Forming wave packets

We can form wave packets from any of the plane waves found above. For example, 

Hence f is the inverse Fourier transform of the C -valued function /(p) cu+,s(p). We can easily calculate the norm of the wave packet j). By the well-known Fourier-Plancherel theorem, it is just given by the norm of the wave packet in Fourier space. We use (43) and find 

We can apply the Dirac operator Hq to this wave packet. 

The wave packet is thus a solution of the Dirac equation 

An arbitrary wave packet can be written as the inverse Fourier transform of a 4-spinor in momentum space  

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We are going to explain the procedure of forming wave packets out of plane waves for the free Dirac equation. The free stationary Dirac equation Ho tp = Erp has no square-integrable solutions at all. But it turns out that for E > me there are bounded oscillating solutions (here bounded means that the absolute value [ipix, t) of the solution remains below a certain constant M for all x and t). As for the Schrodinger equation, it is comparatively easy to find these solutions. They are similar to plane waves with a fixed momentum (wavelength), that is, they are of the form... 

Plane waves with phase velocity form wave packets with group velocity Vg, such that v Vg = (= 1/eoMo). The phase and group velocities are the same only... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

In this section we represent wave packets as vectors v and refer to the individual components of v in the computer-friendly form of multidimensional arrays with indices, for example, v i,j,k), where the indices i,j, and k refer to the various degrees of freedom. These vectors may be real, such as the real vector V of the previous sections, or they may be complex representations of the... 

Gray and Wozny [101, 102] later disclosed the role of quantum interference in the vibrational predissociation of He Cl2(B, v, n = 0) and Ne Cl2(B, v, = 0) using three-dimensional wave packet calculations. Their results revealed that the high / tail for the VP product distribution of Ne Cl2(B, v ) was consistent with the final-state interactions during predissociation of the complex, while the node at in the He Cl2(B, v )Av = — 1 rotational distribution could only be accounted for through interference effects. They also implemented this model in calculations of the VP from the T-shaped He I C1(B, v = 3, n = 0) intermolecular level forming He+ I C1(B, v = 2) products [101]. The calculated I C1(B, v = 2,/) product state distribution remarkably resembles the distribution obtained by our group, open circles in Fig. 12(b). 

We note that the wave packet (x, t) is the inverse Fourier transform of A k). The mathematical development and properties of Fourier transforms are presented in Appendix B. Equation (1.11) has the form of equation (B.19). According to equation (B.20), the Fourier transform A k) is related to (x, t) by... 

In order to obtain a specific mathematical expression for the wave packet, we need to select some form for the function A k). In our first example we choose A(k) to be the gaussian function... 

A similar uncertainty relation applies to the variables t and o . To show this relation, we write the wave packet (1.11) in the form of equation (B.21)... 

Equation (1.27) relates the wave packet F(x, t) at time t to the wave packet F(x, 0) at time f = 0. However, the angular frequency mfk) is dependent on k and the functional form must be known before we can evaluate the integral over k. 

By way of contrast, recall that in treating the free particle as a wave packet in Chapter 1, we required that the weighting factor A(p) be independent of time and we needed to specify a functional form for A(p) in order to study some of the properties of the wave packet. 

Another popular and convenient way to study the quantum dynamics of a vibrational system is wave packet propagation (Sepulveda and Grossmann, 1996). According to the ideas of Ehrenfest the center of these non-stationary functions follows during a certain time classical paths, thus representing a natural way of establishing the quantum-classical correspondence. In our case the dynamics of wave packets can be calculated quite easily by projection of the initial function into the basis set formed by the stationary eigen-... 

In Fig. 3 we present the stick spectrum [corresponding to infinite resolution see eq. (12)] generated from a Gaussian wave packet of the form... 

Time-Independent Wave Packet Forms of Schrodinger and Lippmann-Schwinger Equations. 

The matter wave function is formed as a coherent superposition of states or a state ensemble, a wave packet. As the phase relationships change the wave packet moves, and spreads, not necessarily in only one direction the localized launch configuration disperses or propagates with the wave packet. The initially localized wave packets evolve like single-molecule trajectories. 

How do we understand and describe this wave-particle duality Clearly a plane wave, A exp[i(kx - cot)], has a well-defined angular frequency, (or energy), and wave vector, (or momentum). But it is infinite in extent, with its intensity, A 2, being uniform everywhere in space. In order to create a localized disturbance we must form a wave packet by superposing plane waves of different wavevectors. Mathematically this is written... 

The use of strong fields to drive the dynamics leads to somehow similar effects than those of ultrafast pulses. If the Rabi frequency or energy of the interaction is much larger than the energy spacing between adjacent vibrational states, a wave packet is formed during the laser action. The same laser can prepare and control the dynamics of the wave packet [2]. Both short time widths and large amplitudes can concur in the experiment. However, the precise manipulation of dynamic observables usually becomes more difficult as the duration of the pulses decreases. 

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