiuum. We migh imagie ha couplig o a coiuous disriuio of saes may i fac lead o expoeial relaxaio, if desrucive ierfereces exis ewee oscillaios a may frequecies represeig exchage of ampliude ewee he iial sae ad coiuum saes. COUPLING TO CONTINUUM Whe we look a he log-ime proailiy ampliude of he iiial sae (icludig depleio ad feedack), we will fid ha we ge expoeial decay. The decay of he iiial sae is irreversile ecause here is feedack wih a disriuio of desrucively ierferig phases. Le s look a rasiios o a coiuum of saes { k } from a iiial sae uder cosa peruraio. These form a complee se; so for H ( ) H + V ( ) wih H E k k (3.4) iiial k coiuum s we go o, you will see ha we ca ideify wih he sysem ad { k } wih he ah whe we pariio H0 H S + H B. We wa a more accurae descripio of he occupaio of he iiial ad coiuum saes, for which we will use he ieracio picure expasio coefficies k ( ) k U I (, 0 ) (3.5) The exac soluio o U I was: U, 0 1 i dτ V I τ U τ, 0 (3.6) I ( ) 0 ( ) ( ) I

4 3-3 For firs-order peruraio heory, we se he fial erm i his equaio U I (τ, 0 ) 1. Here we keep i as is. k () I ( ) ( 0 i k dτ k V τ U τ, I 0 ) (3.7) Iserig he projecio operaor 1, ad recogizig k l, k () i τ 0 i d e ω τ k V τ k ( ) (3.8) Noe, here V k is o a fucio of ime. Equaio (3.8) expresses he occupaio of sae k i erms of he full hisory of he sysem from 0 wih ampliude flowig ack ad forh ewee he saes. Equaio (3.8) is jus he iegral form of he coupled differeial equaios, ha we used efore: i k e iω k V k ( ) (3.9) These exac forms allow for feedack ewee all he saes, i which he ampliudes k deped o all oher saes. Now le s make some simplifyig assumpios. For rasiios io he coiuum, le s assume ha rasiios i he coiuum oly occur from he iiial sae. Tha is, here are o ieracios ewee he saes of he coiuum: k kv 0. This ca e raioalized y hikig of his prolem as a discree se of saes ieracig wih a coiuum of ormal modes. Moreover we will assume ha he couplig of he iiial o coiuum saes is a cosa for all saes k: V k V k cosa. So sice you oly feed from io k, we ca remove he summaio i (3.8) ad express he complex ampliude of a sae wihi he coiuum as i k V d 0 τ ω τ e i τ (3.10) ( )

5 3-4 We wa o calculae he rae of leavig, icludig feedig from coiuum ack io iiial sae. From eq. (3.9) we ca separae erms ivolvig he coiuum ad he iiial sae: Now susiuig (3.10) io (3.11), ad seig 0 0: i e iω k V k k + V (3.11) k 1 V 0 k ( ) τ e iω ( τ ) i dτ V () (3.1) This is a iegro-differeial equaio ha descries how he ime-developme of depeds o eire hisory of he sysem. Noe we have wo ime variales for he wo propagaio roues: τ : : k k (3.13) The ex assumpio is ha varies slowly relaive o ω, so we ca remove i from iegral. This is effecively a weak couplig saeme: ω >> V. is a fucio of ime, u sice i is i he ieracio picure i evolves slowly compared o he ω oscillaios i he iegral. 1 V e iω τ k 0 ( ) i dτ V (3.14) Now, we wa he log ime evoluio of, for imes >> 1 ω, we will ivesigae he iegraio limi. Complex iegraio of (3.14): Defiig τ d dτ e iω ( τ ) dτ e iω d 0 0 (3.15) The iegral lim T +i e ω d is purely oscillaory ad o well ehaved. The T 0 sraegy o solve his is o iegrae:

6 3-5 lim e ( iω + ε ) 1 d lim ε ε 0 + iω+ ε lim ε ω + i + ω + ε ε 0 ω + ε 1 +πδ ( ω ) ip ω (3.16) I he fial erm we have used he Cauchy Priciple Par: P 1 1 x x 0 x 0 x 0 (3.17) This leads o π i V δ ( ω ) V + P V (3.18) k k E k E erm 1 erm Term 1 is jus he Golde Rule rae, wrie explicily as a sum over coiuum saes isead of a iegral δ ω ( k ) ρ (Ek E ) (3.19) k w de δ (E k E ) k E k ρ ( ) π V (3.0) Term is jus he correcio of he eergy of E from secod-order ime-idepede peruraio heory, ΔE. kv Δ E V + (3.1) k E k E So, he ime evoluio of is govered y a simple firs-order differeial equaio w i Δ E (3.)

7 3-6 Which ca e solved wih 0 1 o give ( ) i () exp w Δ E (3.3) We see ha oe has expoeial decay of ampliude of! This is a maer of irreversile relaxaio from couplig o he coiuum. ω Swichig ack o Schrödiger Picure, c e i, we fid c () exp w + i E (3.4) wih he correced eergy E E +ΔE (3.5) ad P c exp[ w ]. (3.6) The soluios o he TDSE are expeced o e complex ad oscillaory. Wha we see here is a real dissipaive compoe ad a imagiary dispersive compoe. The proailiy decays expoeially from iiial sae. Fermi s Golde Rule rae ells you aou log imes! Now, wha is he proailiy of appearig i ay of he saes k? Usig eq.(3.10): i 0 V e iω τ k ( τ) () dτ w 1 exp i ( E Ek ) V E E + i w / k 1 c ( ) V E E + i w / k (3.7) If we ivesigae he log ime limi ( ) we fid V P (E E k k ) +Γ /4 (3.8)

8 3-7 wih Γ w (3.9) The proailiy disriuio for occupyig saes wihi he coiuum is descried y a Lorezia disriuio wih a widh give y he relaxaio rae. Noe ha he fial saes wih maximum proailiy of eig occupied is ceered a he correced eergy of he iiial sae E. Readigs 1. Cohe-Taoudji, C., Diu, B. & Lalöe, F. Quaum Mechaics (Wiley-Iersciece, Paris, 1977) p. 1344; Merzacher, E. Quaum Mechaics, 3rd ed. (Wiley, New York, 1998), p Niza,. Chemical Dyamics i Codesed Phases (Oxford Uiversiy Press, New York, 006), p. 305.