Hilbert space techniques

Traditional wisdom in the field of nonequilibrium statistical mechanics has it that one gets rid of recurrences, in the thermodynamic limit, because the spectrum of the Hamiltonian becomes dense or continuous. This kind of statement must be taken cum grano salts. First, it is next to impossible to exhibit, in the usual formalism, an operator that would be mathematically well defined and that would be obtained from a finite system Hamiltonian by the limiting procedures that Hilbert space techniques (as commonly used) have to offer. Second, if this problem were to be ignored for a while, it would still be of interest to know how dense or continuous things become in the thermodynamic limit. And third, it would be useful to link these how much questions and answers with the rate of approach to equilibrium. 

We have established in Chapter 2 that, in the case where >1 is a linear operator, D and M are Hilbert spaces, and s(m) is a quadratic functional, the solution of the minimization problem (4.99) is unique. Let us find the equation for the minimum of the functional F (m). We will use the same technique for solving this problem that we considered above for the misfit functional minimization. 

Although in the semiclassical model the only dynamical variables are those of the molecule, and the extended Hilbert space. if = M M and the Floquet Hamiltonian K can be thought as only mathematically convenient techniques to analyze the dynamics, it was clear from the first work of Shirley [1] that the enlarged Hilbert space should be related to photons. This relation was made explicit by Bialynicki-Birula and co-workers [7,8] and completed in [9]. The construction starts with a quantized photon field in a cavity of finite volume in interaction with the molecule. The limit of infinite volume with constant photon density leads to the Floquet Hamiltonian, which describes the interaction of the molecule with a quantized laser field propagating in free space. The construction presented below is taken from Ref. 9, where further details and mathematical precisions can be found. 

Since in the Floquet representation the Hamiltonian K defined on the enlarged Hilbert space is time-independent, the analysis of the effect of perturbations (like, e.g., transition probabilities) can be done by stationary perturbation theory, instead of the usual time-dependent one. Here we will present a formulation of stationary perturbation theory based on the iteration of unitary transformations (called contact transformations or KAM transformations) constructed such that the form of the Hamiltonian gets simplified. It is referred to as the KAM technique. The results are not very different from the ones of Rayleigh-Schrodinger perturbation theory, but conceptually and in terms of speed of convergence they have some advantages. 

In order to apply the methods and techniques of (nonrelativistic) quantum theory to an evolution equation of the form (8), one has to define a Hilbert space as an appropriate state space. At every instance of time, the solution has to be an element of this Hilbert space. 

Note that in above deductions the double (independent) averages technique was adopted, exploiting therefore the associate sums inter-conversions to produce the simplified results (Park et al., 1980 Blanchard, 1982 Snygg, 1982). Yet, this technique is equivalent with quantum mechanically factorization of the entire Hilbert space into sub-spaces, or at the limit into the subspace of interest (that selected to be measured, for instance) and the rest of the space being thus this approach equivalent with a system-bath sample this note is useful for latter better understanding of the stochastic phenomena that underlay to open quantum systems, being this the physical foundation for chemical reactivity. 

However, the Brillouin-Wigner-based techniques can be applied to the calculation of first order properties, such as dipole moments and multipole moments, as well as second order properties, such as polarizabilities and hyperpolarizabilities. In calculations of properties such as ionization potentials and electron affinities to use of a Fock space formulation is more appropriate that the Hilbert space formulation that we have followed in this monograph. Some progress has been made [94,103] in formulating the necessary Fock space BrillouinWigner methodology. The interested reader can find further details in the original publications [94,103]. 

There is another important conceptual, or even philosophical, difference between the orbital/wavefunction methods and the required density functional methods. In the case of orbitals, the theoretical entities are completely unobservable, whereas electron density, which is featured in density functional theories, is a genuine observable. Experiments to observe electron densities have been routinely conducted since the development of X-ray and other diffiaction techniques. Orbitals cannot be observed either directly or indirectly since they have no physical reality, a state of affairs dictated by quantum mechanics. The orbitals used in ab initio calculations are just mathematical constructs that exist in a multidimensional Hilbert space, while electron density is altogether different, as indicated, since it is a well-defined observable and exists in real three-dimensional space. ... 

DMBE) methods. Regarding the interpolation techniques, we focus on the global reproducing kernel Hilbert space method,... 

It is not the aim of this section to review scattering theory which can be found in textbooks, for instance, [1-3] and in review articles [10-12]. Here we shall only recall the expressions of the transition operator in the framework of the partition technique. In a second step we will use the model Hamiltonian of Part I to derive new exact expressions of the transition matrices. The Hilbert space is partitioned into the space of the resonances (the n-dimensional model space) and the space of the collision states (the infinite-dimensional complementary space). The projectors onto these two spaces are Pq and Qo respectively Pq + Qo = 1- The exact resolvent can be written in the form (see Part I and Ref. [13])... 

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