Quantum Mechanics for Scientists and Engineers. David Miller

Transcription

1 Quatum Mechaics fr Scietists ad Egieers David Miller

2 Time-depedet perturbati thery

3 Time-depedet perturbati thery Time-depedet perturbati basics

4 Time-depedet perturbati thery Fr time-depedet prblems csider sme time-depedet perturbati Hˆ p t t a uperturbed Hamiltia Hˆ that is itself t depedet time The ttal Hamiltia is the Hˆ Hˆ ˆ H pt T deal with such a situati we use the time-depedet Schrödiger equati i Hˆ t where w the ket is geerally time-varyig

5 Time-depedet perturbati thery With ad E as the eergy eigefuctis ad eigevalues f the time-idepedet equati Hˆ E we expad the sluti f the time-depedet Schrödiger equati as exp / a t ie t Nte we icluded the time-depedet factr exp iet / explicitly i the expasi leavig the time depedece f a t t deal ly with the additial chages

6 Time-depedet perturbati thery Nw we substitute exp / a t ie t it the Schrödiger equati i Hˆ gives t ia a E exp ie t/ ˆ ˆ a a H H p t exp iet/ where a t Replacig Hˆ with E ad cacellig gives ˆ ia exp ie t/ a H t exp ie t/ p

7 Time-depedet perturbati thery Nw premultiplyig ia exp ie t/ a Hˆ t exp ie t/ p q by bth sides leads t exp / exp / ˆ ia t ie t a t ie t H t q q q p We have made apprximatis s far This is merely a restatemet f Schrödiger s timedepedet equati i Hˆ t

8 Time-depedet perturbati thery Nw we csider a perturbati series We itrduce the expasi parameter as befre w writig ur perturbati as Hˆ p As befre, we ca set this t 1 at the ed We w express the expasi cefficiets a as a pwer series 0 1 a a a a ad we substitute this expasi it ia ˆ q t exp ieqt/ a t exp iet/ q H pt where we w have H istead f just Hˆ p ˆ p

9 Time-depedet perturbati thery I ia ˆ q t exp ieqt/ a t exp iet/ q H p t equatig pwers f bth sides first we btai the zer rder term a 0 q t The zer rder sluti simply crrespds t the uperturbed sluti ad hece there is chage i the expasi cefficiets i time t zer rder 0

10 Time-depedet perturbati thery 0 1 With a ad a a a ia exp / exp / ˆ q t ieqt a t iet q H pt fr the first rder term we have a ˆ q t a exp iqt q H p t i where we have itrduced the tati E E / q q

11 Time-depedet perturbati thery Nte here i a ˆ q t a exp iqt q H p t i 0 we already kw tha the are all cstats a They give the startig state f the system at t = 0 We te w that, if we kw the startig state, the perturbig ptetial ad the uperturbed eigevalues ad eigefuctis we ca itegrate t btai 1 the first rder, time-depedet crrecti, a q t t the expasi cefficiets

12 Time-depedet perturbati thery After itegratig a ˆ q t a exp iqt q H p t i we kw the ew apprximate expasi cefficiets 0 1 a q aq aq t s we kw the ew wavefucti ad ca calculate the behavir f the system frm this ew wavefucti

13 Time-depedet perturbati thery We ca prceed t higher rder i this time-depedet perturbati thery Equatig pwers f prgressively higher rder gives p1 1 p a ˆ q t a exp iqt q H p t i We see that this perturbati thery is als a methd f successive apprximatis just like the time-idepedet perturbati thery We calculate each higher rder crrecti frm the precedig crrecti

14

15 Time-depedet perturbati thery Simple scillatig perturbatis

16 Simple scillatig perturbatis Oe very useful case is fr scillatig perturbatis where a perturbati is varyig siusidally i time als called a harmic perturbati as i the harmic scillatr fr example, a mchrmatic electrmagetic wave with a electric field i, say, the z directi E t E expit expit E cst where is a psitive (agular) frequecy We csider this here i first-rder time-depedet perturbati thery

17 Simple scillatig perturbatis With E t E exp it exp it E cs t fr a electr, the electrstatic eergy i this field, relative t psiti z = 0 gives a perturbig Hamiltia Hˆ ˆ p t ee t z H p expit expit where, i this case Hˆ p ee z which is a time-idepedet peratr This perturbig Hamiltia is called the electric diple apprximati

18 Simple scillatig perturbatis We will presume that this perturbig Hamiltia is ly fr sme fiite time Fr simplicity, we presume that the perturbati starts at time t = 0 ad eds at time t = t s frmally we have Hˆ t 0, t 0 p Hˆ pexp i t exp i t, 0 t t 0, t t

19 Simple scillatig perturbatis We are iterested i the case where fr times befre t = 0 the system is i sme specific eergy eigestate Time-depedet perturbati thery will tell us with what prbability the system will make trasitis it ther states With this chice 0 all f the iitial expasi cefficiets a are zer 0 except a m which has the value 1 m

20 Simple scillatig perturbatis With this simplificati f the iitial state t the first rder perturbati sluti a exp ˆ q t a iqt q H p t i m 1 1 becmes a ˆ q t exp iqmt q Hp t m i Nw we substitute the perturbig Hamiltia Hˆ t 0, t 0 p Hˆ pexp i t exp i t, 0 t t 0, t t

21 Simple scillatig perturbatis With that substituti ad itegratig ver time frm time 0 t time t t 0 1 ˆ exp 1 q q p m qm a t t H t i t dt i 0 t Hˆ exp i t exp i t dt i q p m qm 1 qm 1 1 0

22 Simple scillatig perturbatis S 1 a t t q 1 Hˆ q p m exp i qm t 1 exp i qm t 1 qm qm si qm t / exp i qm t / t qm / ˆ t q H p m i si / exp qm t i qm t / qm t /

23 Simple scillatig perturbatis The fucti sic x si x / x 1 peaks at 1 fr x = 0 It is ly large fr x si qm t / s, e.g., qm t / 0 is strgly resat with relatively strg 0.5 ctributis ly fr frequecy clse t qm sicx sic x x

24 Simple scillatig perturbatis We have w calculated the ew state fr times t > t which is, t first rder 1 exp ie t / a ( t t )exp ie t / m m q q q q 1 with the a give by ur precedig expressi q t t Nw that we have established ur apprximati t the ew state we ca start calculatig the time depedece f measurable quatities

25

26 Time-depedet perturbati thery Trasiti prbabilities

27 P j t Trasiti prbability calculati P j I this mdel the prbability f fidig the system i state j is i.e., Hˆ j p m P j a j 1 si t / si t / / t t / si / si t t / cst / t t /

28 P j t Trasiti prbability calculati si(x)/x falls ff rapidly fr argumets 1 Hece, fr sufficietly lg t either e r the ther f the tw si(x)/x fuctis i the last term will be small Hˆ j p m si t / si t / / t t / si / si t t / cst / t t /

29 P j t Trasiti prbability calculati As the time t is icreased these tw si(x)/x lie fuctis get sharper ad they will evetually t verlap fr Hˆ j p m si t / si t / / t t / si / si t t / cst / t t /

30 t Trasiti prbability calculati Presumig we take t sufficietly large, we are left with P j si t / si t / ˆ j H p m / t t / We w have sme fiite prbability that the system has chaged state frm its iitial state m t ather fial state j

31 Trasiti prbability calculati P j t si t / si t / ˆ j H p m / t t / This prbability depeds the stregth f the perturbati squared, ad the mdulus squared f the perturbati matrix elemet betwee the iitial ad fial states

32 Trasiti prbability calculati P j t si t / si t / ˆ j H p m / t t / With a scillatig electric field actig a electr this prbability is the square f the field amplitude E which is prprtial t the itesity I (Pwer/Area) s the prbability f makig a trasiti is prprtial t the itesity I

33 t Absrpti ad emissi terms I P j si t / si t / ˆ j H p m / t t / what is the meaig f the tw differet terms?

34 t Absrpti ad emissi terms P j si t / si t / ˆ j H p m / t t / The first term abve is sigificat if i.e., if Ej Em Sice we chse t be a psitive quatity this term is sigificat if we are absrbig eergy raisig frm a lwer eergy state m t a higher eergy state j

35 t Absrpti ad emissi terms P j Hˆ j p m We te that si t / si t / / t t / the amut f eergy we are absrbig is This first term behaves as we wuld require fr absrpti f a pht

36 t Absrpti ad emissi terms P j si t / si t / ˆ j H p m / t t / The secd term abve is sigificat if i.e., if Em Ej Sice we chse t be a psitive quatity this term is sigificat if we are emittig eergy fallig frm a higher eergy state m t a lwer eergy state j

37 t Absrpti ad emissi terms P j si t / si t / ˆ j H p m / t t / We te that the amut f eergy we are emittig is This secd term crrespds t stimulated emissi f a pht the prcess used i lasers The sptaeus emissi f rmal light requires quatizig the electrmagetic field as well

38