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A N N A L E N D E R P H Y S I K 7. Folge. Band 48. 1991. Heft 7, S. 423-502
Interference in Phase Space
J. P. DOWLWG, W. P. SCHLEICH Max-Planck-Institut fur Quantenoptik, Garching bei Miinchen
J. A. WHEELER Department of Physics, Princeton University, Princeton, USA
Abstract . A central problem in quantum mechanics is the calculation of the overlap, that is, the scalar product between two quantum states. In the semiclassical limit (Bohrs correspondence principle) we visualize this quantity as the area of overlap between two bands in p h s e space. In the case of more than one overlap the contributing amplitudes have to be combined with a phase differ- ence again determined by an area in phase space. In this s e w the familiar double-slit interference experiment is generalized to an interference in phase space. We derive this concept by the WKB approximation, illustrate it by the example of Franck-Condon transitions in diatomic molecules, and compare it with and contrast it to Wigners concept of pseudo-probabilities in phase space.
Interfcrenz im Phnsenrsum
Inhaltsubersicht. Ein zentrales Problem der Quantenmechanik ist die Berechnung des Skalar- produktes zweier Quantenzustiinde. Daa Bolusche Korrespondenzprinzip erlaubt eine Darstellung der beiden Zustinde ale Bander im Phasenraum. Die den beiden Biindern gemeinsame Flache ist ein Ma13 fur das Skalarprodukt. Die Quadratwunel aus dieser, geeignet normierten, Fliche represen- tiert den Absolutbetrag der korrespondierenden Wahracheinlichkeitsamplitude. Im Falle mehrerer Durchschnittszonen interferieren diem, wobei deren jeweilige Phasendifferenzen ebenfslls durch Phasenraumgebiete gegeben sind. Dies verallgemeinert das Youngsche Doppelspaltexperiment zu Znterferenz im Phasenraum. Wir leiten dieses Konzept mittels der WKB NiSherung ab, erlautern es am Beispiel von Franck-Condon tfbergiingen in diatomischen Molekiilen und vergleichen es mit der Methode der Wignerschen Phasenraumfunktion.
1. From SchrGdingers Wave Function via Hramcrs Phase Space Orbits to Plsnck-Bohr-Sommerleld Bands and Wignor Crests
To look anew at some of the path-breaking papers on quantum theory that appeared in Annulen der Physik in its early days (Fig. 1) is to be reminded of how much these papers taught us. Coming to understand what it is that fixes energy levels was a despe- rate business; this we realize when we look at Plancks courageous but ill-fated attempt [l] to find a rule to fix the energy levels of a system with many degrees of freedom. To e-uplain the electronic, vibrational, or rotational spectrum of a molecule  represented an impossibility until the advent of Schrodingers Undulationsmechanik , giving birth to the Born-Oppenheimer approximation . Ehrenfests paper  on adiabatic inva-
424 Ann. Physik Leipzig 48 (l!J91) 7
Fig. 1. Some seminal articles on the structure of phase space and early quantum mechanics that have appeared in Annulen der Physik.
riants RY quantized quantities - expanding his sudden flash of insight a t the first Solvay Congress - shows us an inspiring glimpse into what quantization is all about. Not one of these problems could be treated with the full confidence of obtaining complete accu- racy, with the methods of the ingenious Bohr-Sommerfeld Atommechanik [6--81 alone - exemplified here by the first page of Sommerfelds original Annulen der Physik article [O]. A fuller understanding of these problems awaited the advent of the approximation method known most briefly as WKB, a stage of development of ideas traced back
J. P. DOWLINQ et al., Interference in Phase Space 425
to Legenclre, Rayleigh and Jeffreys, through Wentzel, Kramers and Brillouin (from which it derives) to Pierce, Klauder and Berry in our own day. They have given this field the name asymptotology [lo].
Asymptotology [ll-201 applies to every field of physics, from atomic and molecular effects [21j, through nuclear phenomena , to modern-day quantum optics with its squeezed state technology [23, 241. Asymptotology normally demands smoothness in the potential or an analogous condition of motion - in brief, it capitalizes on problems where rrature does not jump: natura non facit sa1tuml) . Therefore nothing might seen1 more paradoxical than using asymptotology to evaluate the quantum mechanical probability of a jump in a sudden transition . To do so however is exactly the pur- pose of this paper. Asymptotology allows us to understand jump probability associated with the Framk-Condon effect, that is, with a sudden radiative transition of a molecule from one vibronic state to another [2, 27-30]. In nuclear physics  i t illuminates the coupling between individual particle motion and collective degrees of freedom. In quantum optics it predicts the now eagerly sought oscillations in the photon count pro- bability distribution of a squeezed state of the electromagnetic field [31, 321. Out of these three representative areaa of physics we choose here the Franck Condon effect as the most suited to illustrate how jump probability operates: by interference in phase space [all.
1.1. Franck-Condon Transitions in a Diatomic Molecule
Ever since the pioneering work2) carried out by James Franck 1271 and Edward Condon [28, 291 (Fig. 2) it has been known that certain vibrational levels are preferen- tially excited in the radiative transition of a diatomic molecule from one electronic state. Why? The reason for - and the requirement of - high transition probability is the
I ) The Homebook of Quotations by B. Stevenson (Sew York: Dodd and Mead 1934) attributes this quotation - translated there as Nature does not proceed by leaps - to Carl Linnaeus ( 1 i O i bis 1778), who made this statement in Sec. 77 of his book, Philosophia botanica. Linnaeus or van Linnh was a Swedish botanist who held a chair in Uppsala as a professor of medicine. More about his life can be found in: Dictionmy of 8cientific Biography (edited by C. C. Gillispie). New York: Scribner 1980.
1) To unravel the occupation probability of electronic, vibrational, or rotational states in a di- atomic molecule - on first sight an insurmountable problem - simplifies considerably when me recall the Born-Oppenheimer approximation : The nuclei, due to their large mass move slowly compared to the electrons. Hence the nuclei remain a t their instantaneous position and keep their momenta during an electronic transition, that is, the intermolecular potential changes suddenly. J. Franck, anticipating this yet to be discovered Born-Oppenheimer approximation, and solely based on this notion of a sudden transition, explained  the radiative dissociation of a diatomic molecule as a sudden change of the binding potential from attractive to repulsive - in todays language, a trsn- sition from the vibrational ground state to the continuum. E. Condon recognized [28, 291 that this explanation appliea to more than the ground state. He argued that the jump from a highly excited vibrational state is equally likely for any phase of the vibratory motion of the nucleus. Since the nucleus spends more time a t the turning points of its oscillatory motions) - because there the mo- mentum is zero - the transition will happen preferentially at the turning points. R. Mulliken  extended this principle to the case of jumps taking place at positions different from the turning points. H e postulated that the transition occurs at positions where the kinetic energy of the final orbit is identical to that of the initial one. His dqjeerencc potential, illustrated in Fig. 11, allows one to find these positions. This requirement of conservation of kinetic energy is identical to the crossing con- dition, Eq. (2.7), of the two Kramers orbita.
426 Ann. Physik Leipzig 48 (1991) 7
Fig. 2. James Franck (left) and Max Born (right) in Gottingen in the twenties (left figure). Edward Condon (left) with H. P. Robertson (right) in Princeton in the thirties. (Courtesy of American Institute of Physics, Niels Bohr Center)
identity of the classical turning points [2, 28, 291 for the vibratory niotions in the initial antl final states, as indicated in Fig. 3 a by the vertical, broken line^.^)
Transition probabilities are not zero for vibrational states such as the states m antl n that fail to meet this demand for identity of the turning points. On the contrary, the probability curve is similar [33--381 to a catenary curve - the same curve a rope follows when it is suspended from both ends. This catenary is, however, modulated by a high frequency variation from one quantum state to the next, as shown in Fig. 3b. Nowhere more than in looking at such a striking pattern of intensity variation are we reminded once again of the interference pattern obtained in the familiar Young double-slit experi- ment 139-411. Interference? Yes! But interference where ? What and where are the two slits in the Franck-Condon problem ?
Is the standard formalism of quantum mechanics  of any immediate help in answering these questions ? This formalism restates  the probability for a transition to occur from the nth vibrational state of a diatomic molecule in one electronic state to the mtli vibrational level of a different electronic state, as
wm,n = I%,nla 9 (1.la)
3) We recall here the familiar parable of the turning point. The four-year old unexpectedly sweeps the aim of the lawn hose to the right - raking the jet of water over the circle of unwary admirers - then again to the left, before he drops it. The turning point of the sweeping motion sepa- rates the wettest victim from her dry neighbor. What a mess. Wetness in this incident models posi- tion probability for a particle reflected at a turning point.
J. P. DOWLING e t al., In