The Time-Dependent Schrödinger Equation with applications to The Interaction of Light and Matter and The Selection Rules in Spectroscopy.

Transcription

1 The Time-Dependent Schödinge Equation with applications to The Inteaction of Light and Matte and The Selection Rules in Spectoscopy. Lectue Notes fo Chemisty 45/746 by Macel Nooijen Depatment of Chemisty Univesity of Wateloo. The Time-Dependent Schödinge equation. An impotant postulate in quantum mechanics concens the time-dependence of the wave function. This is govened by the time-dependent Schödinge equation x t ih Ψ (, ) = H$ Ψ ( x, t) (.) whee H $ is the Hamiltonian opeato of the system (the opeato coesponding to the classical expession fo the enegy). This is a fist ode diffeential equation in t, which means that if we specify the wave function at an initial time t 0, the wave function is detemined at all late times. Let me emphasie that this means the wavefunction has to be specified fo all x at initial time t 0. These initial conditions ae familia fom wave equations as discussed in MS Chapte. In classical physics we often deal with second ode diffeential equations and in addition the time deivative Ψ( x, t)/ would then need to be specified fo all x. Let me emphasie hee that although the expeimental esults that can be pedicted fom QM ae statistical in natue, the Schödinge equation that detemines the wave function as a function of time is completely deteministic.

2 ======= Special solutions: Stationay States (only if $ H is time-independent). If we assume that the wave function can be witten as a poduct: Ψ( x, t) = φ( x) γ ( t) we can sepaate the time dependence fom the spatial dependence of the wave function in the usual way. The sepaation constant is called E and will tun out to be the enegy of the system fo such solutions dγ () t... = () () = ( )/ ih E t t e ie t t 0 h γ γ (.) dt H$ φ( x) = Eφ( x) H$ φ ( x) = E φ ( x) (.3) n n n equation (.) is called the time-independent Schödinge equation and plays a cental ole in all of chemisty. Since the opeato H $ is Hemitean the eigenfunctions fom a complete and (can be chosen to be an ) othonomal set of functions. Using these eigenfunctions of H $ special solutions to the time-dependent Schödinge equation can be expessed as ien ( t t0 )/ h Ψ( xt, ) = φ ( xe ) ; Ψ( xt, ) = φ ( x) n 0 n (.4) Fo these special solutions of the Schödinge equation, all measuable popeties ae independent of time. Fo this eason they ae called stationay states. Fo example the pobability distibution but also n Ψ( xt, ) = Ψ( xt, ) = φ ( x), (.5) 0 A$ = A$ A$, (.6) t as is easily veified, by substituting the poduct fom of the wave function. Also the pobabilities to measue an eigenvalue a k ae independent of time, as seen below t0 $ * Aϕk( x) = akϕk( x) Pk( t) = ϕk( x) Ψ( x, t) = Pk( t0 ) (.7) The common element in each of these poofs is that the time-dependent phase facto cancels because we have both Ψ( x, t ) and Ψ * ( xt, ) in each expession. Let me note that the stationay solutions ae detemined by the initial condition. If you stat off with a stationay state at t 0, the wave function is a stationay state fo all time.

3 The geneal solution of the time-dependent Schödinge equation (TDSE) fo timeindependent Hamiltonians can be witten as a time-dependent linea combination of stationay states. If we assume that the initial state is given by Ψ( x, t ) = cφ ( x) + c φ ( x) +... (.8) 0 (it can always be witten in this fashion as the eigenfunctions of H $ fom a complete set), then it is easily veified that ie( t t0)/ h ie( t t0)/ h Ψ( xt, ) = ce φ ( x) + ce φ ( x) +... (.9) satisfies the TDSE and the initial condition. Fo this geneal linea combination of eigenstates of H $ (the geneal case) popeties do depend on time. This is tue fo expectation values and pobabilities, and is due to the fact that diffeent 'components' in the wave function oscillate with diffeent time factos. In calculating expectation values we get coss tems and the time-dependent phase factos do not cancel out. Independent of the initial wave function, enegy is always conseved (as would be expected fom classical physics), and also pobabilities to measue a paticula enegy E k : * iek ( t t )/ 0 h φ k ( x) Ψ( x, t) dx = c k e = c k (.0) Futhe Remaks: - All depends on the initial wave function, which is abitay in pinciple. The most common way is to specify it by means of a measuement! - Stationay states: the inital state is defined to be an eigenfunction of H $. In this case nothing moves except the phase of the wave function. In ou cuent vesion of QM we would find infinite lifetimes of excited states! (This is because the e.m. field is missing fom ou teatment and we have assumed a time-independent H $ ) - In geneal popeties oscillate in time (e.g. electon density) adiation!? Again thee is a need to include e.m. field. In the eal wold, sytems do not satisfy ou timedependent Scödinge equation indefinitely. The system inteacts with the 3

4 electomagnetic field, and in this way makes a tansition to the gound state (eventually). This occus even if no adiation field is pesent (spontaneous emission). These ae the easons that stationay states and in paticula the gound state is so impotant.. Inteaction of Light and Matte. The total Hamiltonian fo a molecule in the pesence of monochomatic adiation is given by H = H0 + H'( t), whee H 0 is the usual molecula Hamiltonian and H'( t) epesents the inteaction with adiation: iωt i t H'( t) = µ E( ω)cos( ω t) = µ E( ω) e + e ω (.) Vey impotantly, this Hamiltonian is explicitly time-dependent and this completely changes the pictue fom the pevious section. Late on we will use that the field is not completely shap in the fequency and we will assume theefoe that the field stength can depend on ω and have a distibution of fequency components. In eqn. () µ is the dipole moment opeato, while E( ω ) is the electic field (a vecto) oscillating at angula fequency ω. (Fo a deivation of this esult see MS poblem 3.49). We will want to use the time-dependent Schödinge Equation. (TDSE). This is a little complicated and we will make the discussion as simple as possible, without losing anything of the essential physics involved. Let us fist eview the time-dependent SE without adiation, also to establish the notation used in this section. We will be using atomic units thoughout. - Case I, no adiation in a -level system (eview). Ψ i H t t = 0Ψ() (.) Let us assume a -level system fo simplicity. The time-independent Schödinge Eqn. has solutions H Φ ( x) = EΦ ( x) 0 H Φ ( x) = E Φ ( x) 0 (.3) 4

5 whee we assume known the enegies and wave functions. Let us assume that the wave function at time t = 0 is given by Ψ( t = 0) = Φ( x) c+ Φ( x) c (.4) whee c and c ae abitay coefficients. Then the wave function at time t is given by Ψ() t Φ ( x) ce ie t ie t = + Φ ( x) c e (.5) Please veify fo youself that this satisfies the TDSE Eqn. (.), assuming (.3). Note that the enegy has units adians/s hee because we have suppessed h. In this case of no adiation we find that if we would measue the enegy of the system we would find E with pobability ie t ce = c, o E with pobability c, independent of time. In paticula excited states would not decay and have infinite lifetimes! Othe popeties would depend on time howeve. Fo long times this pictue is clealy deficient (we have not included spontaneous emission in this desciption which esults in finite lifetimes fo excited states), but it follows fom the QM we taught you sofa. - Case II. Geneal teatment of adiation in -level system. Without loss of geneality we can wite Ψ() t Φ ( x) c () t e ie t ie t = + Φ ( x) c () t e, (.6) whee the coefficients c, c depend on time now. Substituting in the TDSE and using the full Hamiltonian, we should satisfy the eqn: Ψ i H Ψ = 0 (.7) o = c Φ ( x) e + [ Ec ( t) Φ ( x) e H c ( t) Φ ( x) e ] H'( t) c( t) Φ ( x) e + i c Φ ( x) e + [ E c ( t) Φ ( x) e H c ( t) Φ ( x) e ] H'( t) c ( t) Φ ( x) e 0 i iet iet iet iet 0 iet iet iet iet 0 (.8) We note that the tems between squae backets cancel (the eason to wite Ψ( t ) as in eqn..6). We will now integate this Eqn. against Φ ( x ) * and Φ ( x ) * espectively. Futhemoe we assume (fo sake of simplicity) that and 0 * ' * ' Φ ( x) H ( t) Φ ( x) dx = Φ ( x) H ( t) Φ ( x) dx = (.9) 5

6 * ' Φ Φ = * ' ( x) H ( t) ( x) dx Φ( x) H ( t) Φ( x) dx = iωt iωt * (.0) iωt iωt [ e + e ] E( ω ) Φ( x) µ Φ( x) dx V( ω)[ e + e ] Pefoming the integation, we get two eqns, that ae fully equivalent to Eqn..8 Multiplying the fist equation by e ie t and defining E = E E > 0 we obtain c i t e iet V e iωt e iωt c t e iet ( ω )[ + ] ( ) = 0 i t i t ie t c V e e c t e i t e ie t + + ω ω ( ω )[ ] ( ) = 0 t and the second by eie we get c i t i t i E E t i V e e c t e ω + ω ( ) ( ω )[ ] = ( ) 0 c i t i t i E E t i V e e c t e ω + ω ( ) ( ω )[ ] = ( ) 0 c i E t i E t i V e e c t t = ( ω ) + ( ω+ ) ( ω )[ ] ( ) c i E t i E t i V e e c t t = ( ω+ ) + ( ω ) ( ω )[ ] ( ) (.) (.) (.3) In pinciple these equations can be solved faily easily on a compute. To get some futhe insight we assume that we ae always close to esonance (the Boh condition) ω E. Only the slowly oscillating tems matte, as the fast oscillations only povide some fine stuctue on top of the slow oscillations. This is tue in paticula if we have a distibution of fequencies aound ω E (why?). In this case the equations can be appoximated as c i E t i V e c t t = ( ω ) ( ω ) ( ) c i E t i V e c t t = ( ω ) ( ω ) ( ) (.4) It is to be noted that in the fist equation we have the e iωt component of H'( t), while in the second equation we keep the e iωt tem. This is the oigin of the equality of the Einstein coefficients fo absoption and stimulated emission of adiation: if ω E then, of couse, E = E ω. In the following we will look at two impotant appoximations. II-A. Pecise esonance ω = E In this case the phase factos ae pecisely unity, leading to 6

7 and hence we get second ode equations with known solutions c i V c t Vc t t = ( ω ) () () c i V c t Vc t t = ( ω ) () () c c = Vc () t = Vc () t (.5) (.6) c () t = cos( Vt + ϕ); c () t = sin( Vt + ϕ ) (.7) whee ϕ is detemined by the initial values of c, c. Unde these conditions the pobabilities to measue the enegy E o E hence oscillate in time as cos ( Vt ) and sin ( Vt ) espectively. On aveage thee is equal pobability to find E o E, independent of the initial conditions. The oscillation time depends on V, i.e. it is popotional to the stength of the field and to the tansition dipole! It does not depend on the enegy diffeence between the two states o the fequency of the applied field. II-B. Nea esonance, shot times, weak fields. If we assume that at t = 0, Ψ( t = 0) = Φ ( x), i.e. c = ; c = 0, we can assume that c emains moe o less unity and integate the equation fo c (). t This yields τ i( ω E iv ) t ( ω ) i E c( ) i V( ) e dt e 0 i( E ) ( ( ω ) τ τ = ω = ω ) (.8) The pobability to find the enegy E upon measuement at time τ would be c ( τ ) V ( ω ) i E i E e e ( E) ( ( ω ) τ )( ( ω ) τ = ω ) V ( ω ) 4V E E ( E) ( cos( ) ) ( ω ) = ω τ = ( E) sin [ ( ω ) τ ω ω ] sin [ ( ω E) τ] = V ( ω) τ V( ω) τ F( ω) [ ( ω E) τ] (.9) 7

8 The function F( ω ) defined above as the faction is famous (meaning you should know what it looks like), and you can find a sketch of it on page 53 of MS. The function is shaply peaked nea esonance and it is the explanation fo Boh's ule that enegy is only absobed if the fequency of the adiation matches the enegy diffeence between two quantum states. In pactice we do not have one pecise fequency, but a ange o distibution. This can be incopoated by taking an integal ove the fequency sin [ ( ω E ) τ] c( τ) = P( τ) = τ V( ω) dω (.0) [ ( ω E) τ] If we use that V ( ω ) will vay slowly ove the egion nea esonance (whee F( ω ) is lage), and shift vaiables by calling x = ( ω E) τ dω = dx the integal educes to τ sin ( x) P( τ) = τv( E) dx = πτv ( E ) (.) x the ate of tansition is obtained as P = π V ( E) (.) τ The analysis would be completely the same if we would stat fom state Φ ( x ) at t = 0, and we would find that the ate of the tansition to state Φ ( x ) is given by P = π V ( E) (.3) τ hence the ate of absobtion equals the ate of (stimulated) emmission in the pesence of adiation. Moeove this ate is popotional to V ( E) = [ Φ( x) µ Φ( x) dx E( E)] (.4) i.e. popotional to the squae of the tansition dipole moment and the intensity of the adiation (E ) at the esonance fequency. The pecise fomulation would involve a spheical aveaging because molecules and hence the tansition dipoles ae oiented at andom. 8

9 Connection with LASERs (see chapte 5 of MS) A LASER is a macoscopic system and equies both quantum and statistical mechanics fo a pope desciption. We have only discussed QM thus fa. In statistical mechanics we assume we have a (vey) lage numbe of individual quantum systems. Fo shot times we use quantum mechanics on individual systems, but due to still pooly undestood decoheences one assumes that on lage time scales each micosystem is in a definite eigenstates of H 0 (no supepositions!). This mystical phenomenon (even fo the quantum afficionado) is closely elated to the ill undestood poblem of measuement in quantum mechanics. The poblem is the smooth connection of the micoscopic wold (QM) to the macoscopic wold (statistical mechanics). Eithe of these disciplines is well defined methematically but thei inteconnection is cumbesome. Collisions between molecules play a vital ole to move fom supepositions of eigenstates to the themal equilibium of statistical mechanics that assumes systems ae in eigenstates of the Hamiltonian. Anyway, with some handwaving we can move ahead. In the above teatment of quantum mechanics in the pesence of a adiation field we have pecisely developed the shot time behavio. It shows that if we have a system in state Φ the tansition ate to a state Φ that is in esonance though the adiation souce is given by P = Bρ( E) (.5) τ whee B is a constant, and ρ( E ) indicates the density of the adiation at the esonant fequency (popotional to E ). This is fo a single quantum system. If, at time t, we have N( t) molecules in state Φ then the change in population of state Φ due to absobtion of adiation would be dn = Bρ( E) N( t) (.6) dt Molecules in states Φ emit unde the influence of this adiation souce at pecisely the same ate, and this diminishes the population. Hence in total we would get (fo a two-level system) dn = Bρ( E)[ N( t) N( t)] (.7) dt 9

10 The stationay state is eached if dn / dt = 0 o N() t = N() t. This is not coect. We ae missing something, namely spontaneous emission of adiation that would occu fom states Φ even in the absence of esonant adiation. Cuiously, this natually looking phenomenon (finite lifetimes of excited states, even in the absence of adiation) is vey had to deive using QM. The theoy that is needed to accomplish a igoous deivation equies a quantiation of the electomagnetic field (i.e. the intoduction of photons). It is called quantum field theoy. A pactical way to account fo many of the obseved phenomena is to define the pocess of spontaneous emission. It is an appoximation though. We get fo the ate of change dn dt = Bρ( E )[ N ( t) N ( t)] AN ( t) (.8) At the steady state (equilibium) we can solve fo the intensity of the adiation A ρ( E) = BN ( / N ) (.9) Fo a black body at tempeatue T that hypothetically would consists of a two-level system in equilibium with the adiation it geneates and absobs, we know the population atio fom the Maxwell-Boltmann distibution law N N e h / / = ν kt (.30) The above equation fo the adiation density then agees with the black-body adiation law if h A = 8 π 3 Bν (.3) c Einstein essentially knew about statistical mechanics, discete enegy levels, and black body adiation and fom this he deduced the concepts of spontaneous and stimulated emission and the idea of lasing. He was emakable, even fo a genius. 0

11 3. Selection ules in spectoscopy. A igoous teatment of electomagnetic adiation (oscillating field) involves the timedependent Schödinge equation, usually in the fom of time-dependent petubation theoy (see MS and lectue notes in pevious section). At a given instant in time the electic field is moe o less constant ove the egion of a (small) molecule, as the wavelength of the adiation (> ~00 nm) is so lage compaed to molecula dimensions. The elevant quantity that detemines intensities in the spectum is the tansition dipole moment between initial and final states int int Ψ $ f µ Ψi dτ (3.) Hee Ψ int indicates the total intenal wave function involving both nuclea and electonic coodinates, but not the otational pat of the wave function. Similaly $µ = qα α, (3.) α the total dipole moment opeato involves both nuclei and electons (indicated though summation ove α ). In ode to make the poblem managable we distinguish electons: i nomal modes: qi (3.3) molecula otation: R and the oveall intenal wavefunction (disegading otations) can be witten (appoximately) as int Ψav = Ψa ( q ; ) Φv( q), (3.4) The electonic wave function Ψ a ( ; q) depends on all of the electons and in addition thee is a paametic dependence of the electonic wavefunction on the nomal modes (intenal coodinates) q. The vibational pat Φ v ( q) = φ v ( q) φ v ( q )... is assumed a poduct of hamonic oscillato functions fo each nomal mode q i. If we do not use a subscipt we lq i l indicate the whole set of coodinates, e.g. q q = q, q,.... The otational wave function Ω JM, ( θ, ϕ) detemines the pobability distibution of the oientation of the molecule in J space. This otational pat is teated diffeently fom the est. Let us fist discuss the poblem at a fixed oientation. We can wite the oveall tansition moment in (3.) as q

12 * * el el µ = Ψ $ µ Ψ dτ = Φ ( q) Φ ( q) Ψ ( ; q)$ µ Ψ ( ; q) ddq (3.5) fi bw av w v b In ode fo tansitions to occu this dipole moment should be non-eo. Denoting the most complicated integal ove the electonic coodinates as µ ab ( q) this can be witten as bw av w v ab Ψ * $ µ Ψ dτ = Φ * ( q) Φ ( q) µ ( q) dq (3.6) Moeove we can assume a Taylo seies expansion of the q-dependent tansition dipole by witing µ ab µ ab( q) µ ab( qe ) + qi +..., (3.7) q whee the fist tem indicates the electonic tansition dipole moment at the equilibium 0 geomety, which we will denote below as µ = µ ( q ). This is all we need to analyse the vaious cases. i ab ab e i q= qe a A. Pue otational tansitions: a = b, w= v. µ = µ ab q q µ + Φ * ( ) Φ ( )( ) 0 fi v v aa i µ aa i q q dq 0 = i µ 0 aa ( R) is the pemanent dipole moment in the electonic state a and µ fi 0 only if the (3.8) molecule has a so-called pemanent dipole moment. Futhe teatment of the inteaction with the field will lead to the selection ule fo diatomics J = ±, M = 0 (see below). B. Vibational tansitions: a = b, w v. ab T q q q q dq fi = Φ * w( ) Φv( )( 0 µ µ aa + i ) i i (3.9) Due to the othogonality of Φ w and Φ v the integal only yields non-eo if only one nomal mode is excited (say q j ). This means that all factos in the poduct function ( q) ( q) ( q )... stay the same except fo the one involving the nomal mode q j. Φ w = φ w φ w The tansition moment then educes to µ aa µ fi = φ * w( qj) qjφv( qj) dqj (3.0) q j

13 As shown below this will lead to the selection ule v = ±, and only excitations of nomal modes that lead to a change in dipole moment in the electonic state ae allowed (non-eo tansition moment)! Fo example in CO the symmetic stetch cannot be excited (within this appoximation), because the dipole moment emains eo. Howeve the two bending modes and the assymetic stetch can be excited. To conclude this analysis we should pove the selection ules v = ± fo pue vibational tansitions in the pesent (lowest ode, hamonic) appoximation. In ode to pove v = ± let us conside the one-dimensional integal ϕ ( w qq ) ϕ ( v qdq ) (3.) The fom of the vibational wave functions is ϕ ( q) = P( q) e v v αq /, whee Pv( q) is a polynomial in q of ode v. Fom the othogonality of the wave functions we then deduce that n αq qe / ϕ v( q) = 0 n= 0,,..., v (3.) Theefoe ϕw( qq ) ϕv( qdq ) = R S T 0, w< v, w> v+ 0, w= v ( integal is odd) finite, w= v-, w= v+ This poofs the selection ule v = ± fo puely (o)vibational tansitions. (3.3) C. Electonic tansitions a b. We only keep the lead tem, and have to ealie that Φ w coesponds to the nomal modes and equilibium geometies of the final (excited) state while Φ v coesponds to the coesponding modes in the gound state. Keeping only the fist tem in Equation (7) we find 0 final initial µ fi = µ ab Φw ( q) Φv ( q) dq (3.4) and see that the tansition moment depends on the electonic tansition dipole at the initial geomety 0 µ ab and the ovelap of the vibational wave functions in initial and final states. If this tansition moment is eo (often by symmety) the tansition cannot be obseved o only vey weakly though highe ode effects. The vibational ovelaps ae the Fank-Condon factos that we discussed qualitatively in class, and which ae discussed in MS (but you 3

14 should include the phase infomation in thei figues!). The actual calculation is quite involved because the integal does not factoie if the nomal modes ae diffeent in the two electonic states. Discussion of otational selection ules. The oientation of the tansition dipole µ fi depends on the oientation of the molecule in space, in paticula with espect to the electic field. The pobability distibution fo the oientation is detemined by the otational wave function Ω, ( θ, ϕ). These functions ae m pecisely the spheical hamonics, which we have denoted befoe as Y ( θ, ϕ ). To facilitate the discussion let us conside puely otational tansitions, so that µ fi is just the pemanent dipole of the molecule. A classical dipole would stat otating unde influence of an oscillating electic field. The enegy is given by µ E cos θ, whee θ is the angle between the tansition dipole moment and the electic field. In the quantum wold this inteaction leads to tansitions between otational eigenstates: Ω JMJ l ( θ, ϕ) Ω ( θ, ϕ) that ae J, M I, M govened by the by the tansition moment µ E Ω θ ϕ θ θ ϕ θ θ ϕ fi I, M(, )cos ΩJ M d d I, (, )sin (3.5) J As discussed in MS, fo diatomics these matix elements ae non-vanishing only fo J = ±, M = 0. These ae popeties of the spheical hamonics, nothing mysteious. J I 4