Wavepackets quantum mechanical

Note that equation (A3.11.1881 includes a quantum mechanical trace, which implies a sum over states. The states used for this evaluation are arbitrary as long as they form a complete set and many choices have been considered in recent work. Much of this work has been based on wavepackets [46] or grid point basis frmctions [47]. 

Marquardt R, Quack M, Stohner J and Sutcliffe E 1986 Quantum-mechanical wavepacket dynamics of the CH group in the symmetric top XgCH compounds using effective Hamiltonians from high-resolution spectroscopy J. Chem. Soc., Faraday Trans. 2 82 1173-87... 

Information about critical points on the PES is useful in building up a picture of what is important in a particular reaction. In some cases, usually themially activated processes, it may even be enough to describe the mechanism behind a reaction. However, for many real systems dynamical effects will be important, and the MEP may be misleading. This is particularly true in non-adiabatic systems, where quantum mechanical effects play a large role. For example, the spread of energies in an excited wavepacket may mean that the system finds an intersection away from the minimum energy point, and crosses there. It is for this reason that molecular dynamics is also required for a full characterization of the system of interest. 

In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, so-called as only a portion of the FES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schtddinger equation should, however, be kept in mind. 

In what is called BO MD, the nuclear wavepacket is simulated by a swarm of trajectories. We emphasize here that this does not necessarily mean that the nuclei are being treated classically. The difference is in the chosen initial conditions. A fully classical treatment takes the initial positions and momenta from a classical ensemble. The use of quantum mechanical distributions instead leads to a seraiclassical simulation. The important topic of choosing initial conditions is the subject of Section II.C. 

The time-dependent Schrddinger equation governs the evolution of a quantum mechanical system from an initial wavepacket. In the case of a semiclassical simulation, this wavepacket must be translated into a set of initial positions and momenta for the pseudoparticles. What the initial wavepacket is depends on the process being studied. This may either be a physically defined situation, such as a molecular beam experiment in which the paiticles are defined in particular quantum states moving relative to one another, or a theoretically defined situation suitable for a mechanistic study of the type what would happen if. .. 

The big advantage of the Gaussian wavepacket method over the swarm of trajectory approach is that a wave function is being used, which can be easily manipulated to obtain quantum mechanical information such as the spechum, or reaction cross-sections. The initial Gaussian wave packet is chosen so that it... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

One can also ask about the relationship of the FMS method, as opposed to AIMS, with other wavepacket and semiclassical nonadiabatic dynamics methods. We first compare FMS to previous methods in cases where there is no spawning, and then proceed to compare with previous methods for nonadiabatic dynamics. We stress that we have always allowed for spawning in our applications of the method, and indeed the whole point of the FMS method is to address problems where localized nuclear quantum mechanical effects are important. Nevertheless, it is useful to place the method in context by asking how it relates to previous methods in the absence of its adaptive basis set character. There have been many attempts to use Gaussian basis functions in wavepacket dynamics, and we cannot mention all of these. Instead, we limit ourselves to those methods that we feel are most closely related to FMS, with apologies to those that are not included. A nice review that covers some of the... 

How well do these quantum-semiclassical methods work in describing the dynamics of non-adiabatic systems There are two sources of errors, one due to the approximations in the methods themselves, and the other due to errors in their application, for example, lack of convergence. For example, an obvious source of error in surface hopping and Ehrenfest dynamics is that coherence effects due to the phases of the nuclear wavepackets on the different surfaces are not included. This information is important for the description of short-time (few femtoseconds) quantum mechanical effects. For longer timescales, however, this loss of information should be less of a problem as dephasing washes out this information. Note that surface hopping should be run in an adiabatic representation, whereas the other methods show no preference for diabatic or adiabatic. 

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. 

Irrespective of whether the photon is considered as a plane wave or a wavepacket of narrow radial extension, it must thus be divided into two parts that pass each aperture. In both cases interference occurs at a particular point on the screen. When leading to total cancellation by interference at such a point, for both models one would be faced with the apparently paradoxical result that the photon then destroys itself and its energy hv. A way out of this contradiction is to interpret the dark parts of the interference pattern as regions of forbidden transitions, as determined by the conservation of energy and related to zero probability of the quantum-mechanical wavefunction. 

Since, in this causal model, the extended wave 0 represents a real physical finite wave with well-defined energy, it seems natural to represent it by a suitable mathematical form. At the time when de Broglie put forth his causal interpretation of quantum mechanics, it was necessary for him to construct a finite wave using the Fourier analysis, namely, the multiplicity of harmonic plane waves, infinite in space and time, summing up and giving origin to a wavepacket. 

The problem of the limitless spreading of matter wavepackets leading to a continuous increase in size as the wavepacket travels in space has always been a source of discomfort. The mathematical expression describing this spread-increase as a function of time can be found in any textbook of quantum mechanics [23]. It has the form... 

Similar transient signals were obtained from time-dependent quantum mechanical calculations performed by Meier and Engel, which well reproduce the observed behavior [49]. They show that for different laser field strengths the electronic states involved in the multiphoton ionization (MPI) are differently populated in Rabi-type processes. In Fig. 13 the population in the neutral electronic states is calculated during interaction of the molecule with 60-fs pulses at 618 nm. For lower intensities the A state is preferentially populated by the pump pulse, and the A state wavepacket dominates the transient Na2+ signal. However, for the higher intensities used in the... 

M. S. Child The comments of Brumer and Quack raise questions about the correspondence between classical and quantum mechanics. In this connection one must first recognize that the classical analogue of a wavepacket is not a single particle but an ensemble. Second, the... 

---