Physically acceptable solution

Since we require %, which is given by (2.23), to be finite everywhere, we do not accept the minus signs in these expressions. Therefore the physically acceptable solutions /( ) and g g) are for small values of and g equal to (1+M)2 and (1+lml)2j respectively, times a power series in and g, respectively, with the constant term different from zero. 

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... 

Problems associated with the quantum-mechanical definition of molecular shape do not diminish the importance of molecular conformation as a chemically meaningful concept. To find the balanced perspective it is necessary to know that the same wave function that describes an isolated molecule, also describes the chemically equivalent molecule, closely confined. The distinction arises from different sets of boundary conditions. The spherically symmetrical solutions of the free molecule are no longer physically acceptable solutions for the confined molecule. 

It will be observed that p, i and p must all be positive for physically acceptable solutions. Multiplying equation (23) by Po/Po yields... 

Fig. 1. Values of A plotted against r for constant t. Curve A separates the physical and unphysical values of A. Curve B gives the values of A for which 8pldp)j< diverges. Only the points below these two curves correspond to physically acceptable solutions of the PY equation.

As has been stated andillustrated, the Schrodinger equation possesses physically acceptable solutions (continuous, finite, and single-valued) only for certain definite values of the energy—the characteristic or proper values (Eigenwerte). These define the quantum states of the system. The way in which vibrational, rotational, and translational quantization follow has already been considered (p. 126). 

The coefficients Oj are complicated expressions of the parameters of the system, and an exact evaluation of the Hopf condition is a very involved and tedious task. However, for the physically relevant regime of small inertia, i.e., r and Xy small, all the coefficients are positive if (10.23) is satisfied, and consequently (10.57) has no physically acceptable solution. The spatial Hopf bifurcation to oscillatory patterns cannot occur in hyperbolic reaction-diffusion systems with small inertia. 

Equation (13.162) has unique, physically acceptable solution if the stoichiometric parameter h is large enough. 

From this equation, we can solve for as a function of the mole fraction Xs = Ns/M, where M = Mi + Mr, taking the limit Ns/M 0. There are two solutions to this equation. We choose the physically accepted solution, the result of which is... 

Taking now k = 5 j 4 in Equation (66) we obtain the following value for the physically acceptable solution of (63), namely. 

It is easily verified that the solution with the minus sign is the physically acceptable solution for our model. This can be checked by taking some limiting cases e.g., for ->0 we must have A->0, or for 0 -> 1, A must tend to H-co, or for 5=1, we return to the solution for the Langmuir case. 

Equating the last two expressions for R, one obtains a quadratic equation for b. The only physically acceptable solution of this quadratic is... 

Petit et al. 2006) compared to B , hence A term can be neglected in (6.22). With this condition, we may well get a physically acceptable solution for Ps only when E 0 (In accordance with the expression for B ). Otherwise, the whole system would diverge to a nonlinear singular response mode at zero-external field, which would further dilute the whole theoretical predictions for both PSFLCs and neat FLCs, respectively. In view of this approximation, the second root of (6.21) results in a zero value of spontaneous polarization (Ps), which may be regarded as the trivial solution for the paraelectric phase. The third root is physically unacceptable as it provides untenable behavior of the spontaneous polarization and hence, discarded altogether. With these considerations, (6.22) can be utilized to get the desired solution for spontaneous polarization Ps in the form of Ps = B -----------... 

The physically acceptable solutions of equation (3.21) are known as associated Legendre functions, 0(0) = P (cos8). These are polynomials in cos6 wliose form depends on the particular values of m and 7. 

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