Quantum mechanics finite-size scaling

For readers who desire more details on the development of the theory and applications, there are many excellent review articles and books on this subject in the literature [22-25]. However, in this review chapter we are going to present only the general idea of finite-size scaling in statistical mechanics, which is closely related to the application of these ideas in quantum mechanics. 

B. Finite-Size Scaling Equations in Quantum Mechanics... 

The third method is a direct finite-size scaling approach to study the critical behavior of the quantum Hamiltonian without the need to make any explicit analogy to classical statistical mechanics [54,88]. The truncated wave function that approximate the eigenfunction Eq. (52) is given by... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

The anisotropy of polymer erystals is eharacteristie of the orientation of intra- and intermoleeular interaetions, which is also reflected in a broad separation on the dynamical time scale. It is especially important in polymer crystals to be cognizant of the limitations imposed by either the assumptions on which a method is based (e.g., the quasiharmonic approximation for lattice dynamics) or the robustness of the simulation method (e.g., ergodicity of the simulation in Monte Carlo or molecular dynamics). In some instances, valuable checks on these assumptions and limitations (e.g., quantum mechanical vs. classical dynamics, finite size effects and anharmonicity) can and should be made by repeating the study using more than one method. 

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