Stationary phase approximation

In fact, (3.22) is the usual stationary phase approximation, performed however for an infinitedimensional path integral, which picks up the trajectories with classical action S. Further, at fixed time t we take the integral over Xi again in the stationary phase approximation, which gives... 

A further simplification of the semiclassical mapping approach can be obtained by introducing electronic action-angle variables and performing the integration over the initial conditions of the electronic DoF within the stationary-phase approximation [120]. Thereby the number of trajectories required to obtain convergence is reduced significantly [120]. A related approach is discussed below within the spin-coherent state representation. 

Employing the stationary-phase approximation to the path integral, the semiclassical spin-coherent state propagator is obtained [139, 140, 143, 281, 282] ... 

Makri and Miller [1987b], Doll and Freeman [1988], Doll et al. [1988] (see also the review by Makri [1991b]) exploited the stationary phase approximation (i.e., the semiclassical limit) as an initial approximation to the path integral. For example, for the multidimensional integral of the form J dx exp[iS(x)], one may obtain the following approximation [Makri, 1991b] ... 

Evaluating p in stationary phase approximation we get something new. The stationary phase condition is given by... 

Solution of the golden-rule expression for the non-adiabatic ET rate constant by applying the saddle-point (or stationary phase) approximation yields [36],... 

The preexponential factor in Eq. (11) can be found either by calculating the transition amplitude within the stationary phase approximation as explained in [2], or by considering the eorrespondence-prineiple (CP) limit of the QCl transition amplitude. We will follow the latter prescription since it allows an easier comparison of classical and quantum rates in Sect. 6. 

A semiclassical estimate of the integral in (44) can be obtained from the stationary-phase approximation. Exchange degeneracy splitting is clearly a case where the Cl approach results in a very simple, practically useful expression. 

The derivation of these fundamental correspondence relations, (3), has been given previously,9 and one should see Ref. 9 for a more detailed discussion. To obtain the results it is necessary to assume only (2) (which is essentially a statement of the uncertainty principle), make use of classical mechanics itself, and invoke the stationary phase approximation14 to evaluate all integrals for which the phase of the integrand is proportional to h l. Since the stationary phase approximation14 is an asymptotic approximation which becomes exact as h -> 0, this is the nature of the classical-limit approximation in (3). In a very precise sense, therefore, classical-limit quantum mechanics is the stationary phase approximation to quantum mechanics. 

If there are no roots to (128), then the primitive stationary phase approximation implies / 0 although it is true that in such cases the value of the integral is small, one often wishes to know how small—10 2, say, or 10-4. To determine the asymptotic approximation to the integral in such cases one analytically continues (129), the mathematical apparatus for which is the method of steepest descent .59 This approach notes that although there are no real values of t which satisfy (128), there will in general be complex values which do so—provided, of course, that it is possible to analytically continue the function /(f) into the complex t-plane. The method of steepest descent then deforms the path of integration in (127) from the real f-axis,... 

It was Ford and Wheeler(1959) who realized that this observation is nothing but the mathematical expression of our classical intuition. They skilfully applied this idea to elastic scattering on a simple potential curve, but Matsuzawa (1968) remarked that it applies as well to scattering on two crossing curves. In the lowest order stationary phase approximation the phase shifts tf b are developed around the points of stationary phase at /, and approximated by a parabola. The inelastic scattering amplitude can be written as a sum over separate contributions ... 

Pick at least ten colonies to inoculate small monoclonal test cultures (3mL SD-citrate-CAA + 50 pg/mL kanamycin per colony). Grow cultures to stationary phase (approximately 48 h in a 30°C shaker incubator). 

Figure 7.3 illustrates the reflection approximation. A detailed discussion of the stationary phase approximation and how that approximation permits the... 

This condition defines the classical path from Qq to Q. More specifically, a stationary phase approximation to Equation 4.26 can be considered by equating to zero the first-order variation of the phase about the classical path Q(t) ... 

That the transition occurs preferentially at the classical turning point can be demonstrated if the integral in Eq. (36) (or better yet Eq. (35)) is evaluated using the stationary phase approximation (51). The points of stationary phase precisely occur at the classical turning points at which the momentum is zero. These expressions can also be generalized to the multidimensional case (59). 

As an illustration of the kinds of / that may occur, consider a simple model where the long-range behavior of the potentials f ss[R(0] and F7.7.[R(t)] arise from van der Waals interactions and vary as t.[R(0] Given the existence of a stationary phase point, (3.16), the stationary phase approximation to (3.10) yields the general result... 

In the classical limit, one can invoke the stationary phase approximation and obtain the following nonequilibrium generalization of the Marcus expression [63] ... 

The stationary phase approximation breaks down near R = Ra (Figure 2), where E, reaches its minimum value. Interference dfecls between transitions at R > Ra and R < Ra may be obsoyed and the spectrum extends somewhat beyond the classical limit. This is important only for situations in whidi V is strongly bound, such as He(2 ) + H or Hg fen He + noble gas atoms, the stationary-phase model has been used in all analyses of chemiionization data. 

This classical model has been extended further by Nakamura > and Miller by including interference effects between the incoming and outgoing portions of the trajectory this resulting semiclassical approach is equivalent to a WKB analysis of the quantum mechanical treatment, employing the stationary-phase approximation. ... 

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