Density Hilbert space

The mixed, v t — % notation here has historic causes.) The Schrodinger equation is obtained from the nuclear Lagrangean by functionally deriving the latter with respect to t /. To get the exact form of the Schrodinger equation, we must let N in Eq. (95) to be equal to the dimension of the electronic Hilbert space (viz., 00), but we shall soon come to study approximations in which N is finite and even small (e.g., 2 or 3). The appropriate nuclear Lagrangean density is for an arbitrary electronic states... 

Meanwhile orbitals cannot be observed either directly, indirectly since they have no physical reality contrary to the recent claims in Nature magazine and other journals to the effect that some d orbitals in copper oxide had been directly imaged (Scerri, 2000). Orbitals as used in ab initio calculations are mathematical figments that exist, if anything, in a multi-dimensional Hilbert space.19 Electron density is altogether different since it is a well-defined observable and exists in real three-dimensional space, a feature which some theorists point to as a virtue of density functional methods. 

One can show (30) that densities are square integrable and thus belong to the Hilbert space I2 (Y) of square integrable functions. This allows one to define... 

In the preceding discussion we have expanded the density in terms of N < M Hilbert space such that their norms are less than or equal to one and the trace of the density is equal to N. All these expansions could in principle be exact there is no need for M = r =, as is clearly demonstrated in the KS procedure, where M = N Ai M <°o and i- = < , then new forms of auxiliary states, i.e. different from single determinantal ones, are implicitly introduced. 

Another class of expansions is also possible, but in these the functions cannot be interpreted as belonging to the Hilbert space of one-particle states, even though they are functions of one space and one spin variable and do belong to a Hilbert space. In such expansions the nwms of the functions are less than or equal to N and 1 < M < °°, while the trace of the density is equal to N. In the extreme case of Af = 1 one can even express the density as p = am, where... 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

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

It is clear that the Mulliken operator works in the Hilbert space of the basis functions, which has repercussions on the way the electron density is assigned to the nuclei. The basis functions are allocated to the atomic nuclei they are centered on the decision as to which portion of the electron density belongs to which nucleus rests on... 

Similarly, let us consider a wavefunction particle functions in Hilbert space is denoted by... 

The density matrix ps(0) is an operator in the Hilbert space Hs of S and represents an arbitrary initial condition. The matrix pB(0) operates in HB as in the preceding section we take for it an equilibrium distribution... 

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. 

By performing a partial Wigner transform with respect to the coordinates of the environment, we obtain a classical-like phase space representation of those degrees of freedom. The subsystem coordinate operators are left untransformed, thus, retaining the operator character of the density matrix and Hamiltonian in the subsystem Hilbert space [4]. In order to take the partial Wigner transform of Eq. (1) explicitly, we express the Liouville-von Neumann equation in the Q representation,... 

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,... 

In Kohn-Sham theory, densities are postulated to be sums of orbital densities, for functions (pi in the orbital Hilbert space. This generates a Banach space [102] of density functions. Thomas-Fermi theory can be derived if an energy functional E[p] = I p + F [ p is postulated to exist, defined for all normalized ground-state... 

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... 

Theoretical chemistry has two problems that remain unsolved in terms of fundamental quantum theory the physics of chemical interaction and the theoretical basis of molecular structure. The two problems are related but commonly approached from different points of view. The molecular-structure problem has been analyzed particularly well and eloquent arguments have been advanced to show that the classical idea of molecular shape cannot be accommodated within the Hilbert-space formulation of quantum theory [161, 2, 162, 163]. As a corollary it follows that the idea of a chemical bond, with its intimate link to molecular structure, is likewise unidentified within the quantum context [164]. In essence, the problem concerns the classical features of a rigid three-dimensional arrangement of atomic nuclei in a molecule. There is no obvious way to reconcile such a classical shape with the probability densities expected to emerge from the solution of a molecular Hamiltonian problem. The complete molecular eigenstate is spherically symmetrical [165] and resists reduction to lower symmetry, even in the presence of a radiation field. 

At this point we have expressed the Hamiltonian, the density operator and the evolution operator in Fourier space. We have introduced an effective Hamiltonian, defined in the Hilbert space of the same dimension 2 as the total time-dependent Hamiltonian itself, and we have shown how to transform operators between the two representations. The definition of the effective Hamiltonian enables us to predict the overall evolution of the spin system, despite the fact that we can not find time-points for synchronous detection, f, where Uint f) = exp —iWe//t In actual experiments the time dependent signals are monitored and after Fourier transformation they result in frequency sideband... 

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. 

For a density matrix pab defined on a tensor producted Hilbert space Ha 2 Ttb this state is unentangled if it may be written as a convex sum of unentangled states, namely... 

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