Teubner, M.

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THE JOURNAL OF CHEMICAL PHYSICS 123, 174107 2005Transition probabilities of a string oscillator subject to impulsive collisionswith a heavy mass pointMax TeubneraMax-Planck-Institut für biophysikalische Chemie, Postfach 2841, 37077 Göttingen, GermanyReceived 1 June 2005; accepted 30 August 2005; published online 2 November 2005Impulsive linear collisions between a string oscillator a one-dimensional particle in a box and amass point are studied quantum mechanically. In the limit of a very heavy mass point whichcorresponds classically to many collisions during a single encounter the transition probabilities aredetermined exactly. The result permits a discussion of the mixed quantum-classical regime wherethe collider becomes almost classical while the oscillator remains quantum mechanical. While theaverage transition probabilities Pm→n are well reproduced by the Ehrenfest mean-fieldapproximation, the prediction for the superimposed high-frequency resonance structure isqualitatively wrong for a genuine quantum oscillator. Only if the oscillator is also almost classicaland if m−n2m, where  is the mass ratio collider/oscillator, this structure is correctlypredicted by the Ehrenfest approximation. © 2005 American Institute of Physics.DOI: 10.1063/1.2084946I. INTRODUCTIONVibrational-translational energy transfer is of central im-portance in understanding reaction dynamics.1,2 The problemremains theoretically challenging, since no simple exactlysolvable model has been found which displays the crucialfeatures of energy transfer. The simplest candidates a har-monic oscillator which interacts with the colliding atom ei-ther by a hard sphere3 or an exponential interaction4 can besolved only numerically.In the vibrational-translational energy transfer the depen-dence of the transition probabilities Prs on two parameters isof particular interest. The first is the oscillator frequency measured with respect to a suitable reference frequency /0 and the second is the ratio =mcollmosc1of collider mass to oscillator mass.While the frequency is generally of great interest, thedependence on frequency becomes trivial for systems withhard-sphere interaction. The reason is that 0 is somethinglike the inverse duration of a collision, which is zero forhard-sphere interactions. For such systems the only non-trivial dependence is on the mass ratio .Small  corresponds to light colliders which onlyweakly perturb the oscillator. This is the domain of Fermi’sgolden rule which is widely used by experimentalists. As increases, the interaction becomes stronger and its durationbecomes longer and longer. For large mass ratios  the in-teraction comprises multiple collisions and the theoreticaltreatment may become rather complicated.From the numerical studies it is well known that thedependence of the Prs on  is oscillatory, i.e., that there is aaElectronic mail: mteubne@gwdg.de0021-9606/2005/12317/174107/5/$22.50 123, 1741Downloaded 23 Mar 2009 to 134.76.223.8. Redistribution subject to sequence of mass ratios n where Prs vanishes see Fig. 2.This is true classically as well as quantum mechanically. Inthe classical picture the quasiperiodic behavior is connectedto the occurrence of multiple collisions during a single en-counter.In order to study the effect of a large mass ratio andclassically multiple collisions on vibrational-translationalenergy transfer, we will choose a model with hard-sphereinteraction.The natural model here is the model of Shuler andZwanzig3 where a harmonic oscillator collides with a masspoint. However, since the numerical procedure suggested byShuler and Zwanzig runs into difficulties at large mass ratios,we will study an even simpler model.The string oscillator a one-dimensional particle in abox is one of the very simplest quantum-mechanical objects.Many years ago Widom5,6 initiated the study of vibrational-translational energy transfer in this model by considering im-pulsive collisions with an impinging mass point.While much is known about the string oscillator in theclassical limit,7–9 the quantum model has only been studiedrarely.10 It is the object of the present paper to show that thismodel can be solved in the limit of large collider mass. Thisopens the possibility of a comparison with the Ehrenfestmean-field approximation in the mixed quantum-classical re-gime.II. THE MODELConsider a particle of mass mosc in a box of length L thestring oscillator. From the right a particle of mass mcoll im-pinges on the oscillator. If R and r are the positions of col-lider and oscillator, respectively, the Hamiltonian reads© 2005 American Institute of Physics07-1AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp174107-2 Max Teubner J. Chem. Phys. 123, 174107 2005H = −22mcoll2R2−22mosc2r2. 2Because the oscillator is confined to the box, the wave func-tion R ,r must satisfyR,0 = R,L = 0. 3The additional boundary conditionr,r = 0 4signifies the hard-core repulsion.Similar to Hanson and Werbelow10 we rescale the coor-dinates by x=mcoll /moscR /L and y=r /L and obtain theHelmholtz equation−2x2 −2y2 = 2 , 5with the dimensionless wave number2 =2moscL2E22. 6The Helmholtz equation Eq. 5 must be satisfied in theregion 0y ,yx tan 	, withtan 	 =moscmcoll= −1/2, 7and must vanish on the boundary see Fig. 1.To the right ofx0 =tan 	8the solution has the formIx,y = e−ikmxmy + n=0Rneiknxny , 9where the ny are the normalized eigenfunctions of theFIG. 1. Equation 5 must be satisfied in the enclosed region and mustvanish on the boundary.oscillatorDownloaded 23 Mar 2009 to 134.76.223.8. Redistribution subject to ny = 2sin ny . 10Here km is the wave vector of the incoming particle whichimpinges on an oscillator initially in state m. Energy conser-vation implieskn2 + n2 = 2. 11Real kn represent outgoing waves which leave the oscillatorin state n. The probability of a transition in which the oscil-lator changes from state m to state n isPm→n =knkmRn2. 12III. METHOD OF SOLUTIONIn the circular segment Or0A, wherer0 =sin 	, 13the general solution in polar coordinates r and  which sat-isfies the boundary conditions has the form10IIr, = n=1AnJn/	 rsinn	. 14Solutions I and II overlap in the region x0r0A where theymust be identical. This may be implemented by demandingthat I and II as well as their normal derivatives coincide ona path  which starts on the x axis between x0 and r0 andends at point A,I = II,  In  =  IIn . 15For  Hanson and Werbelow10 use the line x=x0 and postu-late Eq. 15 on 2n points on this line. This leads to a systemon 2n linear equations for the Rk and Ak which is solvednumerically.An alternative approach is to consider for  the circularsegment r0A. On this path we multiply Eq. 15 bysink /	 and integrate over  from 0 to 	. This leads tothe linear systemAkJk/	r0 = 2	n SknRn + fk , 16AkJk/	 r0 = 2	n TknRn + gk , 17with matricesSkn = 21 eiknr0 cos 	x sinnr0 sin 	xsinkxdx , 180AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp174107-3 Transition probabilities of an oscillator J. Chem. Phys. 123, 174107 2005Tkn = 201eiknr0 cos 	xikn cos	xsinnr0 sin 	x+ n sin	xcosnr0 sin 	xsin kx dx 19and vectorsfk = 201e−ikmr0 cos 	x sinmr0 sin 	xsinkxdx , 20g = 201e−ikmr0 cos 	x− ikm cos	xsinmr0 sin 	x+ m sin	xcosmr0 sin 	xsin kx dx . 21Eliminating the Ak we obtainQS − TR = g − Qf , 22where Q is the diagonal matrix with elementsqk = Jk/	 r0Jk/	r0. 23Truncating Eq. 22 at some finite level n supplies an alter-native procedure for the numerical solution. For the samematrix size the accuracy is much higher than in the methodof Hanson and Werbelow.10 The drawback is, however, thatthe matrices S and T and vectors f and g must be calculatednumerically.IV. TRANSITION PROBABILITY FOR LARGECOLLIDER MASSIf the mass of the collider is much larger than the oscil-lator mass, the angle 	 is very small, and Eq. 22 may besolved by expanding in 	.One finds after a brief calculatione−iknr0Skn = kn + 	Skn1 + O	2 , 24e−iknr0Tkn = iknkn + 	Tkn1 + O	2 , 25whereSkn1 = − ikn01x2 sinnxsinkxdx 26andTkn1 = 2n01x cosnxsinkxdx + iknSkn1 . 27The fk and gk are given byeikmr0fk = km − 	Skm1 + O	2 , 28eikmr0gk = − ikmkm + 	Tkm1 + O	2 , 29where m is again the index of the initial state.Finally, we need the qn for small 	. From Ref. 11 weobtain for small 	qn  un + 	 n + O	2 , 30withDownloaded 23 Mar 2009 to 134.76.223.8. Redistribution subject to un =  ikn,   n− kn tan n, n   , 31provided that r0 stays away from the roots of Jn/	z. n isgiven byn =		2 − n2 − n arccosn − 4. 32We now insert Rk=Rk0+	Rk1 into Eq. 22 and compareterms. At order 0Rk0 = − e−2ikmr0um + ikmum − ikmmk. 33At this order there is only a phase shift without transitions.At the next order one finds after some algebra for nmRn1 = − 1n+m4ie−ikm+knr0+im+nmnm2 − n222kmknsmsn + km2 + kn2cmcnkn	 , 34withsn = sin n, cn = cos n. 35This implies for the transition probabilitiesPm→n =1knkm	 4mnm2 − n222kmknsmsn+ km2 + kn2cmcn2	2 36for nm. In particular, Pm→n= Pn→m as expected from de-tailed balance.Figure 2 shows some transition probabilities Pn→m vs	−1. The full curves are numerically exact values and thebroken curves correspond to Eq. 36. For moderate valuesof  the broken curves approach the true P1→2 even for quitelarge 	 upper half of Fig. 2. For other indices Eq. 36becomes applicable only for rather small 	 lower half ofFIG. 2. Transition probabilities vs 	−1. The full curves are numerically exactvalues and the broken curves correspond to Eq. 36. Upper left: P1→2 for2=6. The dotted curve which ends at 	−1=0.9 is Widom’s approximationRef. 6. Upper right: P1→2 for 2=12. Lower left and right: P1→3 for 2=12.Fig. 2.AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp174107-4 Max Teubner J. Chem. Phys. 123, 174107 2005V. COMPARISON WITH THE EHRENFESTAPPROXIMATIONEquation 36 supplies the exact transition probabilitiesin the limit of large collider mass. Systems where a parttypically an oscillator is studied quantum mechanically,while the rest is assumed to behave classically, have foundmuch interest recently see, for example, Refs. 12 and 13.The string model is an attractive toy to study the mixedquantum-classical dynamics on a very simple but nontrivialmodel. It permits an exact discussion of a model where thelimit of large mass is performed on one, but not both of theparticles. It differs from other similar models in the nonana-lyticity of the interaction potential. As a consequence, theprobability of nonadiabatic transitions decays only algebra-ically instead of exponentially.14,15Consider the string model in the limit of large collidermass. Since the energy is fixed, the collider moves veryslowly. The interaction with the quantum oscillator is onlyvia the boundary conditions which also change very slowly.This suggests to compare the exact result of Eq. 36 with theEhrenfest mean-field approach see, for example, Tully12.If the collider can be regarded as a classical particle, itinteracts with the quantum oscillator only if its position Rtintrudes into the box 0, L where the oscillator lives. Theoscillator then is subject to a moving wall during the timeinterval 0 tT of intrusion where it satisfiesit= −22mosc2r2, 37with the boundary conditions0,t = Rt,t = 0, 0  t  T . 38If initially the oscillator is in state m, adiabaticity indicatesthat it remains in this state during intrusion and thereforer,t = 2RtsinmrRte−ift, 39withft =2moscm220tR−2tdt. 40The function Rt may be found from energy conserva-tionmcoll2Ṙ2 +22moscm22Rt2= E , 41where the second term is the energy eigenvalue of the adia-batic wave function.16 We write the solution in the formRt = Lz	 t , 42where =moscL2 43is the characteristic frequency of the undisturbed string os-cillator andDownloaded 23 Mar 2009 to 134.76.223.8. Redistribution subject to zx = 1 + 2x2 − 2kmx . 44In particular, this implies for the duration of the interactionT =2km2 . 45To determine the transition probabilities we insert theansatz = n2LantsinnrRt46into Eq. 37. This leads toȧk + i2mosck22Rt2ak = n2ncknanṘtRt, 47withckn = 01x cosnxsinkxdx . 48Denoting akt by ak, with =t /, and integrating weobtainakei/2k2w − ak0= n2nckn0 zzanei/2k2wd, 49wherew = 0 dz2. 50On the right side we employ the assumption of adiabaticity.Then only the term with n=m survives and for km,akei/2k2w= 2mckm0 zz3/2ei/2k2−m2wd. 51The Ehrenfest prediction for the transition probability Pm→kEis ak*2, where *=2km /2. It becomes for large Pm→nE = 	 8mnm2 − n22km cos m2−1, 52withm =2n2 − m2w* = 2mn2 − m2 arctankmm.53The Ehrenfest prediction has the correct  dependence re-2 −1call that 	  for large  but differs from the exactAIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp174107-5 Transition probabilities of an oscillator J. Chem. Phys. 123, 174107 2005result of Eq. 36 in the prefactor. In particular, detailed bal-ance is not satisfied. Deviations between numerically exactquantum calculations and the Ehrenfest prediction have alsobeen found for other oscillators.17–19Are there any conditions where the Ehrenfest approxi-mation to the transition probabilities becomes exact? Appar-ently it is not sufficient to let the mass of the collider tend toinfinity.It is well known that large mass is not sufficient for aparticle to be “classical.” From the WKB approximation orthe uncertainty relation xp it is obvious that no par-ticle is classical near turning points. It is also known19 thatthe position uncertainty of the classical particle must besmall if the Ehrenfest approximation is to be valid.In the present case the kinetic energy of the collidershould be large versus the energy of the oscillator. This im-plies kn for all n. Replacing the kn by  one finds thatEqs. 36 and 52 become identical up to rapidly oscillatingterms with frequency  and zero mean. Even if the col-lider is almost classical, the Ehrenfest approximation pre-dicts the transition probabilities only in some averaged sense.It fails to correctly predict the high-frequency resonancestructure for a genuine quantum oscillator.Are there any conditions for a large but finite mass ratio where the Ehrenfest approximation is correct also for thehigh-frequency structure? This will be the case if n−m−m1.A brief calculation yieldsn − m − m 24mm − n2 , 54provided that m ,n. In particular, we must haveDownloaded 23 Mar 2009 to 134.76.223.8. Redistribution subject to m   55and m and n must be large. The Ehrenfest high-frequencystructure is correct only if the oscillator is almost classicaland ifm − n2  m . 561 J. Chesnoy and G. M. Gale, Adv. Chem. Phys. 70, 297 1988.2 J. C. Owrutsky, D. Raftery, and R. M. Hochstrasser, Annu. Rev. Phys.Chem. 45, 519 1994.3 K. E. Shuler and R. Zwanzig, J. Phys. Chem. 33, 1778 1960.4 D. Secrest and B. R. Johnson, J. Chem. Phys. 45, 4556 1966.5 B. Widom, J. Chem. Phys. 28, 918 1955.6 B. Widom, J. Chem. Phys. 30, 238 1959.7 M. P. Hanson and M. Blomme, J. Chem. Phys. 50, 4324 1969.8 H. K. Shin, J. Chem. Phys. 62, 4130 1975.9 D. Gutkowicz-Krusin, J. Chem. Phys. 67, 2830 1977.10 M. P. Hanson and L. G. Werbelow, J. Chem. Phys. 58, 3669 1973.11 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Prod-ucts Academic, New York, 1980.12 J. C. Tully, Faraday Discuss. 110, 407 1998.13 K. Drucker, J. Comput. Phys. 153, 225 1999.14 H. Nakamura, Nonadiabatic Transition World Scientific, Singapore,2002.15 M. Wilkinson and M. A. Morgan, Phys. Rev. A 61, 062104 2000.16 Equation 41 can also be derived using the Hellman-Feynman force.Replacing the region behind the right wall by a constant potential UE, we have RU=−UR2. A short calculation shows that thisbecomes −mosc−1 22m2R−3 in the limit of large U.17 P. Bala, P. Grochowski, B. Lesyng, and J. A. McCammon, in QuantumMechanical Simulation Methods for Studying Biological Systems, LesHouches Workshop, 1996, edited by D. Bicout and M. Field SpringerVerlag, Berlin/Heidelberg, 1996.18 H. J. C. Berendsen and J. Mavri, in Quantum Mechanical SimulationMethods for Studying Biological Systems, Les Houches Workshop, 1996,edited by D. Bicout and M. Field Springer Verlag, Berlin/Heidelberg,1996.19 F. A. Bornemann, P. Nettesheim, and C. Schütte, J. Chem. Phys. 105,1074 1996.AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

Transition probabilities of a string oscillator subject to impulsive collisions with a heavy mass point

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Transition probabilities of a string oscillator subject to impulsive collisions with a heavy mass point

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