Classically allowed region

A sum over all final vibration-rotation states f lying below e, for which the geometry Qo is within the classically allowed region of the corresponding vibration-rotation wavefunction Xf (Q) (so that vq is real) of... 

Within the classically allowed region, the wave function and the probability density oscillate with n nodes outside that region the wave function and probability density rapidly approach zero with no nodes. 

Consider a particle of mass wr in a parabolic potential well. Calculate the probability of finding the particle in the classically allowed region when the particle is in its ground state. 

At its maxima, p must satisfy p" < 0, which suggests that such locations must lie within the classically allowed region. The same applies for the existence of inflection points in p. At the nodes of E, Equation 33.12 implies that p" > 0. At such points, p in fact has a minimum. Finally, when the potential is reflection symmetric then at the origin (x = 0), either E 0 (even function) or E 0 (odd function). For even... 

Consider the case of a piecewise-constant potential, as shown in Fig. 1.3. In the classically allowed region, E> U,Eq. (1.2) has solutions... 

The / and g functions are commonly termed the regular and irregular coulomb functions.4,5 In the classically allowed region the / and g functions are real... 

An important property of the wavefunction is its normalization, and we have yet to normalize the radial coulomb radial functions. Following the approach of Merzbacher, we can find an approximate WKB radial wavefunction, good in the classically allowed region, given by6... 

An alternative, physically appealing way of calculating the bound wavefunctions was used by Bhatti et al.9 In the classically allowed region, the kinetic energy of the electron is given approximately by Hr, thus the wavelength of the wavefunction varies as Vr. If we make the substitution x = Vr and increment x by a uniform step size h, we should always have nearly the same number of steps in each lobe of the wavefunction. Making the substitutions... 

A convenient solution to this problem is provided by a certain uniform approximation [30], which is built on pairs of solutions and automatically takes care of the discarding . The result requires the same input as (4.9), namely the action and its second derivatives at the saddle points, and is hardly more complicated. Within the classically allowed region, it is... 

If in (4.8a) the first ionization potential E01 is neglected, the electron starts its orbit in the continuum with an initial velocity viiJ) = k + A(t ) = 0 and t, t, and k are all real inside the classically allowed region. The classical boundary then is characterized by the fact that the two solutions of the afore-mentioned pair merge. (There are no such real solutions outside the classically allowed region.) This can be exploited to obtain (approximate) closed formulas for the cutoffs (corresponding to the well known cutoff E + 3.177, for high-order harmonic generation) see [31]. 

We assume that the classically allowed region delimited by t is also delimited by another turning point t" [simple zero of Q2(z), to the right of which there is a classically forbidden region extending to +oo. In this region the wave function is... 

When I = —1/2 and B is positive, there is on the positive part of the real axis a generalized classically allowed region Q2(x) > 0] delimited to the left by the origin. In this region the wave function is according to (4.29) given by the formula... 

With Q2 rj) given by (5.2) we recall (4.19) and (4.29) and normalize the physically acceptable solution g(m,ni,EmynuS rj) of the differential equation (2.32a,b) such that in the classically allowed region to the left of the barrier in Figs. 5.1b,c and Figs. 5.2b,c the phase-integral expression for this solution, with the use of the short-hand notation defined in (4.17), is... 

Quasiclassical perturbative calculations to be discussed here are based on the Landau method of calculation of the transition matrix elements [2] and the recovery of the Landau quasiclassical exponent from the classical encounter time [4-6]. The quasiclassical VP rate constants, contain classical dynamical parameters that characterize a vdW complex motion in the classically allowed regions. In this respect, quasiclassical Landau approach indicates a possible way of transition from quantum to classical dynamics. This transition is not at all trivial, since the correspondence principal limit ( ft —> 0) of yields zero rate. 

For potential functions with a single minimum (or a very shallow second minimum), location of the origin in the minimum is probably the best choice to obtain most rapid convergence of the variation functions. Again, as for the symmetric cases, some care should be exercised in choosing the harmonic scale factor for the basis, to insure that the truncated basis has sufficient flexibility to produce amplitude in the classically allowed regions and to cancel amplitude in the unallowed regions. 

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