Giles Henderson Eastern Illinois University Charleston. IL 61920
I
How a Photon is Created or Absorbed
A student typically encounters the concept of a quantum transition in his first year of chemistry or physics. A transition is usually depicted as a vertical arrow between two quantum states with emphasis on conservation of energy, i.e., the energy of the photon absorbed or emitted must exactly equal the change in energy experienced by the atom or molecule. At this point, the natural questions of the student are, "How is a photon created or absorbed? What is the mechanism of this process and how long does i t take?" The usual instructor response may be that a transition involves a quantum jump which is an instantaneous process and the Uncertainty Prrnciple prohibits us from ohsenring or descrihing in classical terms the details of the transition, or he may evade the question hv claimine the conceots are hevond the scoDe of an introdu&ry couke and w i i he deveioped later in quantum chemistw. After com~letinea Bachelor .uhvsics - or .~hvsical . degree, our student has heenkxposed to aiot of the prescriptive formalism of auantum mechanics with h e a w e m ~ h a s i s on finding eigenv&es and solutions to the time-iidependent Schrodinaer eauation and ~ossiblvmodest exnosure to the time-dependent equation A d peiturbation theory for the purpose of developing transition probabilities. However, to his great disappointment, his freshman questions probably still remain unanswered. By now the complexity and ahstractness of quantum mechanics has either discouraged him from pursuing the answers or convinced him the Uncertainty Principle really does prohibit a conceptual understanding of the process. The stage is set for the cycle to repeat itself for the uocomine eeneration of his students. ~ h state k itiffairs has been greatly influenced by over 40 years of popular belief that since a bound system exhibits only certain discrete energies and a transition from one to another cannot proceed through any observable intermediate levels, then the corresponding wavefunction must also evolve in a similar discontinuous manner. This interpretation has been shown to be incorrect ( I ) . To illustrate the problem, consider a two-state svstem described by the stationary state functions $l(q.t) where q and t correspond the spatial and temporal variables, respectively (see Fig. 1).Schrodinger (2) interpreted the time-dependent state functions as.standing deBroglie matter waves which are solutions to the differential wave equation
I t is perhaps unfortunate that the time dependence has been highly neglected in the traditional undergraduate texts. This is undoubtedlv a conseauence of the im~ortanceof the eigenvalues and probability function to most problems of interest. Since xl(q) and xz(q? are eigenfunctions of the Hamiltonian fie., Hx;(q) = Ejxj(ql) the energy eigenvalues are constants of motion and are stationary or invariant in time. The corresponding probability functions arealso time independent or stationary since Pi = $i*(q,t)$i(q,t) = x,*(q)xi(q). However, it will be shownin this paper that the time dependence of the wavefunction is of crucial importance to understanding the nature of quantum transitions. If IL, rsuch that they. eive . . and il.? . - are of the o. r i.~ esvmmetw .. ;I no]!-7ero transition moment integral (3),then rlectromagnetic radi:~tionwhirh satisfies the Hohr frequency condition
Figure 1. The stationary state energies and expectation value of the energy CwrespoMling to the superposition function of a two state system are depicted above.The time evoiutim of the notmaliredSUDWPOS~~~OII coefficientsare shown during the transition period below
w = (Ez - E d l h can stimulate an absorption or emission transition between these states. The formal description of our system during this period of perturbation is given by alinear combination of the stationan, state functions sometimes called
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where the coefficientsc l and c2 must satisfy the normalization reauirement c12 cq2 = 1.For the sake of simolicitv. we will both ass&el that durinithe transition period (At t l c12and cz2me linear functions of time ( 6 ) ,see Figure 1.Since the superposition function is not an eigenfunction of the Hamiltonian [fl'F(q,t) Z E'P(q,t)] the energy is not a constant of motion during the transition period. Hence, the system is now said to be in a non-stationarv state. Indeed. if an experiment were devised to interrogate the energy of our system during the transition period, only one of two possible answers would be observed, E l or Ez, and the.coefficients el2 and cz2 give the respective probabilities of one or the other result. This has commonly been interpreted to mean that there is an instantaneous quantum jump in the energy at some
+
io)
This assumption is in accord with the usual dipole approximation ~ E ,(flIlz) ~ ~ ~is thetranin whichc2*(t)cz(t)= 4a2/h2 ( , L , ~ ~ )where sition moment integral and Ex0is the maximum electric field strength of x-pol'arizedradiation. A reasonable alternative would be the Rabi approximation (5): e 2 * ( t ) c 2 ( t ) = sin2 Vtnh where V is the amplitude of the perturbation. Volume 56, Number 10, October 7979 1 631
unpredictable time during the transition period. This may in turn suggest that the superposition function is merely a mathematical formalism: if one could observe the evolution of the state it u,ould also exhihit un ahrupt diswntinuity or chance f n m J I to $ 2 or vice versa. Conventional spectruscupic studies are ikapahle of revealing the time dependent properties of non-stationary states because they employ incoherent thermal sources which exhibit completely random phase properties. The first experimental characterization of bulk samples undergoing spectroscopic transitions were obtained from nuclear magnetic resonance observations of the transient nutation effect (7) and spin echoes ( 4 9 )using coherent radiation produced by a single radio frequency oscillator. More; recently, the analogous transient nutation effect ( 1 0 , l l ) and so called "photon echoes" (12-14) have been observed in molecular spectra using pulsed coherent laser radiation. These experiments confirm that there are no "quantum jumps" in the non-stationary state; rather there are smooth, continuous periodic changes in the magnetic and electric properties of a system undergoing a transition. In view of these observations it is clear that the superposition function (eqn. (2)) may be regarded as more than just a formal description; it is indeed real and contains experimentally observable information on the non-stationary, transition species. However, since a superposition function is not an eigenfunction of the Hamiltonian, it is improper to expect that an energy measurement will give an intermediate, time dependent result. The measurement itself will cause the system to change to either its initial or final stationary state with probabilities consistent with c12 and ~22.We can, however, ask for the expectation value of the energy which does change monotonically with time during the transition period (see Fig. 1) ( E ) I= S [ c ~ ( t ) h ( v ,+ t )cz(t)+z(q,t)l*A
X [c~(t)il.~(q,t) + ca(t)+r(q,t)ldq= ci2(t)El+ cz2(t)Ez (3) This result can be interpreted as the energy of an individual atom or molecule at a specified time during its transition period. Of course, neither a single atom nor the energy of an ensemble of transient snecies can be observed directlv. What can he experimentally observed is the distribution of a macroscooic collection of atoms or molecules over the stationarv eigen8tates. Therefore, a different hut equivalent interpretation is that clz, and cz2 may he regarded as the probability of observing El and E z from a single measurement of an ensemble of atoms or molecules a t a specific time during the transition period. ~~~
~
Method It is now very instructive to examine the time dependence of the non-stationary probability function. From our past experience with quantum mechanics, we can anticipate that a physical understanding of a system is most clearly "seen window" (15). In this case we can expect that through the will provide us with a statistical view of the dynamics of nuclear or electronic motion during a transition. David McMillin (16) has recently shown that this approach clearly reveals the origin of an oscillating dipole moment during an electronic transition of a one-electron atom. The probability function is obtained from the superposition wavefunction in the usual manner
vfi
vfi
P(q,t)= **(q,t)B(q.t) = ~ c ~ ( t ) x ~ ( q ) e - ' ~ l * ~ ~
+e ~ ( t ) X ~ ( q ) e - ' h ~ t ~ h ] * [ ~ ~ ( t ) x ~ ( q ) e e e E ~ t ~ h
+ c ~ ( t ) x d q ) e - ~=~clZ(t)x~Yq) ~ ~ ' ~ ] + czZ(t)xz2(q)
+ 2e,(t)o(t)x,(q)xz(4)Cos[(Ez- Edtlhl (4) The first two terms in eqn. (4) vary in time directly with the rate of change in c12 and C Z respectively. ~, However, the last term arises as an interference from the superposition of fil and $2 and exhibits periodic oscillations a t their beat frequency. Since this term is modulated by the product of c l and cz the beat amplitudes will systematically build during the beginning 632 / Journal of Chemical Education
n
1
3 5 7
l
l
i
1
1
1
1
1
1
1
1
1
9 i i 13 15 17 19 21 23 25
FRM ~WERISOS.~O~SE~/FRAIIEI
Figure 2. The contributionof the interference term to me dynamic probability function is governed by the product of the timdependent coefficients. Thus the amplihlde of beat frequency increases during the baginning and decays during theendof the transition period. The time scale given below the figure Is only relevant to the discussion of then = 2 n = 1 transition in h y d r w n (see section Hydrogen Atom in Results and Discussion) and gives the frame number.
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of a transition reaching a maximum when el = cz = 1 1 6 a n d then decay during the end of the transition period (see Fig. 2). It is this interference term which gives rise to charge oscillations precisely in resonance with the electromagnetic radiation absorbed or emitted during the transition. In order to graphically depict the dynamics of typical prohability functions we will again assume that c12and cz2 vary linearly with time (5, 6). Three simple model systems will be considered: the rigid rotor, the harmonic oscillator, and the hydrogen atom. In each case dynamic probahility functions will be computed for transitions from the respective mound state to the first excited state. In reality, the appropriate superposition functions will contain contributions from mixing of higher excited states. However, including these higher order terms unnecessarily complicates the treatment and is of limited value in understanding the fundamental process. For convenience, we will deliberately assume a transition period (At) equal to three times the period of the electromagnetic radiation (7). This is, of course, not realistic for transitions induced by ordinary laboratory source intensities in which At E (lo6-lo7)?, but it is obviously impractical to illustrate 106oscillations. Results and Discussion
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Ricrid Rotor
Equation (4) was evaluated for a rigid n>tmusing normalized sllhrricd harmonic uavefunctions for thestates X I (J =
where PJMI(cos 8 ) are the associated Legendre Polynomials (171, and the molecular orientation is defined by the angles 0 and r$ (see Fig. 3). Three dimensional computer graphics (18) were used to plot the dynamic probability as a function of the spatial and temporal variables (Fig. 4). For this particular case ( M = 0) the functions vary only with@and are constant for all values of 6. On the left side of Figure 4 at time = 0, J = 0, and c2 = 0, the orientational probahility ('P*'P) is constant for all values of 8. This result corresponds to the familiar spherical symmetry of the J = 0 state usually depicted as an infinitesimallv thin snhere of constant radius and is in accord with the Uncertainty Principle, i.e., in the J = 0 stationary state, the aneular momentum is known ~reciselv(L2 = 0). but the spatiarorientation or "position" is uncert% (all vklues of 0 are equally probable). During the transition (0 < time < At and 0 < c2 < 1)the orientational probability surface exhibits ' surtwo distinct "ridges" which periodically cross the @ ,t ime face. We can imagine a maximum prohability (highest eleva-
X/ Figure 3. The spatial orientation of a rigid rotor is defined by the conventional spherical polar coordinates. 0 and 4. tion) journey across the surface, starting a t time = 0 and J = 0 in the left foreeround. which. as we advance in time. takes u s d i d g o ~ ~ a l l y a r rthesurfirrr ~~is t o 0 = ;lain the hackground. This iournr\. cmtinues from this same ~ o i n in t time from 0 = 2sL 0 in the foreground2across the &face again and again until the end of the transition period. We are clearlv obsemina a quantum mwhanirnl, itntistically fnvored tm:&ors dictinr a clockaisr rot,ttion tin lhv direction of increasinr 0 ) . here-is also another exactly symmetrical set of diagonal ridges which cross the surface in the opposite direction describing equally probable counterclockwise rotation. Moreover, the time required for a probability ridge to traverse an angle of B = 2n corresponds to precisely the period of the microwave energy, T. Thus the analysis confirms a resonance condition in which the frequency of the radiation is exactly equal to the rotational frequency of the rigid rotor. If the rotating molecule possesses a permanent dipole moment there is clearly a mechanism for the oscillating electric field comoonent of the microwave enerev to imoart a toraue on the kdecule. The resonance conztion whl insure constant ohase coherence between the oscillatine field and the rotatine hipole. The coupling of the external geld with the rotating d i ~ o l eis maximized when their mutual nhase anele is 7r/2. since this orientation results in maximum torque. The statistical auantum t r a i e c t o ~ can he compared directlv with the dns\ical traj~:ct~,ry shown as diayunsl 1inc.s in the N.rrmc plane rlir~.ctlvbelou, the prdx~l~ilitv surface in Figure 4. 'l'ht! process de.crihed a l w e ran n d i l y he reversed to desrrihc aimulatvd emiss~on( J = 1 J = UI. In this case a photon (an elrrtromaynctic wavr uf tinite dur;ition) is "created" at the expense of mdrculsr nrt;~tional t!~~ergy hy the p e r i d i c rotation of an (.lwtric dipole, not unlike radiowaves created 11ythe periodic tsrillation dchdrge in an anttmnn. The frequency of the radiation is eaunl t~ rhr uneulur freuuenrv of the rl9tor. It mieht also he notLd that the rakation 'ill he"composed of an equal mixture of right and left circularlv components . oolarized . corrkspondingY to equally probable clockwise and k u n t e r clockwise molecular rotations.
a
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Harmonic Oscillator
The methods described above can also he applied to the harmonic oscillator. For this case the wavefunctions used in eqn. (4) for x,(u = 0) and xz(u = 1)are
where [ is the departure from equilibrium hond length: [ = - re), a = 4r2uO/h,andH,([) are the Hermite polynomials (19).A plot of the dynamic probability surface for a
&(r
Figure 4. The dynamic qwnlum trajectories Ot a rigid rotor undergoing a transition between its ground stale(J= 0)and the first excited state ( J = 1) are piottedas a probability surfaceaver the orientation (nand time. The periodof rotation is Shown to equal the period of me interacting microwave radiation (r). The classical trajectories are shown as diagonal lines in the 0. time plane.
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harmonic oscillator undereoine .. . . a transition from u = 1 u = 0 is given ill Figure 5 . In this figure, the s ~ ~ r f w gives c the quantum probabilities of bond lrngths as s function of time where y = r - r, On t h 1 ~ 1 ~side f t uit ht, diagram,ar rime = 0, I, = 0:md CY = 0. thc most prol~i~hle r is the r~ruilihriumb m d length or q = 0. If the system is perturbed h; an oscillating electric field of the correct frequency, w = [E(u = 1) - E ( v = O)]/h, then the ground state wavefunction becomes mixed with the excited state function. The resulting time dependent orohabilitv function clearlv reveals ~ e r i o d i cmolecular vibrations. maximum probihility ( h i b e s t elevation) journey takes us periodically hack and forth tonegative and positive values of q. Indeed, the statistically favored trajectory is an oscillation of hond length a t precisely the same period (7) and frequency (u,,, = T-') as the infrared radiation. An obvious ~rereauisitefor this resonance interaction is an oscillatina kolechar dipole. Again we can correlate the quantum tray jectory with the classical trajectory shown in the q,time plane of Figure 5. In both the quantum and classical description, the a m ~ l i t u d eof the oscillations increase d u r i n ~an absomtion transition as the molecules vibrational energy (or more properly, the expectation value of the molecules vibrational kneigyjincreasei, consistent with the increase in separation of the classical turning points. However, before and after the transition period, the quantum description is very different from the classical description. Classically the molecules continues to oscillate a t a fixed amplitude, amplitude = *[2E(u)/k]'/2and frequency, u = (1/2a)(k/p)1'2 where k and u are the force constant and reduced mass of the oscillator. respectively. In contrast, the quantum rit~scriptionof the srnti~marvstatrs rives a time indeoendent r~rotsnhilitvof hmd length add proviaes no specific details about the dynamics of trajectories.
A
Hydrogen Atom
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The dynamics of the Lyman electronic transition (n = 2 n = 1) for atomic hydrogen will he considered in this section. If we neglect spin, the stationary state wavefunctions of interest include (20)
'Since the pnhhility is shmw on a Cartesian rather th:m s polor rmmlirutr s\st..m, the h = ;2r wirn~atimin the harkxruund 1s equivalent to 0 = 0 orientation in the foreground. Volume 56, Number lo, October 1979 / 633
Figure 5. ThedyMmic quamumbajectwiesof aharmonicoscillatmundergoing a transition between its round state (u = 0) and the first excited state (u = I ) are plotted as a pmbabiliw sxface over q ( W band i e w ) and time. The vibrational period is shown to equal the period of the interactinginfrared radiation ( 7 ) .The classical trajectory is shown as a solid line in the q, time plane.
where ao = 0.5292 A (Bohr radius). Transitions between these levels are governed by the selection rule A1 = f1. Therefore the transition 2s 1s is forbidden while the transition 2p 1s is allowed. The A1 selection rule can be rationalized on the basis of inversion symmetry considerations (3) or on the basis of conservation of angular momentum (21). The temporal behavior of the superposition function can also provide auseful insight to the. origin of the dipole selection rule (16).We will first consider the forbidden 2s 1s transition. Equation (4) was evaluated at regular time intervals corresponding to [E(n = 2) - E(n = l)]Atlh = ?r/4 where At = 5.06 X 10-l7 sec (see Fig. 6)a t a fixed value of r$ to give a cross section of electric charge density in the y,z plane. The charge density was then plotted and photographed on a Tektronix graphics terminal as aconventional probability density dot diagram. The 2s 1s series is depicted in Figure 6. The 1s state appears in the upper left corner and the sequence advances from left to right to the final 2s state in the lower right corner. This series shows the charge density pulsating a t the Bohr frequency, w = [E(2s) E(1s)llfi , ... with charge "tunnelina" back and forth across the 2s nodal surface a
