Hilbert space physical properties

The quantum states of Schrodinger s theory constitute an example of Hilbert space, and their scalar product has a direct physical meaning. Any such example of Hilbert space, where we can actually evaluate the scalar products numerically, is called a representation of Hilbert space. We shall continue discussing the properties of abstract Hilbert space, so that all our conclusions will apply to any and every representation. 

A. —The states of any physical system have the same properties as vectors in abstract Hilbert space, and there exists a correspondence between the states of a physical system and the elements of which are in what follows to be called the state vectors of the system. 

It is clear that the various density functional schemes for molecular applications rely on physical aiguments pertaining to specific systems, such as an electron gas, and fitting of parameters to produce eneigy functionals, which are certainly not universal. By focusing on the energy functional one has given up the connection to established quantum mechanics, which employs Hamiltonians and Hilbert spaces. One has then also abandoned the tradition of quantum chemistry of the development of hierarchies of approximations, which allows for step-wise systematic improvements of the description of electronic properties. 

The electronic Coulomb interaction u(r 12) = greatly complicates the task of formulating and carrying out accurate computations of iV-electron wave functions and their physical properties. Variational methods using fixed basis functions can only with great difficulty include functions expressed in relative coordinates. Unless such functions are present in a variational basis, there is an irreconcilable conflict with Coulomb cusp conditions at the singular points ri2 - 0 [23, 196], No finite sum of product functions or Slater determinants can satisfy these conditions. Thus no practical restricted Hilbert space of variational trial functions has the correct structure of the true V-electron Hilbert space. The consequence is that the full effect of electronic interaction cannot be represented in simplified calculations. 

The difference with respect to the standard approach lies in the nature of the quantum state. Spin is not taken as a property of a particle. Spin quantum state is sustained by material systems but otherwise a Hilbert space element. A quantum state can be probed with devices located in laboratory (real) space thereby selecting one outcome from among all possible events embodied in the quantum state. The presence of the material system is transformed into the localization of the two elements incorporated in the EPR experiment. If you focus on the localization aspect from the beginning, one is bound to miss the quantum-physical edge. 

The mathematical formalism jofitjuantum mechanics is expressed in terms of linear operators, which rep resent the observables of a system, acting on a state vector which is a linear superposition of elements of an infinitedimensional linear vector space called Hilbert space. We require a knowledge of just the basic properties and consequences of the underlying linear algebra, using mostly those postulates and results that have direct physical consequences. Each state of a quantum dynamical system is exhaustively characterized by a state vector denoted by the symbol T >. This vector and its complex conjugate vector Hilbert space. The product clT ), where c is a number which may be complex, describes the same state. 

Any statement concerning the physical properties of the system has thus to be expressed in terms of mathematical operations defined on this Hilbert space. 

If the creation and annihilation operators anticommute properly, they change the occupation numbers in an abstract Hilbert space of particle number representation which can be considered the same even if the physical orbitals do move (change). For this reason the true fermion operators need not be varied either. Their algebraic properties are determined by the relevant commutation rules, which are also independent of the physical properties of the system or of the nature of the basis orbitals. These anticommutation properties of the operators are the same after and before the variation. 

Successful model building is at the very heart of modern science. It has been most successful in physics but, with the advent of quantum mechanics, great inroads have been made in the modelling of various chemical properties and phenomena as well, even though it may be difficult, if not impossible, to provide a precise definition of certain qualitative chemical concepts, often very useful ones, such as electronegativity, aromaticity and the like. Nonetheless, all successful models are invariably based on the atomic hypothesis and quantum mechanics. The majority, be they of the ah initio or semiempirical type, is defined via an appropriate non-relativistic, Born-Oppenheimer electronic Hamiltonian on some finite-dimensional subspace of the pertinent Hilbert or Fock space. Consequently, they are most appropriately expressed in terms of the second quantization formalism, or even unitary group formalism (see, e.g. [33]). 

A way to overcome the difficulties in the definition of the Hermitian phase operator has been proposed by Pegg and Barnett [40,45]. Their method is based on a contraction of the infinite-dimensional Hilbert-Fock space of photon states Within this method, the quantum phase variable is determined first in a finite 5-dimensional subspace of //, where the polar decomposition is allowed. The formal limit, v oc is taken only after the averages of the operators, describing the physical quantities, have been calculated. Let us stress that any restriction of dimension of the Hilbert-Fock space of photons is equivalent to an effective violation of the algebraic properties of the photon operators and therefore can lead to an inadequate picture of quantum fluctuations [46]. 

Whenever the same parameters are available from two different curves (e.g., wq aiid t from Figure 1 or Figure 4a), there is some mathematical relation between the curves. For the "linear" system we have considered (i.e., displacement is proportional to driving amplitude Fq) the time-domain and frequency-domain responses are connected by a Fourier transform. Similarly, absorption and dispersion spectra both yield the same information, and are related by a Hilbert transform (see Chapter 4). In this Chapter, we will next develop some simple Fourier transform properties for continuous curves such as Figures 1-4, and then show the advantages of applying similar relations to discrete data sets consisting of actual physical responses sampled at equally-spaced intervals. 

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