Quantum mechanics interference

The H2O molecules are cooled in a supersonic expansion to a rotational temperature of 10K before photodissociation. The evidence for pathway competition is an odd-even intensity alteration in the OH product state distribution for rotational quantum numbers V = 33 45. This intensity alternation is attributed to quantum mechanical interference due to the N-dependent phase shifts that arise as the population passes through the two different conical intersections. 

By making use of classical or quantum-mechanical interferences, one can use light to control the temporal evolution of nuclear wavepackets in crystals. An appropriately timed sequence of femtosecond light pulses can selectively excite a vibrational mode. The ultimate goal of such optical control is to prepare an extremely nonequilibrium vibrational state in crystals and to drive it into a novel structural and electromagnetic phase. 

From the deflection function we calculate the differential cross section which is needed in equation (1). We note that there could be several different trajectories (two different impact parameters) that produce the same scattering angle, leading to quantum mechanical interference of their nuclear wave functions. We thus... 

Only much later it was realized that the excellent coherence of laser light offers another, maybe much more powerful control parameter, which allows us to make use of quantum mechanical interference. This principle forms the basis of what is today generally referred to as coherent control. 

We begin with a broad-stroke description of the different strategies for using quantum mechanical interference to control product selectivity in a chemical reaction. 

A note of caution must be inserted at this point. It appears, at first sight, that there is a meaning that can be attached to the absolute phase of the field and to the phases of the molecular expectation values. However, it must be remembered that the phase of the molecular quantity is induced by the radiation field prior to the present time. Therefore all phases must be related to the phase of a previous pulse that synchronizes the molecular clock with the field clock. With this synchronization it is possible to understand how quantum mechanical interference between events induced in the past propagates and can be used to control energy and/or population transfer at a later time. 

Under normal conditions for an atom-molecule collision the summation over J extends over many (of the order of one hundred or even more) so-called partial waves which makes the practical calculation rather cumbersome. An even more serious problem is the substantial blurring of distinct dynamical structures such as quantum mechanical interferences or resonances. Since these structures depend parametrically on J, the summation over all possible J values rapidly washes out finer details. 

Because of the lack of quantum mechanical interference effects classical mechanics is well suited for the treatment of direct dissociation. Very few trajectories actually suffice to construct the rotational and the vibrational excitation functions which establish the unique relation between (ro,7o) and (n,j). /(70) and N(ro) are the links between the multi-dimensional PES on one hand and the final state distributions on the other. 

Figure 12.9 depicts a comparison between classical trajectory results and exact close-coupling calculations for He--Cl2 and Ne- -Cl2, respectively. In both cases, the classical procedure reproduces the overall behavior of the final state distributions satisfactorily. Subtle details such as the weak undulations particularly for He are not reproduced, however. As shown by Gray and Wozny (1991), who treated the dissociation of van der Waals molecules in the time-dependent framework, the bimodality for He CI2 is the result of a quantum mechanical interference between two branches of the evolving wavepacket and therefore cannot be obtained in purely classical calculations. 

The decay of the individual quasi-bound (metastable) resonance states follows an exponential law. The wave packet prepared by an ultrashort pulse can be represented as a (coherent) superposition of these states. The decay of the associated norm (i.e., population) follows a multi-exponential law with some superimposed oscillations due to quantum mechanical interference terms. The description given above is confirmed by experimental data. 

While the distinction of the electron trajectories as being either direct or indirect and the observation of quantum mechanical interferences among the trajectories can be understood in terms of the DC electric field strength and the photoelectron kinetic energy with respect to the saddlepoint in the Coulomb + DC electric field potential, this is not the only quantity that characterizes the photoelectrons that are emitted. Above the saddlepoint in the Coulomb + DC potential there exists a continuation of the Stark manifold. This Stark manifold manifests itself in the excitation spectrum of the atom, which shows pronounced peaks in the photoionization efficiency as a func-... 

Many semi-classical and quantum mechanical calculations have been performed on the F + H2 reaction, mainly being restricted to one dimension [520, 521, 602]. The prediction of features due to quantum-mechanical interferences (resonances) dominates many of the calculations. In one semi-classical study [522], it was predicted that the rate coefficient for the reaction F (2P1/2) + H2 is about an order of magnitude smaller than that for F(2P3/2) 4- H2, which lends support to the conclusion [508] that the experimental studies relate solely to the reaction of ground state fluorine atoms. Information theory has been applied to many aspects of the reaction including the rotational energy disposal and branching ratios for F + HD [523, 524] and has been used for transformation of one-dimensional quantum results to three dimensions [150]. Linear surprisal plots occur for F 4- H2(i> = 0), as noted before, but non-linear surprisal plots are noted in calculations for F + H2 (v < 2) [524],... 

The measurement of quantum mechanical interference effects in the velocity dependence of scattering cross-sections provides data which, together with accurate second virial coefficients, yield information on the intermolecular poten-... 

From theoretical discussions involving the molecular eigenstates picture questions have arisen as to whether particular quantum mechanical interference effects can be observed by the use of suitably monochromatic radiation for excitation of the molecules 13>. (See Sect. 7.) Of course, it is also necessary to settle the controversies as to whether the BO or molecular eigenstates are correct, and if the former is indeed correct, which particular version of the BO approximation is to be employed for the calculation of nonradiative decay rates. 

We briefly note in passing that because of the sum over molecular states 9j in (9), there is the possibility that cross-terms may contribute, thereby leading to quantum mechanical interference effects, 3 1S>. 

The conceptual framework underlying the control of the selectivity of product formation in a chemical reaction using ultrashort pulses rests on the proper choice of the time duration and the delay between the pump and the probe (or dump) step or/and their phase, which is based on the exploitation of the coherence properties of the laser radiation due to quantum mechanical interference effects [56, 57, 59, 60, 271]. During the genesis of this field. 

Another strategy for optical control of chemical reaction dynamics has been investigated by Brumer and Shapiro. The simplest version of their scheme relies on quantum mechanical interference between one and three photon absorption pathways to a given final continuum state [16]. Thus a fixed relative optical phase between continuous wave light sources of different colors must be maintained. Some essential features of Brumer and Shapiro s proposed method have been implemented in recent experimental work [17]. 

The first two terms in Eq. (5.13) arise from the cooperative mechanism, while the distributive mechanism gives rise to the third and fourth terms. Deriving the general rate for a proximity-induced two-photon absorption process from the square modulus of the result is an elaborate procedure producing sixteen terms, including cross-terms associated with quantum mechanical interference between the cooperat ve and distributive mechanisms. However, in view of the selection rules discussed earlier, it is not generally necessary to perform this calculation since each of the four specific mechanisms for two-photon absorption under consideration can, at most, have only two terms of Eq. (5.13) contributing to the matrix element. 

The discussion in Section 6.2.1 was based on the assumption of either r = oo or Tjo = 0. When two interacting basis functions have comparable transition probabilities to a common level, 1% I20, then the perturbed intensities, J+0 and / 0, exhibit a form of anomalous behavior that is explicable as a quantum-mechanical interference effect. 

Brumer and Shapiro (1989) proposed a frequency domain control scheme based on quantum mechanical interference between two transition paths. The two paths involve 1-photon and 3-photon transitions where the radiation that excites the 1-photon transition, uq, is generated by frequency tripling the laser... 

Chapters 2, 3, and 5 form the core of this book. Perturbations are defined and simple procedures for evaluating matrix elements of angular momentum operators are presented in Chapter 2. Chapter 3 deals with the troublesome terms in the molecular Hamiltonian that are responsible for perturbations. Particular attention is devoted to the reduction of matrix elements to separately evaluable rotational, vibrational, and electronic factors. Whenever possible the electronic factor is reduced to one- and two-electron orbital matrix elements. The magnitudes and physical interpretations of matrix elements are discussed in Chapter 5. In Chapter 4 the process of reducing spectra to molecular constants and the difficulty of relating empirical-parameters to terms in the exact molecular Hamiltonian are described. Transition intensities, especially quantum mechanical interference effects, are discussed in Chapter 6. Also included in Chapter 6 are examples of experiments that illustrate, sample, or utilize perturbation effects. The phenomena of predissociation and autoionization are forms of perturbation and are discussed in Chapters 7 and 8. 

In summary, Miller s 1970 papers [1, 2] on classical 5-matrix theory had a profound influence on the theory of molecular collisions and related topics such as photodissociation. Following earlier work on elastic scattering, they demonstrated how the results of classical mechanics can be built into a quantum mechanical framework of inelastic collisions. In my view the greatest asset of the classical 5-matrix theory is its interpretative power. The general shape of transition probabilities or collisional cross sections can be easily understood in terms of classical trajectories and their quantum mechanical interference. Exact quantum mechanical programs are like black boxes and the results are often difficult to understand without the help of classical mechanics or semiclassical analyses. The new developments such as the IVR are likely to become major tools for systems consisting of many atoms. 

Rainbow scattering and the quantum mechanical interference of different trajectories... 

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