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 (Bohr's 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 difference again determined by an area in phase space. I n 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 Wigner's concept of pseudo-probabilities in phase space. 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 [22], to modern-day quantum optics with its squeezed state technology [23, 241. Asymptotology normally demands smoothness inthe potential or an analogous condition of motionin brief, it capitalizes on problems where rrature does not jump: "natura non facit sa1tum"l) [25]. Therefore nothing might seen1 more paradoxical than using asymptotology to evaluate the quantum mechanical probability of a jump in a sudden transition [26]. To do so however is exactly the purpose 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]. I n nuclear physics [22] it 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 probability 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 t o illustrate how jump probability operates: by interference in phase space [all.
Franck-Condon Transitions in a Diatomic MoleculeEver 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 preferentially excited in the radiative transition of a diatomic molecule from one electronic state.Why? The reason forand the requirement ofhigh transition probability is the I ) The Homebook of Quotations by B. Stevenson (Sew York: Dodd and Mead 1934) attributes this quotationtranslated 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. M...