Extended Hilbert spaces

The density operator in the extended Hilbert space is given by... 

Here, for notational convenience, we have assumed that Vnm = We would like to emphasize that the mapping to the continuous Hamiltonian (88) does not involve any approximation, but merely represents the discrete Hamiltonian (1) in an extended Hilbert space. The quantum dynamics generated by both Hamilton operators is thus equivalent. The Hamiltonian (88) describes a general vibronically coupled molecular system, whereby both electronic and nuclear DoF are represented by continuous variables. Contrary to Eq. (1), the quantum-mechanical system described by Eq. (88) therefore has a well-defined classical analog. 

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. 

The First idea of quantum error-correction, which we have already employed in the bit flip code, is to "give space" to the system by adding extra qubits, which play the role of ancillary qubits this ancilla adding procedure is highly related to the notion of redundancy in classical error-correction. Then, one encodes the information onto a well-chosen subspace C, the code space, of the extended Hilbert space of the system comprising the initial plus extra qubits. In other words, one applies a well-chosen unitary transformation C, the coding matrix, which "delocalizes" information on all the qubits of the system. That is exactly what we did in the bit flip code, when encoding information onto the subspace spanned by 0l) = 000), 1 l) = 111). ... 

It will be important to continue the development of the effective Hilbert space formalism (58) for converting problems with explicitly time-dependent Hamiltonians to stationary problems in an extended Hilbert space. This feature will permit the use of propagation algorithms now used for problems with time-independent Hamiltonians to also be used for pulsed laser-molecule interactions. Peskin and Moiseyev (162) have recently developed and applied a version of the extended space formalism (they refer to it as the (t, / ) formalism). 

Density Functions play a fundamental role in the definition of Quantum Theory, due to this they are the basic materials used in order to define Quantum Objects and from this intermediate step, they constitute the support of Quantum Similarity Measures. Here, the connection of Wavefunctions with Extended Density Functions is analysed. Various products of this preliminary discussion are described, among others the concept of Kinetic Energy Distributions. Another discussed set of concepts, directly related with the present paper, is constituted by the Extended Hilbert Space definition, where their vectors are defined as column structures or diagonal matrices, containing both wavefunctions and their gradients. The shapes of new density distributions are described and analysed. All the steps above summarised are completed and illustrated, when possible, with practical application examples and visualisation pictures. 

The extended Hilbert space wavefunction formalism discussed in this work becomes a rich source of new concepts. Among others, it can be mentioned the deduction of KE DF and several alternative DF, like the functions constructed using Quadrupole or Angular Momentum operators. Such new DF forms are based upon EH spaces, where initially the wavefunction and its gradient are taken simultaneously into account. The new DF can be described and visualised too, producing useful suggestive pictures of the molecular structure environment. It is set in this manner a new source of DF able to be used in QO definition and, thus, in QSM studies. 

The round brackets indicate the so-called /i-product which is a generalisation of the usual Hilbert space inner product with an indefinite metric. The extended operator H takes the role of an excitation energy operator which lives, like the extended states A,B), in an extended Hilbert space Y. These objects will be defined rigorously in this chapter in a very formal manner. In first reading these formal definitions may well be skipped. 

Although we have addressed up to now mostly the electron density, several studies have reported the use of some derived entity in lieu of the electron density. In fact, such a case has already been introduced in the so-called kinetic MQSM of Eq. [27], where we use the gradient of the electron density. This process is also true for those situations in which extended density functions are used in which several new density functions were derived from extended Hilbert space. ... 

Attention is directed to a previous discussion of what happens when the electronic basis is extended to the complete Hilbert space, [79] p. 60 especially Eqs. (2.17)-(2.18). It is shown there that in that event the full symmeti of the invariance group is regained (in effect, through the cancellation of the... 

We prove our statement in two steps First, we consider the special case of a Hilbert space of three states, the two lowest of which are coupled strongly to each other but the third state is only weakly coupled to them. Then, we extend it to the case of a Hilbert space of N states where M states are strongly coupled to each other, and L = N — M) states, are only loosely coupled to these M original states (but can be stiongly coupled among themselves). 

In Section n.B, it was shown that the condition in Eq. (10) or its relaxed form in Eq. (40) enables the construction of sub-Hilbert space. Based on this possibility we consider a prescription first for constmcting the sub-Hilbert space that extends to the full configuration space and then, as a second step, constructing of the sub sub-Hilbert space that extends only to (a finite) portion of configuration space. 

As stated in the introduction, we present the derivation of an extended BO approximate equation for a Hilbert space of arbitary dimensions, for a situation where all the surfaces including the ground-state surface, have a degeneracy along a single line (e.g., a conical intersection) with the excited states. In a two-state problem, this kind of derivation can be done with an arbitary t matrix. On the contrary, such derivation for an N > 2 dimensional case has been performed with some limits to the elements of the r matrix. Hence, in this sence the present derivation is not general but hoped that with some additional assumptions it will be applicable for more general cases. 

In any practical application of the Hohenberg-Kohn theory, a specified density functional / s[p] restricted to ground-state densities defines an equivalent orbital functional A [ ,, ] that can be extended to all functions in the orbital Hilbert space. The OEL equations for occupied orbitals of the reference state of an A-electron ground state take the form, for i < N,... 

The Schlosser-Marcus variational principle is derived for a single surface a that subdivides coordinate space 9i3 into two subvolumes rm and rout. This generalizes immediately to a model of space-filling atomic cells, enclosed for a molecule by an external cell extending to infinity. The continuity conditions for the orbital Hilbert space require i>out =a i>in This implies a vanishing Wronskian surface integral... 

If, further, T is a linear operator defined on the L2 space , one says that an element F = F(X) belongs to the domain D(T) of the operator T, if both T and its image TT belong to L2 the set T P is then referred to as the range of the operator T. It should be observed that a bounded operator T as a rule has the entire Hilbert space as its domain—or may be extended to achieve this property—whereas an unbounded operator T has a more restricted domain. In the latter case, we will let the symbol C(T) denote the complement to the domain D(T) with respect to the Hilbert space. A function F = (X) is hence an element of C(T), if it belongs to L2, whereas this is not true for its image 7 F. 

This result was extended by Samokhin (1998) to the more general case of the linear operator equation in a complex Hilbert space M (see Appendix A), which is extremely important for many geophysical applications, for example, in electromagnetic forward and inverse problems. The main difference between the real and the... 

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