Time-dependent states

Much of quantum chemistry is concerned with stationary states, for which is a product of a space term (an eigenfunction of H) and a time-dependent factor exp(—/Et/h), which we usually ignore because it has no effect on particle probability distribution. Sometimes, however, it becomes necessary to consider time-dependent states. In this section we illustrate how some of these may be treated. 

There are two types of situation to distinguish. One is situations where the potential is changing as a function of time, and hence the hamiltonian operator is time dependent. An example is a molecule or atom in a time-varying electromagnetic field. The other is situations where the potential and hamiltonian operator do not change with time, but the particle is nonetheless in a nonstationary state. An example is a particle that is known to have been forced into a nonstationary state by a measurement of its position. We deal here with the second category. 

As our first example, consider a particle in a one-dimensional box with infinite walls. Suppose that we measure the particle s position and find it in the left side of the box (i.e., between x = 0 and T/2 we will be more specific shortly) at some instant that we take to be f = 0. We are interested in knowing what this imphes about a future 

Our knowing that the particle is on the left side at f = 0 means that the wavefunction for this state is not one of the time-independent box eigenfunctions we saw in Chapter 2, because those all predict equal probabilities for finding the particle on the two sides of the box. If the state function is not stationary, it must be time dependent, and it must satisfy Schrddinger s time-dependent equation (6-1). We have, then, that the state function is time dependent, and that I s q/p is zero everywhere on the right side of the box when f = 0. (We have not yet been specific enough to describe p in detail on the left side of the box.) 

Schrddinger s time-dependent equation (6-1) is not an eigenvalue equation. However, Eq. (6-2) shows that Schrodinger s time-dependent equation is satisfied by time-independent eigenfunctions of H if they are multiplied by their time-dependent factors fit) = Q i—iEt/h). Furthermore, Eq. (6-2) continues to be satisfied if the term i if) fit) is replaced by a sum of such terms. (See Problem 6-9.) This means that we can seek to express the time-dependent state function, Pfx, 0. as a sum of time-independent box eigenfunctions as long as each of these is accompanied by its time factor fit). When i = 0, all the factors / it) equal unity, so at that point in time becomes the same as the sum of box eigenfunctions without their time factors. 

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Modem photochemistry (IR, UV or VIS) is induced by coherent or incoherent radiative excitation processes [4, 5, 6 and 7]. The first step within a photochemical process is of course a preparation step within our conceptual framework, in which time-dependent states are generated that possibly show IVR. In an ideal scenario, energy from a laser would be deposited in a spatially localized, large amplitude vibrational motion of the reacting molecular system, which would then possibly lead to the cleavage of selected chemical bonds. This is basically the central idea behind the concepts for a mode selective chemistry , introduced in the late 1970s [127], and has continuously received much attention [10, 117. 122. 128. 129. 130. 131. 132. 133. 134... 

The corresponding zero-order time-dependent states are (from Chapter 0)... 

In this section I will write h for the time-dependent states and ijr for the time-independent ones. The jri may themselves depend on the space and spin variables of all the particles present. 

The traditional treatment of molecules relies upon a molecular Hamiltonian that is invariant under inversion of all particle coordinates through the center of mass. For such a molecular Hamiltonian, the energy levels possess a well-defined parity. Time-dependent states conserve their parity in time provided that the parity is well defined initially. Such states cannot be chiral. Nevertheless, chiral states can be defined as time-dependent states that change so slowly, owing to tunneling processes, that they are stationary on the time scale of normal chemical events. [22] The discovery of parity violation in weak nuclear interactions drastically changes this simple picture, [14, 23-28] For a recent review, see Bouchiat and Bouchiat. [29]... 

The basic equations (l)-(8) also predict the existence of free time-dependent states, in the form of nontransverse wave phenomena in vacuo. Combination of... 

Thus we have found the distribution of the fluctuations around the macroscopic value. They have been computed to order Q 1/2 relative to the macroscopic value n, which will be called the linear noise approximation. In this order of Q the noise is Gaussian even in time-dependent states far from equilibrium. Higher corrections are computed in X.6 and they modify the Gaussian character. However, they are of order 2 1 relative to n and therefore of the order of a single molecule. 

They determine the variances and the covariance of the fluctuations of nx and nY around the values of the macroscopic solution of (5.5). Rather than study the general time-dependent state, however, we concentrate on the stationary state. 

In regard to control of energy flow, one can think of defining and trying to prepare time-dependent states closely related to these subpolyads (which contain time-independent levels of the spectroscopic Hamiltonian). If the proper time-dependent states could be prepared, energy could be made to flow with confinement to the chosen subpolyad. 

Fig. 8 shows time-dependent state populations as obtained from quantum dynamical (MCTDH) calculations. While the full (here, 24 dimensional) model exhibits an ultrafast XT decay, no net decay is observed for the reduced 3-mode model truncated at the lowest level of the effective mode hierarchy. The dynamics is strongly diabatic if confined to the high-frequency subspace (Heff ) and involves repeated coherent crossings [51]. The dynamical interplay between the high-frequency and low-frequency modes is apparently a central feature of the process. To account for these effects, a treatment at the level of is necessary, i.e., a six-mode model including the low-frequency modes. At the level of the dynamics is found to be essentially exact. 

Thus, a time-dependent state stands for a system where one cavity photon is no longer available. When state C3( >"f) = 1 where of characterizes a weak interaction one has a possibility to "release" one photon o> to the cavity the base state required to complete the description is Rb 61d5/2>i 10 >410 )2- The quantum state takes on the form ... 

This chapter is organized as follows. In Section II, we show how quantum chaos systems can be controlled under the optimal fields obtained by OCT. The examples are a random matrix system and a quantum kicked rotor. (The former is considered as a strong-chaos-limit case, and the latter is considered as mixed regular-chaotic cases.) In Section III, a coarse-grained Rabi state is introduced to analyze the controlled dynamics in quantum chaos systems. We numerically obtain a smooth transition between time-dependent states, which justifies the use of such a picture. In Section IV, we derive an analytic expression for the optimal field under the assumption of the CG Rabi state, and we numerically show that the field can really steer an initial state to a target state in random matrix systems. Finally, we summarize the chapter and discuss further aspects of controlling quantum chaos. 

As shown in Section II.A, the overlap in the controlled dynamics rapidly oscillates because the system contains many states. To analyze this complicated behavior more easily, we introduce the following two time-dependent states,... 

We introduce another quantum state by a linear combination of the two time-dependent states,... 

In Section II.A, we have already obtained the optimal field e t) by the numerical calculation for the random matrix systems, Eq. (6). However, only the overlap between the time-evolving controlled state < )(t)) and the target state (pj) was shown there. In this section, we show the overlaps between the time-dependent states defined by Eq. (15) and < )(t)), and we find a smooth transition picture. 

Figure 6.1-4 Illustration of Eq. 6.1-19, the time-dependent approach to continuum resonance Raman scattering. Shown is a 2 > 1> vibrational Raman transition in Bra for Aq = 457.9 nm excitation. As examples, (A), (B) and (C) show the potential curves of the relevant ground (X = continuous line) and excited (B = 7o+m, dashed line, and 77 = 7T , dotted line) electronic states, together with the absolute values of the coordinate representations of the initial state It >= 1 >, final state ]f >= 2 >, and the time-dependent state i(r) > at times / = 0, 20 and 40 fs, respectively. The excitation and de-excitation processes and the related unimolecular dissociations are indicated schematically by vertical and horizontal arrows. For clarity of presentation, the energy gap between state (> and f> is expanded (Ganz et al., 1990).

The Hamiltonian H of a system may be represented in terms of the differential operator d/dt. This is the analogue in special relativity of the representation of the momentum in terms of V. Operating with the observable H on an arbitrary time-dependent state 4 (t)) we have... 

When A is substituted with the unity operator, Eq. (1.108) shows that acceptable wavefunctions should be normalized to 1, that is, (/1 /) = I.A central problem is the calculation of the wavefunction, 4>(r, Z), that describes the time-dependent state of the system. This wavefunction is the solution of the time-dependent... 

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