The time-dependent view

Inserting (6.9) into (4.8) yields for the short-time behavior of the wave-packet 

This derivation contains several interesting aspects which are rather general and which are valuable for the overall understanding of dissociation dynamics  

Within the short-time approximation, the center of the wavepacket remains at Re while its center in momentum space, V t, moves outward with constant velocity Vr = —dV/dR. 

The steeper the potential the faster the autocorrelation function diminishes and the broader is the spectrum. The width of S(t) is related to the breadth of the spectrum by 

---

Which approach offers the better interpretation of the photodissociation dynamics in general and of the dissociation cross sections in particular The answer to this question depends very much on the particular system and cannot be decided in a unique way. There are cases which are better described by the time-independent picture while other systems may be better understood within the time-dependent view. In the course of this monograph we will always attempt to combine both views in order to reveal all facets of a particular system. 

The classical picture of photodissociation closely resembles the time-dependent view. The electronic transition from the ground to the excited electronic state is assumed to take place instantaneously so that the internal coordinates and corresponding momenta of the parent molecule remain unchanged during the excitation step (vertical transition). After the molecule is promoted to the potential energy surface (PES) of the upper electronic state it starts to move subject to the classical equations... 

L. The time-dependent view of spectroscopy (Heller, 1981). Suppose that at time t = 0 we make a Franck ondon transition to an upper electronic state. This takes the initial vibrational wave function up in energy and onto a different potential for the motion of the nuclei. This initial state is no longer stationary and it starts evolving in time. The first thing that will happen is tiiat the wave function will depart from the Franck-Condon region. What we want to know is how quickly it will do so. If the molecule is a diatomic, the initial state will periodically revive. But for a polyatomic, IVR will, over time, reduce tiie revival. Hence for a polyatomic we also want to know where the wave-packet goes to. This... 

The temporal width ATq is determined by the speed with which the wave packet moves out of the FC position and therefore by the slope of the PES near the FC point the steeper the potential, the more rapidly the wave packet leaves its place of birth, the broader is the spectrum and vice versa. The maximum of the spectrum is determined by the vertical excitation energy, i.e the difference between the energy of the initial state and the (potential) energy in the upper state evaluated at the ground-state equilibrium. In the time-dependent view the general shape of the absorption spectrum can be explained by the reflection principle. ... 

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 a subsequent treatment from the time-dependent response point of view, connection with the Greens function... 

With time-dependent computer simulation and visualization we can give the novices to QM a direct mind s eye view of many elementary processes. The simulations can include interactive modes where the students can apply forces and radiation to control and manipulate atoms and molecules. They can be posed challenges like trapping atoms in laser beams. These simulations are the inside story of real experiments that have been done, but without the complexity of macroscopic devices. The simulations should preferably be based on rigorous solutions of the time dependent Schrddinger equation, but they could also use proven approximate methods to broaden the range of phenomena to be made accessible to the students. Stationary states and the dynamical transitions between them can be presented as special cases of the full dynamics. All these experiences will create a sense of familiarity with the QM realm. The experiences will nurture accurate intuition that can then be made systematic by the formal axioms and concepts of QM. 

Gulyaev and Polak [Kinetics and Catalysis, 6 (352), 1965] have studied the kinetics of the thermal decomposition of methane with a view toward developing a method for the commercial production of acetylene in a plasma jet. The following differential equations represent the time dependence of the concentrations of the major species of interest. 

The time dependence result evidently from the angle 6 and, considering from a quantum mechanical point of view the interaction between a magnetic moment and the static magnetic field B0 (Zeeman term), we can invoke a local field of the form... 

The time-dependence of the SSP process has to be viewed in connection with the temperature and the factors concerning diffusion. After a certain period into the reaction, no increase in the IV is observed and thermal decomposition becomes the dominant process. 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

The transport equations for laminar motion can be formulated, in general, easily and difficulties may lie only in their solution. On the other hand, for turbulent motion the formulation of the basic equations for the time-averaged local quantities constitutes a major physical difficulty. In recent developments, one considers that turbulence (chaos) is predictable from the time-dependent transport equations. However, this point of view is beyond the scope of the present treatment. For the present, some simple procedures based on physical models and scaling will be employed to obtain useful results concerning turbulent heat or mass transfer. 

During the production phase, the positions, velocities, and accelerations created at each step in time were put on magnetic tapes. These tapes were later analyzed for the time-dependent and independent properties of the system. From a statistical mechanical standpoint, the data on these tapes may be viewed in one of two ways ... 

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

In connection with the time-dependent picture of electronic transition a missing mode effect (MIME) has been postulated [75] trying to explain the vibrational progressions when they are measured in quanta which do not occur in the set of normal vibrational modes in the molecule. It has been shown that the total wave packet < being a product of overlap factors, Eq. (47), of several displaced modes can lead, when Fourier transformed, to a spectrum with a progressional interval which is a mixture of the original normal modes. The spectrum of W(CO)5(py) on which this effect has been exemplified is, however, insufficiently resolved [75] to be used as a proof that the MIME in view of the uncertainty of the damping factor exists in reality. 

Fig. la c. Illustration of the time dependent theory of emission spectroscopy for one-dimensional harmonic potential energy surfaces, a schematic view of the emission transition, b time dependence of the overlap < (f> t) >, e calculated emission spectrum... 

Equation (9.5) can be viewed as first a modulation of the window to frequency cq thus producing a bandpass filter w(n)eJ an followed by a filtering of x(n) through this bandpass filter. The output is then demodulated back down to baseband. The temporal output of the filter bank can be interpreted as discrete sine waves that are both amplitude- and phase-modulated by the time-dependent Fourier transform. 

The time dependence of p and consequently of a in Eq. (267a) is actually stipulated by a motion of some microscopic object, to which we conditionally may assign the coordinate o(cp) and the velocity d(cp). In view of Eqs. (237), (267a), and (267b), this motion can equivalently be characterized by the phase f and energy ... 

---