Exactly Solvable Models

This Introductory Section was intended to provide the reader with an overview of the structure of quantum mechanics and to illustrate its application to several exactly solvable model problems. The model problems analyzed play especially important roles in chemistry because they form the basis upon which more sophisticated descriptions of the electronic structure and rotational-vibrational motions of molecules are built. The variational method and perturbation theory constitute the tools needed to make use of solutions of... 

So, our main purpose is attained, since we have solved the equations of motion for the case, which is one of the most general cases of exactly solvable models. This aim can be enlarged in the sense, that all quadratic cases will be solved in the following papers. 

Instead of this nonlinear coupling, a linear coupling provides an exactly solvable model with the Hamiltonian (S.P. Kim et.al., 2000 2002 2001)... 

Historically, it had been believed that the finite lamellar thickness observed in experiments is completely due to kinetic control and that, if sufficient time is granted for the lamellae, the thickness would progressively grow to the extended chain value. The exact enumeration calculations and an exactly solvable model... 

O. Mazonka and C. Jarzynski, Exactly solvable model illustrating far-from-equihbrium predictions. Preprint, arXiv cond-mat/9912121. 

B. Derrida, Random-energy model an exactly solvable model of disordered systems. Phys. Rev. B 24, 2613-2626 (1981). 

And so, even at large Reynolds number in a simple, exactly solvable model, the diffusion increase is quite limited (cf. Howell s result at j3 = 0 [1]). The reason, obviously, lies in the regularity of the motion in which a particle moving along the x-axis unfailingly returns back after every halfperiod. 

Jiri Cizek s research program centers on the quantum theory of molecular electronic structure and related developments in quantum chemical methodology, coupled-cluster approaches to many-electron correlation problems,105 large-order perturbation theory,106 dynamical groups and exactly solvable models, lower bounds, and the use of symbolic computation language in physics and in chemistry. 

Very few exactly solvable models of polymer systems with topological constraints have been described in the literature till now ... 

Baxter RJ (1982) Exactly solvable models in statistical mechanics, Academic, NY... 

The standard scheme has been the most popular one for the last 20 years or so. The harmonic scheme gets its name because if the bath is harmonic (more precisely, if F can be represented as a linear combination of harmonic coordinates), then this scheme is exact (16,19). By comparing these schemes to results from exactly solvable models we have concluded that the standard scheme is really not very accurate at high frequencies and that the harmonic and Egelstaff schemes are more promising (5,6). A fourth scheme satisfying detailed balance,... 

P. Ullersma, An exactly solvable model for Brownian morion. Physica 32,27,56,74,90 (1966). 

An instructive study of the JE for an exactly solvable model of an ideal gas under expansion (where the probability distribution of the work can be derived) verifies the relationship and shows how its predictions compare with numerical results, providing insight into its application. " With similar motives, a model for effusion of the ideal gas is considered, and more recently, this group has extended this analysis to... 

B. Derrida, Random-energy Model An Exactly Solvable Model of Disordered Systems,... 

Qian and Sahni [111] have used the Holas-March route to the exchange-correlation potential Vxc r) to construct this quantity in an exactly solvable model. This is the so-called Hooke s atom, in which two electrons, mutually repelling Coulombically, move in an external harmonic potential. For a particular choice of spring constant, the ground-state wave function is analytically known. In the study of Qian and Sahni (see their Fig. 12), the exchange-correlation potential Vxc is calculated exactly from the Holas-March [52] approach. In particular, Qian and Sahni make a full study of the contribution to VXc from the correlation kinetic energy. 

Such a composite will be further referred to as the Hashin-Strikman composite. The elastic properties of the Hashin-Strikman composite are described by the formulae that are obtained from the exactly solvable model of a single spherical inclusion of one phase in an infinite matrix of the second phase and depend only on the volume concentrations and elastic properties of the constituent phases. The properties of the Hashin-Strikman composite do not depend on the scale chosen. 

It is believed to have non-universal critical exponents, depending on the ratio A /J. A related exactly solvable model, the 8-vertex model (Baxter, 1971, 1972) can be written rather similarly in spin representation as (Kadanoff and... 

For ID structures, the interaction between CTEs can be taken into account only for nearest neighbors. Besides, as follows from (7), due to strong self-trapping of CTEs the bandwidth of CTEs can be neglected. Due to these assumptions an exactly solvable model was obtained in (19). 

The dependence of Tc on the interchain coupling constant is characterized by the ID susceptibility exponent, a result obtained also by Klemm and Gut-freund (7) who studied the phase transitions in a quasi ID metal on the basis of the two exactly solvable models of the ID electron gas (6), (8) following Scalapino et al. s (9) treatment of other pseudo-one-dimensional systems. 

These findings are also supported by the results of a simple, exactly solvable model (Hubbard model for H2) where the ground state is correlated while the singly excited states are not [21]. For this model, the RPA yields bad results collapsing to complex eigenvalues for certain choices of the model parameters while the FOSEP approximation even gives the exact excitation energies. 

We have shown in the previous sections that an exactly solvable model can provide generic results concerning line profiles and dynamics. The physics was discussed in terms of resonances and effective Hamiltonians. These concepts are also of fundamental importance for real systems. Here we recall the simplest one a hydrogen atom in its ground state exposed to a static electric field described in Refs. [10, 35]. 

F. W. Wiegel, in Phase Transitions and Critical Phenomena, C. Domb and J. L. Lebowitz, Eds., Academic press, London, Vol. 7,1983. Conformational Phase Transitions in a Macromolecule Exactly Solvable Models. 

In [96] exactly solvable models based on Dyck paths and Motzkin paths in two dimensions, and a partially directed walk model in three dimensions are given. Orlandini et al. [96] observe re-entrant behaviour in three dimensions, but not in two. Reentrant behaviour is shown in figure 10 below ... 

Keywords Schrodinger equation Biconfluent Heun equation Harmonium Exactly solvable models... 

Kestner NR, Sinanoglu O (1962) Study of electron correlation in helium-like systems using an exactly solvable model. Phys Rev 128 2687-2692... 

Ghosh SK, Samanta A (1991) Study of correlation effects in an exactly solvable model two-electron system. J Chem Phys 94 517-522... 

In this section, we shall consider an exactly solvable model, that of driven Brownian oscillator (DBO) systems. The exact formulations will be presented in this subsection, while the POP-CS-QDT counterpart will be outlined in the next. We will show in Sec. 5 that the POP-CS-QDT and the CODDE agree with the exact QDT very well however, the COP-CS-QDT results in contrary and unphysical behaviors, for example, on its temperature dependence of the equilibrium properties. 

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