Incoming states

Here we have collected the N independent g that correspond to different incoming states for N open chaimels into a matrix g (where the sans serifhoM notation is again used to denote a square matrix). Also we have the matrices... 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

Due to time-reversal symmetry, every resonance solution has an antiresonance solution, also known as antiresonance states, that occurs at k = —kr — iki. These antiresonance states are incoming states that also diverge at the asymptotes. 

We now use the Lippmann-Schwinger equation to explore the long-time behavior fif the continuum part of the wave packet P (0 that we have created with the laser Jilise [Eq. (2,28)]. We can use either the outgoing or incoming states as the basis set f exfianding P (r). In what follows we shall expose the different boundary condi-hs and see which type of solution is best suited for which purpose. Substituting (2 52) in Eq. (2.28), we obtain that... 

If we expand (f) in the outgoing states instead of the incoming states, the closed contour integration encircles a pole at E = E — ie. Hence the integration yields... 

The above results deal with photodissociation. A similar formulation can be applied to an inelastic scattering event, for example, the scattering of A + BC from an initial state E, m 0) into a final state E, n 0) of A + BC. This problem can be phrased as our asking for the amplitude of making a transition from a free state E, m 0) at t —> —oo to free state E, n 0 at t — oo. Since E, m+) is known to have evolved from E, m 0) and E, n ) is known to evolve to state E, n 0), the answer is given by the expansion coefficients of an outgoing state in terms of the incoming states. That is,... 

Bguation (12.21) uses the fact that we can prove, in complete analogy to the proof given in the context of the material incoming states [see Eq. (2.57)], that as t oo tt fhlly interacting states go over to the states wherein the matter and radiation are decoupled, but where the matter itself is interacting. That is,... 

The incoming states planar wave states sustained by the Rubidium atom. The collimators allow separation in space base states 10) and 20) propagate along channels 1 and 2 < i(x) and 02(x) are the scattered states at DS-1 and, as the source state is the same and we take identical slit interaction potentials, these functions differ only in space origin. After interaction with the laser beam the base states label from 1) to 10> are sufficient for discussing several possibilities... 

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We have already argued that the phenomena of Raman scattering and fluorescence cannot be distinguished from each other unless the system interacts with its thermal environment. Next we extend the model discussed in Section 18.3 to explicitly include thermal relaxation effects. Our model now consists of four levels The incoming state zrt) = 11, vi, ki) with energy fCin = -F ha>i mi = c ki ... 

We reexamine the example of Section V for incoming states which differ in their initial angular momentum, where we exemplarily choose 7 = 1 and Ji = 6. The LCT field is taken from the s-wave calculation (see Fig. 10). As a... 

The full scattering incoming states E, n ) are eigenstates of the Schrodinger equation... 

Thus, the expectation value of f with respect to the incoming states is given by... 

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We assume that all the dump laser fields are in exact resonance with the corresponding transitions, coo, t = En - Eq. We also assume that all the pump laser fields are in near resonance with the transitions from a given energy E (or Em,) of the incoming states. The detunings are defined as... 

The above procedure resembles the Lippmann-Schwinger method [106] for constructing the incoming state in scattering theory. 

Vp is defined by equation (1), m is the mass of the projectile atom, and Vq(z) is the surface-averaged potential. We assume that Vq(z) has an attractive region capable of supporting bound states. The "distorted wave" Hamiltonian, H, has both continuum- and bound-state vectors denoted by Kq,Icq > and, n> -respectively, where and are the surface-parallel projection and surface-perpendicular projection of the outgoing momentum vector Icq. We define also the incoming state vector kj and the outgoing momentum k= (Kjk ) of the specular (G= 0) channel. Because of the periodicity in the parallel direction, only those states which satisfy the following surface-potential momentum conservation equation can couple to the specular channel ... 

The summation extends over all discrete states nearly in resonance with the incoming state. The projection operator P which contains all states out of resonance is defined by I = Q + P where I is the identity operator. Since P contains all continuum states, the behavior of j ] .> at large z will be contained in Using the... 

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