Asymptotical phase

The analysis underlying the evaluation of the S matrix elements was formulated for the J = 0 (and 1 = 0) case [75] and did not take proper account of the correct asymptotic phases of the spherical Bessel functions [161]. This phase should have been exp(—— I n/l) rather than the phase given in Eq. (4.47). To correct for this omission in both the reactant and product channels, we must multiply by a phase of exp(—i7 ti/2) = for the products and for the reactants. These factors are included on the RHS of Eq. (4.47). 

In the present context only the factor b/K) is of further interest. Since it contains the asymptotic phase shift Ae and the nominal wavenumber k which also is defined by its asymptotic value, the expression for b/K) holds everywhere even though it has been derived in the asymptotic limit (in contrast, the scattering function can be defined only in the asymptotic limit). Hence, the correct wavefunction P[Pg.288]

The asymptotic phase is given in terms of the relative momentum k( and relative position r, of each pair i by... 

For water boiling at 1 atm with a superheat of 1°C, the right-hand side of Eq. (21) has the value 10 cm, and for 10°C, 10" cm. Hence, the inertial terms are generally negligible in the visible bubble range, which justifies focusing attention on the asymptotic phase of bubble growth. 

The differential cross section for elastic scattering of a charged particle by a nucleus is given in terms of the asymptotic phase shifts di of the partial wave of angular momentum 1% by the formula (Schiff )... 

One may adopt the following definition of 0(A), which seems to be most natural and convenient. Our definition is essentially the same as what is called asymptotic phase (Coddington and Levinson, 1955). Let P denote a point... 

The notion of asymptotic phase or isochrons has a certain relationship to the eigenvectors, defined for the linearized system of (2.1.1) about its periodic solution Xo(t). Consequently, various quantities which appear in our perturbation results, e.g., a and p, may be reinterpreted in terms of those eigenvectors. This new interpretation may sometimes prove convenient for analytical or numerical calculations based on a specific model (Sect. 3.5). Moreover, such a reformulation opens the way to a more systematic method of phase description as developed in Chap. 4. 

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