Other Approximate Methods

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

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All of these have some limitations and have not been thoroughly compared against the Colburn method, which is considered to be within 10% of any correct solution. Cairns has compared his proposal with 6 different systems and 4 other approximation methods. In general, the agreement with the Colburn-Hougen method is excellent. 

The essence of the LST for one-dimensional lattices resides in the fact that an operator TtN->N+i could be constructed (equation 5.71), mapping iV-block probability functions to [N -f l)-block probabilities in a manner which satisfies the Kolmogorov consistency conditions (equation 5.68). A sequence of repeated applications of this operator allows us to define a set of Bayesian extended probability functions Pm, M > N, and thus a shift-invariant measure on the set of all one-dimensional configurations, F. Unfortunately, a simple generalization of this procedure to lattices with more than one dimension, does not, in general, produce a set of consistent block probability functions. Extensions must instead be made by using some other, approximate, method. We briefly sketch a heuristic outline of one approach below (details are worked out in [guto87b]). 

This property may not be possessed by many other approximate methods based on, e.g., mean field or semielassieal approaehes. Also, in low dimensional systems, the above property is not true for CMD, so to apply CMD to such systems is not consistent with spirit of the method (though perhaps still useful for testing purposes). 

If modeling or other approximate methods are not applicable, then a number of experiments should be conducted in order to examine the effect of superficial velocity on the performance of the bed, and more specifically on die breakpoint volume. Keeping the same contact time and particle size, one can study the effect of linear velocity by changing just the length of the bed accordingly, and in this way examining the controlling step. For solid... 

Accuracy. The results must be sufficiently accurate to interpret the experiments of interest. In a complete quantum-mechanical calculation, this accuracy can be verified by convergence tests within the calculation. In classical, or other approximate methods, accuracy and reliability generally must be judged by experience with test comparisons with complete quantum-mechanical calculations. The numerical stability of the method must also be considered. 

The value of the polarizability a of an atom or molecule can be calculated by evaluating the second-order Stark effect energy — %aF2 by the methods of perturbation theory or by other approximate methods. A discussion of the hydrogen atom has been given in Sections 27a and 27e (and Problem 26-1). The helium atom has been treated by various investigators by the variation method, and an extensive approximate treatment of many-electron atoms and ions based on the use of screening constants (Sec. 33a) has also been given.3 We shall discuss the variational treatments of the helium atom in detail. 

As noted above, these authors also proved the general validity, for the Rayleigh problem, of the principle of exchange of stabilities. Further, by formulating the problem in terms of a variational principle, Pellew and South-well devised a technique which led to a very rapid and accurate approximation for the critical Rayleigh number. Later, a second variational principle was presented by Chandrasekhar (C3). A review by Reid and Harris (R2) also includes other approximate methods for handling the Benard problem with solid boundaries. 

Many features of this fascinating problem have not been mentioned here, but can be found in the comprehensive and up-to-date review of random copolymers in [2]. In particular, many of the rigorous results and the numerical results obtained both by other approximation methods and by Monte Carlo methods which have not been discussed here can be found there... 

Numerical or direct integration and other approximate methods... 

In the sections that follow several relatively simple transport problems are analyzed rigorously in the hope that the light thrown by these investigations will lead to a fuller basic understanding in cases of greater physical interest. There is no recourse to diffusion theory or other approximate methods. In general, while the ideas of proofs are discussed, the details are omitted. The interested reader is referred to the original research papers for such details. 

Other approximate methods are available for gases containing suspended luminous flames, clouds of nonblack particles, refractory walls and absorbing gases present, and so on(Ml,P3). 

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