Hilbert space Hamiltonian

Then, two things (that are actually interdependent) happen (1) The field intensity F = 0, (2) There exists a unique gauge g(R) and, since F = 0, any apparent field in the Hamiltonian can be transformed away by introducing a new gauge. If, however, condition (1) does not hold, that is, the electronic Hilbert space is truncated, then F is in general not zero within the tmncated set. In this event, the fields A and F cannot be nullified by a new gauge and the resulting YM field is true and irremovable. 

The action of the Hamiltonian, H, can be expressed as a superoperator mapping the Hilbert space 2 (iK j into itself by... 

It applies for both formulations above that the expansion in principle contains an infinite number of terms. The convergence to a few lowest order terms relies on the ability to orderly separate influences of the dominant rf irradiation terms (through a suitable interaction frame) from the less dominant internal terms of the Hamiltonian. In principle, this may be overcome using the spectral theorem (or the Caley-Hamilton theorem [57]) providing a closed (i.e., exact) solution to the Baker-Campbell-Hausdorf problem with all dependencies included in n terms where n designates the dimension of the Hilbert-space matrix representation (e.g., 2 for a single spin-1/2, 4 for a two-spin-1/2 system) [58, 59]. 

The previous argument is valid for all observables, each represented by a characteristic operator X with experimental uncertainty AX. The problem is to identify an elementary cell within the energy shell, to be consistent with the macroscopic operators. This cell would constitute a linear sub-space over the Hilbert space in which all operators commute with the Hamiltonian. In principle each operator may be diagonalized by unitary transformation and only those elements within a narrow range along the diagonal that represents the minimum uncertainties would differ perceptibly from zero. 

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

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. 

Consider the extreme case where H is diagonal in the base set Qjk(q,Q). Accordingly, in absence of external fields, no time evolution is to be expected except for changes in time-phases. But, by hypothesis, we took H to be the generator of time displacement in Hilbert space. Such a situation is not useful in molecular physics because one search after a Hamiltonian that is able to generate the time evolution. This suggests the idea that the time generator H of interest contains two classes of operators H = + V. The Hamiltonian Ho is assumed to... 

As a consequence of eq. (10), the molecular Hamiltonian H cannot by itself be a generator of time evolution in Hilbert space, for this reason it was assigned to Ho in eq. (5). 

Alternatively, Eq. (4.160) can be written as an overlap of system and bath matrices, which allows a more direct physical interpretation. To do so, we assume /-dimensional Hilbert space and expand the interaction Hamiltonian as a sum of products of system and bath operators. 

The starting point for Lowdin s PT [1-6] and Eeshbach s projection formalism [7-9] is the fragmentation of the Hilbert space H = Q V, of a given time-independent Hamiltonian H, into subspaces Q and V by the action of projection operators Q and P, respectively. The projection operators satisfy the following conditions ... 

Thus the eigenstates of HSF are labeled by the partitions [ASP] associated with the irreducible representations of S%v. These [Asp] labels are called permutation quantum numbers. The spin-free Hilbert space Fsp of the Hamiltonian may be decomposed into subspaces FSF([ASF]) invariant to 5 p... 

The decomposition eq. (2-6) of the spin-free space FSP induces a decomposition on the Pauli-allowed portion of the Hilbert space of the Hamiltonian H of eq. (2-1). The Hamiltonian H which includes spin interactions may operate on any ket of the space Fsp V", where V is the electronic spin space. Here the symbol indicates a tensor product, so that Fsp Va consists of all spatial-spin kets which are composed of linear combinations of a simple product of a spatial ket of FSP and a spin ket of Va. The Pauli-allowed portion of the total A-electron Hilbert space of is... 

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. 

To describe a chemical reaction from a physical standpoint at the nonrelati-vistic level, one must first construct the Hilbert space associated with all quantum states related to the system defined by its molecular hamiltonian, Hm. For the isolated system the time-dependent Schrodinger equation... 

The formulas of Appendix A are easily transformed to time-dependent forms via the Fourier-Laplace transform. For the purpose of upcoming extensions to the nonself adjoint case, we introduce the self-adjoint Hamiltonian H = FT = Htt. Furthermore the function 0 considered above is said to belong to the domain of the operator H, i.e.,

self-adjoint problems D(W) = D(H) and for bounded operators, the range R(H) and the domain D(H) are identical with the full Hilbert space however, in general applications they will differ as we will see later. 

E. Eliav, A. Borschevsky, K.R. Shamasundar, S. Pal, U. Kaldor, Intermediate Hamiltonian Hilbert space coupled cluster method Theory and pilot application, Int. J. Quantum Chem. 109 (13) (2009) 2909. 

Using the Dirac notations a) = ipa( ) and assuming that ipa( ) are or-thonormal functions (a (3) = 5ap we can write the single-particle matrix (tight-binding ) Hamiltonian in the Hilbert space formed by 4>a ( )... 

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,... 

For fixed total energy E, Equation (2.59) defines one possible set of Nopen degenerate solutions I/(.R, r E, n),n = 0,1,2,..., nmax of the full Schrodinger equation. As proven in formal scattering theory they are orthogonal and complete, i.e., they fulfil relations similar to (2.54) and (2.55). Therefore, the (R,r E,n) form an orthogonal basis in the continuum part of the Hilbert space of the nuclear Hamiltonian H(R, r) and any continuum wavefunction can be expanded in terms of them. Since each wavefunction (R, r E, n) describes dissociation into a specific product channel, we call them partial dissociation wavefunctions. 

One of the limiting problems of numerically solving the eigenvalue problem of a given spin Hamiltonian for a system with a finite number of spin centers is that the Hilbert space dimension Q that translates into the dimension of matrices that need to be diagonalized increases with... 

Some comments about nonlinearities in the Hamiltonian may be added here. The case we are considering here is called scalar nonlinearity (in the mathematical literature it is also called nonlocal nonlinearity ) [7] this means that the operators are of the form P(u) = (An, u)Bu where A, B are linear operators and<.,.>is the inner product in a Hilbert space. The literature on scalar nonlinearities applied to chemical problems is quite scarce (we cite here a few papers [2,8]) but the results justified by this approach are of universal use in solvation methods. 

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