Phase integral

Gutzwiller M C 1967 Phase-integral approximation in momentum space and the bound states of an atom J Math. Phys. 8 1979... 

A proper calculation requires that a ( ) and x( ) should be evaluated in terms of semiclassical phase integrals, but it is sufficient for illustrative purposes to... 

Finally, the stationary phase integration over time yields the identity... 

Design Letters. Single-phase integral-horsepower motors may be one of the following ... 

Design L. A Design L motor is a single-phase integral-horsepower motor designed to withstand full-voltage starting and to develop a breakdown torque as shown in MG 1-10.33 with a locked-rotor current not to exceed the values shown in MG 1-12.33. 

An anharmonic correction for the density of states was also evaluated by solving the phase integral for the Cl-—CH3C1 intermolecular complex 39 i.e. ... 

Sampling and Sample Preparation for Field and Laboratory Countercurrent Chromatography The Support-Free Liquid Stationary Phase Integrated Analytical Systems... 

Fig. 2.17 DDT mass [t] and mass fractions [%] bound to colloidal and particulate phases integrated for all continental shelves. Grey lines show mass fractions, and black lines mass dashed lines show results from the SAT, and solid lines results from the AGG experiment.

According to equation (21) the systems of the canonical ensemble are conservative. Each system moves independently of all others and the phase integral exists for each of them. Each system therefore moves on a surface of constant energy and corresponds to a microeanonical ensemble. In this sense the canonical ensemble is built up from a multitude of microeanonical ensembles. Quantities defined for the microeanonical ensemble may therefore be averaged over the canonical ensemble. The original system which is represented by the canonical ensemble however, cannot be described, even approximately, as conservative. It is customary to denote the Hamiltonian of the systems of the canonical ensemble as the Hamiltonian of the original system, which is not justified. 

The quantities appearing in Eq. (16.2) are not independent. They are related by a Gibbs-Duhem equation, which is obtained in the same way as in the ordinary thermodynamics of bulk phases integrating with respect to the extensive variables results in Ua —TSa — pVa + 7Aa + E/if Nf. Differentiating and comparing with Eq. (16.2) gives ... 

Constant D is probably a good approximation in so far as the degree of melting is significantly smaller than the proportion of the least abundant mineral phase. Integrating the differential equation gives expressions for the solid and the instantaneous liquid in equilibrium with it... 

When the volume dV2 of the liquid evaporates, the volume of the vapor increases by dVt the two partial differentials refer to the same mass of substance. Thus (3 V2/d Vl)P2 = —Pi ip2, Pi and p2 being the densities of the two phases. Integration of the equation (3p1/3p2)K1 = P1/P2 affords- p0 = (p,/p2) (p2 - Pa)-The pressure p0 is that on both sides of a plane liquid surface. Pressure p2 is different from p0 whenever the liquid surface is curved. If its two principal radii of curvature are/ and/ 2, then... 

Under this condition, the reactant A is unlikely to reach the emulsion phase. Integrating the material balance for the bubble phase, eq. (3.534) yields the desired performance expression in terms of conversion ... 

As for the theory of this phenomenon, it was first observed by Onsager [27a] that, since in the limit a — 0 an LCD a is expected to yield a singularity of the type —surface potential, the statistical-mechanical phase integral for counterions should diverge for a greater than some critical value, characteristic of a given valency. Indeed consider a counterion (for definiteness anion) of valency z. The appropriate phase integral is of the form... 

The phase integral s(r) has the derivative p(r) and it satisfies the secular equation... 

This equation shows that only the closed classical trajectories (x(f) = x(0) and x(t) = x(0)) should be taken into account, and the energy spectrum is determined by these periodic orbits [Balian and Bloch, 1974 Gutzwiller, 1967 Miller, 1975b Rajaraman, 1975]. Finally, the stationary-phase integration over time yields the identity... 

On the surface, the integration of UCB Pharma and Celltech was very similar to the integration of UCB Chemical and UCB Films with Solutia s resins, additives, and adhesives business. It covered about EUR 2.1 billion in revenue and 8,500 people the aspiration was to achieve the same improvement target (five percent on sales before the merger, increased to around ten percent on sales in the planning phase) and the plan was to realize this through a similar four-phase integration approach (Fig. 26.3). 

But no fine structure - yet - until in 1915 Bohr considered the effect of relativistic variation of mass with velocity in elliptical orbits under the inverse square law of binding, and pointed out that the consequential precessional motion of the ellipses would introduce new periodicities into the motion of the electron, whose consequences would be satellite lines in the spectra. The details of the dynamics were worked out independently by SOMMERFELD [38] and WILSON [39] in 1915/16 based on a generalisation of Bohr s quantization, namely, the quantization of action the values of the phase integrals Jf = fpj.d, - of classical mechanics should be constrained to assume only integral multiples of h. 

We shall consider mainly publications in which asymptotic methods are used, but we also mention numerical methods, since we use numerical results for comparison with our phase-integral results. For a general review of the field we refer to Bethe and Salpeter (1957), Ryde (1976), Bayfield (1979), Koch (1981), Gallas, Leuchs, Walther and Figger (1985), Lisitsa (1987) and Gallagher (1988, 1994). 

We share the opinion expressed by Farrelly and Reinhardt (1983) that discrepancies between Stark effect results obtained by the use of the Carlini (JWKB) approximation and by accurate numerical calculations cannot be attributed to the break-down of the approximation, but are due to a failure to use the approximation in a correct way. An appropriate approach based on the phase-integral approximation of arbitrary order generated from an appropriately chosen base function is a still more efficient and often highly accurate method for the treatment of several problems, not only in quantum mechanics, but in various fields of theoretical physics. With... 

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