5.74 Introductory Quantum Mechanics II

Transcription

1 MIT OpenCourseWare Introductory Quantum Mechancs II Sprng 9 For nformaton about ctng these materals or our Terms of Use, vst:

2 INTERACTION OF LIGHT WITH MATTER One of the most mportant topcs n tme-dependent quantum mechancs for chemsts s the descrpton of spectroscopy, whch refers to the study of matter through ts nteracton wth lght felds (electromagnetc radaton). Classcally, lght-matter nteractons are a result of an oscllatng electromagnetc feld resonantly nteractng wth charged partcles. Quantum mechancally, lght felds wll act to couple quantum states of the matter, as we have dscussed earler. Le every other problem, our startng pont s to derve a Hamltonan for the lght-matter nteracton, whch n the most general sense would be of the form H = H + M H + L H. (4.1) LM The Hamltonan for the matter H M s generally (although not necessarly) tme ndependent, whereas the electromagnetc feld H L and ts nteracton wth the matter H LM are tme-dependent. A quantum mechancal treatment of the lght would descrbe the lght n terms of photons for dfferent modes of electromagnetc radaton, whch we wll descrbe later. We wll start wth a common semclasscal treatment of the problem. For ths approach we treat the matter quantum mechancally, and treat the feld classcally. For the feld we assume that the lght only presents a tme-dependent nteracton potental that acts on the matter, but the matter doesn t nfluence the lght. (Quantum mechancal energy conservaton says that we expect that the change n the matter to rase the quantum state of the system and annhlate a photon from the feld. We won t deal wth ths rght now). We are just nterested n the effect that the lght has on the matter. In that case, we can really gnore H L, and we have a Hamltonan that can be solved n the nteracton pcture representaton: H H H t M LM + ( ) + V () = H t (4.) Here, we ll derve the Hamltonan for the lght-matter nteracton, the Electrc Dpole Hamltonan. It s obtaned by startng wth the force experenced by a charged partcle n an electromagnetc feld, developng a classcal Hamltonan for ths system, and then substtutng quantum operators for the matter: Andre Tomaoff, MIT Department of Chemstry, /7/8

3 4- p ˆ (4.3) x xˆ In order to get the classcal Hamltonan, we need to wor through two steps: (1) We need to descrbe electromagnetc felds, specfcally n terms of a vector potental, and () we need to descrbe how the electromagnetc feld nteracts wth charged partcles. Bref summary of electrodynamcs Let s summarze the descrpton of electromagnetc felds that we wll use. A dervaton of the plane wave solutons to the electrc and magnetc felds and vector potental s descrbed n the appendx. Also, t s helpful to revew ths materal n Jacson 1 or Cohen-Tannoudj, et al. > Maxwell s Equatons descrbe electrc and magnetc felds( E, B ). > To construct a Hamltonan, we must descrbe the tme-dependent nteracton potental (rather than a feld). > To construct the potental representaton of E and B, you need a vector potental A( r, t) and a scalar potentalϕ (r, t). For electrostatcs we normally thn of the feld beng related to the electrostatc potental through E = ϕ, but for a feld that vares n tme and n space, the electrodynamc potental must be expressed n terms of both A andϕ. > In general an electromagnetc wave wrtten n terms of the electrc and magnetc felds requres 6 varables (the x,y, and z components of E and B). Ths s an overdetermned problem; Maxwell s equatons constran these. The potental representaton has four varables ( A x, Ay, A z andϕ ), but these are stll not unquely determned. We choose a constrant a representaton or guage that allows us to unquely descrbe the wave. Choosng a gauge such that ϕ = (Coulomb gauge) leads to a plane-wave descrpton of E and B : 1 A r t A( r, t) (, ) + = (4.4) c t A = (4.5) 1 Jacson, J. D. Classcal Electrodynamcs (John Wley and Sons, New Yor, 1975). Cohen-Tannoudj, C., Du, B. & Lalöe, F. Quantum Mechancs (Wley-Interscence, Pars, 1977), Appendx III.

4 4-3 Ths wave equaton allows the vector potental to be wrtten as a set of plane waves: ( ) r, = A εˆ e ( A r t t) + A * εˆ e ( t). (4.6) r Ths descrbes the wave oscllatng n tme at an angular frequency and propagatng n space n the drecton along the wavevector, wth a spatal perod λ = π. The wave has an ampltude A whch s drected along the polarzaton unt vector εˆ. Snce A =, we see that εˆ = or ε ˆ. From the vector potental we can obtan E and B A E = t = A ˆ e ( (4.7) t) r e ( t) ε r B = A = εˆ A e r ( ) r e t ) (4.8) ( ) t ( If we defne a unt vector along the magnetc feld polarzaton as bˆ = ( εˆ) = ˆ εˆ, we see that the wavevector, the electrc feld polarzaton and magnetc feld polarzaton are mutually orthogonal ˆ εˆ bˆ. Also, by comparng eq. (4.6) and (4.7) we see that the vector potental oscllates as cos(t), whereas the feld oscllates as sn(t). If we defne then, Note, E B = = c. 1 E = A (4.9) 1 B = A (4.1) E( r, t ) = E εˆ sn ( r t) (4.11) B (r, t) = B bˆsn ( t) r. (4.1)

5 4-4 Classcal Hamltonan for radaton feld nteractng wth charged partcle Now, let s fnd a classcal Hamltonan that descrbes charged partcles n a feld n terms of the vector potental. Start wth Lorentz force 3 on a partcle wth charge q: F = q ( E + v B ). (4.13) Here v s the velocty of the partcle. Wrtng ths for one drecton (x) n terms of the Cartesan components of E, v and B, we have: F = q E + v B v B. (4.14) x ( x y z z y In Lagrangan mechancs, ths force can be expressed n terms of the total potental energy U as ) U d U F x = + (4.15) x dt vx Usng the relatonshps that descrbe E and B n terms of A andϕ, nsertng nto eq. (4.14), and worng t nto the form of eq. (4.15), we can show that: U = q ϕ qv A (4.16) Ths s derved n CTDL, 4 and you can confrm by replacng t nto eq. (4.15). Now we can wrte a Lagrangan n terms of the netc and potental energy of the partcle The classcal Hamltonan s related to the Lagrangan as L= T U (4.17) L = 1 mv + qv A q ϕ (4.18) H = p v L = p v 1 mv q v A qϕ (4.19) L Recognzng p = = mv + qa (4.) v we wrte v = 1 m ( p q A ). (4.1) Now substtutng (4.1) nto (4.19), we have: 3 See Schatz and Ratner, p Cohen-Tannoudj, et al. app. III, p. 149.

6 4-5 = 1 A 1 A q H p p q p q m p qa A+ qϕ (4.) m ( ) m ( ) ( ) 1 H = m p qa( r,t) + q ϕ ( r, t) (4.3) Ths s the classcal Hamltonan for a partcle n an electromagnetc feld. In the Coulomb gauge (ϕ = ), the last term s dropped. We can wrte a Hamltonan for a collecton of partcles n the absence of a external feld and n the presence of the EM feld: = p H + V ( r ) m. (4.4) H = Expandng: H = H 1 p q A( r ) + V r m ( ) q m ( + ( ). (4.5) q m p A A p ) + A (4.6) Generally the last term s consdered small compared to the cross term. Ths term should be consdered for extremely hgh feld strength, whch s nonperturbatve and sgnfcantly dstorts the potental bndng molecules together. One can estmate that ths would start to play a role at ntensty levels >1 15 W/cm, whch may be observed for very hgh energy and tghtly focused pulsed femtosecond lasers. So, for wea felds we have an expresson that maps drectly onto solutons we can formulate n the nteracton pcture: H = H + V ( t) (4.7) q V () t = ( p A A p. (4.8) m + ) Quantum mechancal Electrc Dpole Hamltonan Now we are n a poston to substtute the quantum mechancal momentum for the classcal. Here the vector potental remans classcal, and only modulates the nteracton strength.

7 4-6 p = (4.9) V t ()= q ( A + A m ) (4.3) We can show that A A. Notce = A + A (chan rule). For nstance, f we are = A ( ) ( ) operatng on a wavefuncton A ψ = A ψ + A ( ψ ). The frst term s zero snce we are worng n the Coulomb gauge( A = ). Now we have: V t ()= q A m (4.31) = q A p For a sngle charge partcle our nteracton Hamltonan s V ()= t q A p m m r t = q A εˆ p e ( (4.3) ) + c.c. m Under most crcumstances, we can neglect the wavevector dependence of the nteracton potental. If the wavelength of the feld s much larger than the molecular dmenson(λ ) ( r ), then e 1. Ths s nown as the electrc dpole approxmaton. We do retan the spatal dependence for certan types of lght-matter nteractons. In that case we defne r as the center of mass of a molecule and expand e = e e ( r ) r r r (4.33) = e r 1 + (r r ) + For nteractons, wth UV, vsble, and nfrared (but not X-ray) radaton, r r = means that e r r <<1, and settng 1. We retan the second term for quadrupole transtons: charge dstrbuton nteractng wth gradent of electrc feld and magnetc dpole. Now, usng A = E, we wrte (4.3) as qe V t t + t ()= m εˆ pe εˆ pe (4.34)

8 4-7 V ()= t qe (εˆ p)snt m = q (Et () p ) m or for a collecton of charged partcles (molecules): (4.35) Ths s nown as the electrc dpole Hamltonan (EDH). V t ()= q (εˆ p ) E sn t (4.36) m Transton dpole matrx elements We are seeng to use ths Hamltonan to evaluate the transton rates nduced by V(t) from our frst-order perturbaton theory expresson. For a perturbaton V ( t) = V sn t the rate of transtons nduced by feld s π w = V δ (E E ) + δ (E E + ) (4.37) Now we evaluate the matrx elements of the EDH n the egenstates for H : V = V qe = εˆ p (4.38) m We can evaluate the matrx element p usng an expresson that holds for any one-partcle Hamltonan: Ths expresson gves So we have m p = [r, H ] = p. (4.39) m rh Hr = m ( r E E r ) (4.4) = m r. V = qe εˆ r (4.41)

9 4-8 or for a collecton of partcles V = E εˆ q r = E ˆ ε μ (4.4) = E μ l μ s the dpole operator, and μ l s the transton dpole matrx element. We can see that t s the quantum analog of the classcal dpole moment, whch descrbes the dstrbuton of charge densty ρ n the molecule: μ = dr r ρ (r ). (4.43) These expressons allow us to wrte n smplfed form the well nown nteracton potental for a dpole n a feld: V ( t) = μ E() t (4.44) Then the rate of transtons between quantum states nduced by the electrc feld s π E μ l π E μ l w = δ (E E ) = ( ) +δ δ ( E E + + ) ( +) (4.45) Equaton (4.45) s an expresson for the absorpton spectrum snce the rate of transtons can be related to the power absorbed from the feld. More generally we would express the absorpton spectrum n terms of a sum over all ntal and fnal states, the egenstates of H : w f = π E μ f δ ( f ) + δ (, f + f ) (4.46) The strength of nteracton between lght and matter s gven by the matrx element μ f f με ˆ. The scalar part f μ says that you need a change of charge dstrbuton between f and to get effectve absorpton. Ths matrx element s the bass of selecton rules based on the symmetry of the states. The vector part says that the lght feld must project onto the dpole moment. Ths allows nformaton to be obtaned on the orentaton of molecules, and forms the bass of rotatonal transtons.

10 4-9 Relaxaton Leads to Lne-broadenng Let s combne the results from the last two lectures, and descrbe absorpton to a state that s coupled to a contnuum. What happens to the probablty of absorpton f the excted state decays exponentally? relaxes exponentally... for nstance by couplng to contnuum P exp [ wn t ] We can start wth the frst-order expresson: t b = d τ V t (4.47) t or equvalently b = e t V t ( ) t (4.48) We can add rreversble relaxaton to the descrpton of b, followng our early approach: Or usng V t ()= E μ snt b = e t V t () t w n b (4.49) b = e t snt V w n b t = E () t e ( +) e ( )t μ w n b () t (4.5) The soluton to the dfferental equaton α t y + ay = be (4.51) s y t α ()= Ae at + be t. (4.5) a+ α

11 4-1 b E μ e ( +)t e ( )t n ()= t Ae w t/ + w / + ( + ) n w n / + ( ) (4.53) Let s loo at absorpton only, n the long tme lmt: b t ()= E For whch the probablty of transton to s ( )t μ e w n / (4.54) E μ b P = = 4 1 ( ) + w n /4 (4.55) The frequency dependence of the transton probablty has a Lorentzan form: The lnewdth s related to the relaxaton rate from nto the contnuum n. Also the lnewdth s related to the system rather than the manner n whch we ntroduced the perturbaton.