Absorption and Emission of Radiation: Time Dependent Perturbation Theory Treatment

Transcription

1 Absorptio ad Eissio of Radiatio: Tie Depedet Perturbatio Theory Treatet Wat Hailtoia for Charged Partile i E & M Field Need the potetial U. Fore o Charged Partile: 1 F e E V B Fore (geeralized for i Lagragia ehais) j th opoet: U d U U is the potetial Fj q j dt q q j are oordiates j Eaple: U d U F dt V sie: d V dt Copyright Mihael D. Fayer, 17

2 Use the two equatios for F to fid U, the potetial of a harged partile i a E & M field Oe we have U, we a write: H H H Where H ' is the tie depedet perturbatio Use tie depedet perturbatio theory. Copyright Mihael D. Fayer, 17

3 Usig the Stadard Defiitios fro Mawell s Eqs. B A 1 E A t A vetor potetial salar potetial Fore o Charged Partile: 1 F e E V B The: 1 A 1 F e V A t Copoets of F, F, et d ( ) d y A A Az V A Vy Vz A y z Copyright Mihael D. Fayer, 17

4 y A A Az V A Vy Vz A y z Addig ad Subtratig V A y z V A Vy Vz V A A A A A A V V V y z y z Total tie derivative of A is da A A A A dt t y z V Vy Vz Due to epliit variatio of A with tie. Due to otio of partile - Chagig positio at whih A is evaluated. Copyright Mihael D. Fayer, 17

5 The: A da A A A V V V t dt y z y z Usig this da A V A V A dt t Sie: A A A V A y Az V AV V Vy Vz Copyright Mihael D. Fayer, 17

6 Substitutig these piees ito equatio for F 1 A 1 F e ( V A ) t aels da A dt t V A V A The 1 1d A F e A V dt A V V AV beause AV AV AV y y AzV z V V A V substitute A V V y y V A V A ot futio of V & V y V z idepedet of V Copyright Mihael D. Fayer, 17

7 Therefore 1 1 d F e A V A V dtv e d F e e AV e AV dt V The geeral defiitio of F is: Sie is idepedet of V, a add i. Goes away whe takig V Therefore: F e U e AV d U U dtv U potetial Copyright Mihael D. Fayer, 17

8 Legragia: L = T U T Kieti Eergy For harged partile i E&M Field: e LT e AV 1 T y z The i th opoet of the oetu is give by P i L q i where: dqi q i V dt q V i Therefore, e P A Copyright Mihael D. Fayer, 17

9 The lassial Hailtoia: Therefore H P P y Pz L y z e P A e L T e AV (, et.) 1 T y z e e e H A y yay z zaz This yields: 1 e y z A ya y za ze 1 H y z e Copyright Mihael D. Fayer, 17

10 Wat H i ters of oetu P, P y, P z sie we kow how to go fro lassial oetu to QM operators. Multiply by / 1 H y z e Usig: e Pi q A the e q P Ai i i i i Classial Hailtoia for a harged partile i ay obiatio of eletri ad ageti fields is: 1 e e e H p A py Ay pz Az e Copyright Mihael D. Fayer, 17

11 QM Hailtoia Make substitutio: The the ter: p i e e e e i A i A p A A The operator operates o a futio,. Usig the produt rule: Sae. Pik up fator of two e i A ( e A i ( ) ) i e A ( ) e e e A e p A A i i A Copyright Mihael D. Fayer, 17

12 The total QM Hailtoia i three diesios is: 1 e i e i e H A A A e This is geeral for a harged partile i ay obiatio of eletri ad ageti fields For light E & M Field = (o salar potetial) Ad sie 1 A t The: A (Loretz Gauge Coditio) Weak field approiatio: A is egligible Copyright Mihael D. Fayer, 17

13 Therefore, for a weak light soure H 1 i e kieti eergy of partile A For ay partiles iteratig through a potetial V, add potetial ter to Hailtoia. Cobie potetial eergy ter with kieti eergy ter to get oral ay partile Hailtoia for a ato or oleule. H j j j j H H H j V Tie idepedet The reaiig piee is tie depedet portio due to light. e H i Aj j The total Hailtoia is Use H + H ' i tie depedet perturbatio alulatio. Copyright Mihael D. Fayer, 17

14 E & M Field Plae wave propagatig i z diretio (-polarized light) uit vetors i j y k z A ia z with A Aos t vetor potetial To see this is E & M plae wave, use Mawell's equatios 1 z E Ai A si t t z B A j A si t Equal aplitude B ad E fields, perpediular to eah other, propagatig alog z. i j k A y z A Copyright Mihael D. Fayer, 17

15 To use tie depedet perturbatio theory we eed: e A P H ' j j j j Dipole Approiatio: Most ases of iterest, wavelegth of light uh larger tha size of ato or oleule 3 1 Å= ato, oleule 1 1Å oleule part of a yle of light Take A ostat spatially two partiles i differet parts of oleule will eperiee the sae A at give istat of tie. Copyright Mihael D. Fayer, 17

16 Dipole Approiatio: Pull A out of braket sie it is ostat spatially e 1 H' A P j j j e 1 i A j j j j does t operate o tie depedet part of ket, pull tie depedet phase fators out of braket. e 1 ( qt, ) H ( qt, ) i Ae ( q) ( q) i ( E E ) t / j j j Need to evaluate j Ca epress i ters of j rather the j Copyright Mihael D. Fayer, 17

17 First for oe partile 's a write followig equatios: d E V d * * (1) ( ) d E V d () ( ) ople ojugate of Shrödiger equatio Left ultiply (1) by Left ultiply () by * * d * * (1) ( ) E V d d * * * () ( ) E V d Subtrat * d d * * ( ) E E d d Copyright Mihael D. Fayer, 17

18 * d d * * ( ) E E d d Traspose d ( E E ) d d d * * * Itegrate * E E d * d * d d d d This is what we wat. Need to show that it is equal to j Copyright Mihael D. Fayer, 17

19 * E E d Itegrate right had side by parts: * d * d d d d u dv d d * d udv uv vdu ad olletig ters Beause wavefutios vaish at ifiity, this is, so we have: usig produt rule * d d d * d d d d d d * * * * d d d d d d d d d d d d d ad 4 th ters ael d d d d * * d * d d Itegratig this by parts equals seod ter d Copyright Mihael D. Fayer, 17

20 Therefore, fially, we have: d E E d This a be geeralized to ore the oe partile by suig over j. Substitutig ito e 1 ( qt, ) H ( qt, ) i Ae ( q) ( q) i ( E E ) t / j j j gives 1 ( qt, ) H ( qt, ) i A( E E) e i( E E ) t/ with: e j j Operator (harge legth) - dipole -opoet of "trasitio dipole." Copyright Mihael D. Fayer, 17

21 Absorptio & Eissio Trasitio Probabilities dc dt i C ( q, t) H ( q, t) Equatios of otio of oeffiiets Tie Depedet Perturbatio Theory: Take syste to be i state ( qt, ) at t = C =1 C = Short tie C Usig result for E&M plae wave: dc dt 1 A E E e / i E E t No loger oupled equatios. e j j Trasitio dipole braket. Copyright Mihael D. Fayer, 17

22 For light of frequey : A A os( t) Therefore: 1 A e e i t i t vetor potetial ote sig differee dc dt 1 A E E e e i E E h t/ i E E h t/ Copyright Mihael D. Fayer, 17

23 Multiplyig through by dt, itegratig ad hoosig ostat of itegratio suh that C = at t = i EEh t/ i EEh t/ i e 1 e 1 C AE E E E h E E h ote sig differees Rotatig Wave Approiatio Cosider Absorptio E > E E This ter large, keep. Drop first ter. E (E E h) as h ( E E ) deoiator goes to Copyright Mihael D. Fayer, 17

24 For Absorptio Seod Ter Large Drop First Ter The, Probability of fidig syste i as a futio of frequey, C C * Usig the trig idetities: i i e 1 e 1 (1 os ) 4si / Get: ( E E h ) t si * 1 CC A E E ( E E h ) E E E eergy differee betwee two eigekets of H E Eh aout radiatio field is off resoae. Copyright Mihael D. Fayer, 17

25 E E E eergy differee betwee two eigekets of H E Eh aout radiatio field is off resoae. Plot of C * C vs E: Maiu at E= Q t ( ) C t Q a ( ) 4/t E 1 Q A E E Maiu probability t square of tie light is applied. Probability oly sigifiat for width ~4/t 1 ps 67-1 Deteried by uertaity priiple. For square pulse: t = ps 3-1 fro uertaity relatio Copyright Mihael D. Fayer, 17

26 The shape is a square of zeroth order spherial Bessel futio. t ireases Height of etral lobe ireases, width dereases. Most probability i etral lobe 1 s pulse Width ~.3-1, virtually all probability t * C C Dira delta futio (E=) ; h = (E - E ) Total Probability Area uder urve * 1 * Q si ( Et/ ) h h E C C d C C de de Q h t si d 1 Q A E E Qt Probability liearly proportioal to 4 tie light is applied. Copyright Mihael D. Fayer, 17

27 Sie virtually all probability at E =, evaluate A (i Q) at frequey ( E E )/ Therefore: * CC A t Related to itesity of light as show below. trasitio dipole braket Probability ireases liearly i t. Ca t let C C get too big if tie depedet perturbatio theory used. * Liit by eited state lifetie. Must use other ethods for high power, o-liear eperiets (Chapter 14). Copyright Mihael D. Fayer, 17

28 Have result i ters of vetor poteti Wat i ters of itesity, I. al, A. Poytig vetor: S E B 4 For plae wave: 4 S k A si ( tz/ ) 4 Itesity tie average agitude of Poytig vetor Average si ter over t fro to 1/ Therefore: I ad * CC I t A Liear i itesity. Liear i tie. Copyright Mihael D. Fayer, 17

29 This is the big result. * Liear i itesity. CC I t Liear i tie. Trasitio dipole braket for polarized light. Ca have light with polarizatios, y, or z, i. e., I, I y, I z The: * CC I Iy y Iz z t, y, ad z are the trasitio dipole brakets for light polarized alog, y, ad z, respetively Copyright Mihael D. Fayer, 17

30 Aother defiitio of stregth of radiatio fields radiatio desity: 1 4 ( ) E ( ) average The E ( ) ( ) A A ( ) Isotropi radiatio A A A 1 A 3 y z For isotropi radiatio * CC ( ) y z t 3 Copyright Mihael D. Fayer, 17

31 Eistei B Coeffiiets for absorptio ad stiulated eissio Probability of trasitio takig plae i uit tie (absorptio) for isotropi radiatio where ( ) ( ) 3 B y z trasitio dipole braket Copyright Mihael D. Fayer, 17

32 For eissio (idued, stiulated) everythig is the sae eept keep first epoetial ter i epressio for probability aplitude. E E stiulated eissio eed radiatio field Previously, iitial state alled. Now iitial state, fial state. E > E. i EEh t/ i EEh t/ i e 1 e 1 C AE E E E h E E h Rotatig wave approiatio. Keep this ter. B ( ) B ( ) Eistei B oeffiiet for absorptio equals B oeffiiet for stiulated eissio. Copyright Mihael D. Fayer, 17

33 Restritios o treatet 1. Left out spotaeous eissio. Treatet oly for weak fields 3. Oly for dipole trasitio 4. Treatet applies oly for C C 5. If trasitio dipole brakets all zero * 1 Higher order ters lost whe we took vetor potetial ostat spatially over oleule: Lose Mageti dipole trasitio Eletri Quadrupole Mageti Quadrupole Eletri Otapole et Oly iportat if dipole ter vaishes. Copyright Mihael D. Fayer, 17

34 Eistei A oeffiiet Spotaeous Eissio B B, Eistei B Coeffiiets idued eissio absorptio Wat: A spotaeous eissio oeffiiet N = uber of systes (oleules) i state of eergy E (upper state) N = uber of systes (oleules) i state of eergy E (lower state) At tep T, Boltza law gives: N N e e E E / k T B / k T B e h / k T B Copyright Mihael D. Fayer, 17

35 At equilibriu: rate of dowward trasitios = rate of upward trasitios N A B ( ) N B ( ) spotaeous eissio stiulated eissio absorptio Usig N N e e E E / k T B / k T B e h / k T B e Solvig for ( ) / B ( ) B A B ( ) h k T ( ) h / kbt A e h / kbt B e B Copyright Mihael D. Fayer, 17

36 B B The: ( ) A B h / kbt e 1 Take saple to be blak body, reasoable approiatio. Plak s derivatio (first QM proble) Gives A ( ) 3 8 h B h A 3 h / kbt e 1 Spotaeous eissio o light eessary, I =, 3 depedee. Copyright Mihael D. Fayer, 17

37 Spotaeous Eissio: 3 depedee No spotaeous eissio - NMR 1 8 Hz Optial spotaeous eissio 1 15 Hz Typial optial spotaeous eissio tie, 1 s (1-8 s) NMR 1 15 optial NMR spotaeous eissio tie 1 13 s (>1 5 years). Atually loger, ageti dipole trasitio uh weaker tha optial eletri dipole trasitio. Copyright Mihael D. Fayer, 17

38 Quatu Treatet of Spotaeous Eissio (Briefly) Radiatio Field Photos State of field sae as Haroi Osillator kets Nuber operator aa uber of photos i field Absorptio: a 1 aihilatio operator Reoves photo probability proportioal to braket squared itesity eed photos for absorptio Copyright Mihael D. Fayer, 17

39 Eissio oe ore photo i field a 1 1 reatio operator Probability + 1 whe very large >> 1, Itesity However, for = a 1 1 Still a have eissio fro eited state i absee of radiatio field. QM E-field operator: E i V a i tikr a i tikr k 1/ ( k/ o ) k k ep( k ) ep( k ) k Eve whe o photos, E-field ot zero. Vauu state. All frequeies have E-fields. Flutuatios of vauu state. Fourier opoet at E idues spotaeous eissio. Copyright Mihael D. Fayer, 17