Finite dimensional Fock space

A Hermitian operator p is a von Neumann density if it is nonnegative and has unit trace. In more concrete terms, if is the finite-dimensional Fock space for a quantum model where electrons are distributed over a finite number of states, then p is a von Neumann density if (i) v,pv) > 0 for all operators v on and (ii) l)g = 1. By the formula v,pv) we mean the trace scalar product of the operators v and pv, that is, v,pv) = traceg(u pu) since (p, l)g = tracegp = 1 we have used this scalar product to express the trace condition. More generally. 

Of course, the exact state cannot be anything but size-extensive and the demonstration given in this subsection may therefore seem somewhat pedantic. However, the discussion of size-extensivity for exact states prepares us for the discussion of size-extensivity for approximate wave functions in Sections 4.3.2-4.3.4. Moreover, the demonstration of size-extensivity given here is valid also for the exact solution in a finite-dimensional Fock space, not just the exact state in a complete basis. 

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]. 

The Fock space dimensionality is finite when one deals with finite model systems. Thus, it appears that we can solve the model Hamiltonians exactly for finite systems and from the finite system properties, infer the behaviour of the system in the thermodynamic limit by suitable scaling techniques. However, the dimensionality increases as (25 4-1) for a spin-S chain and as 4 for fermions, where N is the number of sites. Thus, it is very difficult to carry out brute force numerical computations on a large system and the exact diag-onalization studies are primarily restricted to quasi-one-dimensional systems with very few sites per unit cell. 

The algebraic approximation results in the domain of the operator being restricted to a finite-dimensional subspace S of Hilbert space. For an AT-electron system, the algebraic approximation may be implemented by defining a suitable orthonormal basis set of M > N) one-electron spin orbitals (most often solutions of the matrix Hartree-Fock equations) and then constructing all unique IV-electron determinants. The number of unique determinants that can be formed is given by... 

---