The Wave Packet

Once assumed (certified) the undulatory shape of the quantum particle, with simple yet general form 

Resolving this anomaly comes from the removal of the limitation of die undulatory representation by a single wave - considering the representation by a wave packet seen as a integral convolution 

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

In terms of pulsation (the other wavy size but also the quantification size), this is considered as varying slightly (in the first order) comparing to its equilibrium value inside the packet, while the amplitude in mutual space (of wave vector) is considered almost the same for all waves from the packet. 

Under these conditions the de Broglie wave packet is sequentially given as 

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In time-dependent quantum mechanics, vibrational motion may be described as the motion of the wave packet... 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

In view of the foregoing discussion, one might ask what is a typical time evolution of the wave packet for the isolated molecule, what are typical tune scales and, if initial conditions are such that an entire energy shell participates, does the wave packet resulting from the coherent dynamics look like a microcanonical... 

Marquardt R and Quack M 1991 The wave packet motion and intramolecular vibrational... 

Molecular spectroscopy offers a fiindamental approach to intramolecular processes [18, 94]. The spectral analysis in temis of detailed quantum mechanical models in principle provides the complete infomiation about the wave-packet dynamics on a level of detail not easily accessible by time-resolved teclmiques. 

Problems arise if a light pulse of finite duration is used. Here, different frequencies of the wave packet are excited at different times as the laser pulse passes, and thus begin to move on the upper surface at different times, with resulting interference. In such situations, for example, simulations of femtochemistry experiments, a realistic simulation must include the light field explicitely [1]. 

Consider the wave packet populating just one vibrational level. This occurs for only a short period of time (the length of the femtosecond pulse). Then we can think of vibration occurring in a classical fashion. The wave packet travels along the vibrational level until it reaches the other extremity when it may be reflected and continue to travel backwards and forwards along the level. Because of the strongly anharmonic nature of the vibration the wave packet is broadened, as shown, as r increases. 

However, because of the avoided crossing of the potential energy curves the wave functions of Vq and Fi are mixed, very strongly at r = 6.93 A and less strongly on either side. Consequently, when the wave packet reaches the high r limit of the vibrational level there is a chance that the wave function will take on sufficient of the character of Na + 1 that neutral sodium (or iodine) atoms may be detected. 

Figure 4, Control of wave packet dynamics in an optimal 5-well structure. The parameters of the structure are designed to maximize tunneling. The wave packet at the target time (solid line) has a large overlap with the target (dashed line).

As the number of eigenstates available for coherent coupling increases, the dynamical behavior of the system becomes considerably more complex, and issues such as Coulomb interactions become more important. For example, over how many wells can the wave packet survive, if the holes remain locked in place If the holes become mobile, how will that affect the wave packet and, correspondingly, its controllability The contribution of excitons to the experimental signal must also be included [34], as well as the effects of the superposition of hole states created during the excitation process. These questions are currently under active investigation. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

Absorption of wave packet amplitude that approaches the edges of the computational grid can be accomplished by periodically damping out the wave packet in small regions at the grid edges [10]. Suppose A represents this (real) operation. Instead of Eq. (2), one has... 

There are two issues that may be confusing in the development above. The first issue, which applies to any/(H), including simply/(H) = H, is how to obtain correct scattering dynamics information if only the real part of the wave packet is available. The second issue is the relation of the wave packet dynamics generated by the/(H) of choice in the RWP method, Eq. (16), to standard wave packet dynamics generated by H. That is, can % u) be related to T(f) in a more explicit manner than in the discussion revolving around Eqs. (13) and (14) ... 

Six-dimensional, numerically accurate four-atom wave packet calculations were pioneered by Zhang and Zhang [17] and Neuhauser [18]. While numerous details of the present RWP implementation differ from these earlier approaches, it should be noted that many of the general ideas remain the same. In applications, finite-sized grids and basis sets are introduced to describe the wave packet, and... 

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