Hilbert space characterized

This analysis is heuristic in the sense that the Hilbert spaces in question are in general of large, if not infinite, dimension while we have focused on spaces of dimension four or two. A form of degenerate perturbation theory [3] can be used to demonstrate that the preceding analysis is essentially correct and, to provide the means for locating and characterizing conical intersections. 

By following Section II.B, we shall be more specific about what is meant by strong and weak interactions. It turns out that such a criterion can be assumed, based on whether two consecutive states do, or do not, form a conical intersection or a parabolical intersection (it is important to mention that only consecutive states can form these intersections). The two types of intersections are characterized by the fact that the nonadiabatic coupling terms, at the points of the intersection, become infinite (these points can be considered as the black holes in molecular systems and it is mainly through these black holes that electronic states interact with each other.). Based on what was said so far we suggest breaking up complete Hilbert space of size A into L sub-Hilbert spaces of varying sizes Np,P = 1,..., L where... 

In case of three conical intersections, we have as many as eight different sets of eigenfunctions, and so on. Thus we have to refer to an additional chai acterization of a given sub-sub-Hilbert space. This characterization is related to the number Nj of conical intersections and the associated possible number of sign flips due to different contours in the relevant region of configuration space, traced by the electronic manifold. 

The general formula and the individual cases as presented in Eq. (97) indicate that indeed the number of conical intersections in a given snb-space and the number of possible sign flips within this sub-sub-Hilbert space are interrelated, similar to a spin J with respect to its magnetic components Mj. In other words, each decoupled sub-space is now characterized by a spin quantum number J that connects between the number of conical intersections in this system and the topological effects which characterize it. 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

Real space algorithms (section 4) allow for mappings between present day computer programs and strict molecular quantum mechanics [10,11]. It is the separability of base molecular states that permits characterizing molecular states in electronic Hilbert space and molecular species in real space. This feature eliminates one of the shortcomings of the standard BO scheme [6,7,12]. Confining and asymptotic GED states are introduced. In section 5 the concept of conformation states in electronic Hilbert space is qualitatively presented. 

Density matrices can be characterized as trace-class linear operators on a Hilbert space, > —> >, such that is self-adjoint, non-negative and of trace equal to one, where trace is defined by... 

In the description of nature afforded by quantum mechanics, one classifies and characterizes the state of a total system in terms of the eigenvalues of a set of commuting observables acting on an element of the Hilbert space, the state vector. Molecular orbital theory in its canonical representation as originally... 

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. 

Consider a system characterized by agiven Hamiltonian operator., an orthonormal basis (/) (also denoted n ) that spans the corresponding Hilbert space and a time dependent wavefunction h (i)—a normalized solution of the Schrodinger equation. The latter may be represented in terms of the basis ftmctions as... 

Quantum mechanics involves the characterization of a physical system by a set of Hermitian operators, one for any observable quantity, in a state space S assumed to be a Hilbert space. In Schrodinger s perspective, S was viewed as a space of complex wave functions with differential operators as tools. In this sense, the operator characterizing the energy of the system, the Hamilton operator H, was one of the most important. However, linear momentum P, coordinated spatial positions Q, rotational (orbital) momentum L, the square of the total momentum L2, and the spin J of... 

In Fig. 1, we have rearranged the wavefunctions in Hilbert space such that in each row we find all wavefunctions which yield the same density. Since pv0 is the exact one-particle density, it follows that the exact wavefunction must lie in the row containing all wavefunctions which yield pv0. In Fig. 1, we identify the exact wavefunction with h In fact, we characterize the vertical line denoted by... 

Some of the basis functions define an active space P and the remaining part of Hilbert space is called the orthogonal space Q = I — P. The active space is spanned by the basis functions that have a filled core and the remaining active electrons distributed over a set of active orbitals. The orthogonal complete space incorporates aU other possible basis functions that are characterized by having at least one vacancy in a core orbital. The state wavefunction in an active space is written as... 

Let us have a set of model functions < spanning an IMS, with the projector P. The complementary functions span the complement of the IMS, with the projector R. P + R) is the projector to the CMS, Pc- The rest of the functions in the Hilbert space are the virtual functions, having at least one inactive occupancy (hole or particle), characterized by the projector Q. (P + Q) = I, the entire Hilbert space of a given A-electron problem in a finite basis. 

The temperature and the chemical potential are direct macroscopic observables par excellence. Since the quantum mechanics say rules that the observable size may be represented by the self-adjoint operators formed on the Hilbert space, is adding the interest to associate the operators for the temperature and the chemical potential for an adequate macroscopic characterization. 

Therefore, the Wigner function is directly related to the density matrix operator, p, which characterizes the quantum system [28] in Hilbert space. 

T. Werther (2003) Characterization of semi-Hilbert spaces with application in scattered data interpolation. Curve and Surface Fitting Saint-Malo 2002, A. Cohen, J.-L. Merrien, and L.L. Schumaker (eds.), Nashboro Press, Brentwood, 374-383. 

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