Conjugate coordinate

The differential quantity of work done by a specific force is defined as the product of the force and the differential change of the conjugate coordinate. The differential quantity of work done by the system on the surroundings is expressed as... 

Then, from Eq. (M.12) and its complex conjugate coordinate, Eq. (M.13) becomes... 

A most recently developed description of the problem interprets the amplitude mode model in terms of a molecular concept of chains with cyclic boundary conditions (Zerbi et al., 1989). An effective conjugation coordinate Qja with a corresponding force constant fja is defined. The vibrational frequencies are calculated as a function of , which plays the same role as A in the amplitude mode model. Accordingly, the Raman intensity is obtained from the respective ja components of individual modes. This new concept, like the amplitude mode model, properly describes the relative intensities of different modes, but the correct line shapes and line intensities for excitation with different laser lines can not be obtained, since transition matrix elements are not evaluated explicitly. 

The slow step in the reduction is the reduction of the first cobalt atom. When the R group contained no other conjugated coordination site, the rates of reduction by Cr +, Eu +, and [RuCNHale] were of similar magnitude. But if a remote conjugated carbonyl was present, for instance, with the ligand 7,... 

The transformation from one pair of canonically conjugate coordinates q and momenta p to another set of coordinates Q = Q(p,q,t) and momenta P = P(p>qT) is called a canonical transformation or point transformation. In this transformation it is required that the new coordinates (P,Q) again satisfy the Hamiltonian equations with a new Hamiltonian H P,Q,t) [35] [43] [52]. 

This eq. (3.11) shows that the amplitude of the scattered wave is proportional to the Eourier transform, defined in eq. (5.A17) of the appendix of Ch. 5, of the density p of scattering centres. This is a particularly favourable situation, because the measurements of the amplitudes of the waves coherently scattered in all directions, that is with all possible wavevec-tors 2, allows in principle to calculate by an inverse Eourier transform, also defined by eq. (5.A17) of the appendix of Ch. 5, the density p(f) of scattering centres. This last equation holds for a one-dimensional (ID) Eourier transform, that is a single coordinate x (or 1) and consequently a single conjugate coordinate k (or v in eq. (5.A17)). The extension to a 3D space defined by f and j, 2 is trivial and can be found in any textbook on X-ray. Without entering the details of calculations of inverse 3D Eourier transforms, we nevertheless have to n te that in a coherent scattering experiment the measured quantities are not amplitudes E (eq. (3.11)) of vectors, which are complex quantities, with real and... 

Let (p, q) be one set of canonically conjugate coordinates and momenta (the old variables) and (P,Q) be another such set (the new variables).13 (P, Q, p and q are IV-dimensional vectors for a system with N degrees of freedom, but for the sake of clarity multidimensional notation will not be used the explicitly multidimensional expressions are in most cases obvious.) In classical mechanics P and Q may be considered as functions of p and q, or inversely, P and Q may be chosen as the independent variables with p and q being functions of them. To carry out the canonical transformation between these two sets of variables, however, one must rather choose one old variable and one new variable as the independent variables, the remaining two variables then being considered as functions of them. The canonical transformation is then carried out with the aid of a generating function, or generator, which is some function of the two independent variables, and two equations which express the dependent variables in terms of the independent variables.13... 

The values of two dynamical quantities f and g of a system can be accurately measured at the same time only if their commutator is zero otherwise these measurements can be made only with an uncertainty Af g whose magnitude is dependent on the value of the commutator. In particular, for a canonically conjugate coordinate... 

Because of the strong coloration, depending on their state of oxidation, intrinsically conducting polymers have frequently been studied with SRRS [507, 508]. Molecular vibrations could be assigned based on various approaches. Most frequently band positions of the monomers and of already known oligomers were compared with those of the polymers. Alternatively, band positions were calculated based on an effective conjugation coordinate [509-516]. In a typical example shown in Fig. 5.84, SRR spectra of poly aniline are displayed as a function of electrode potential. 

Proof Introduce on local regular coordinates xi,..., Xn. Then the covector ( T M, which is the 1-form on is given by conjugate coordinates fi,..., Prom the definition of the form u it follows that u = da = d i AdxiH-l-d nAdXn. 

This coordinate reflects the degree of bond alternation. This coordinate is used in the elfective-conjugation-coordinate model proposed by Zerbi et al. [15] for the purpose... 

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