Basis states

A partial acknowledgment of the influence of higher discrete and continuum states, not included within the wavefunction expansion, is to add, to the tmncated set of basis states, functions of the fomi T p(r)<6p(r) where dip is not an eigenfiinction of the internal Flamiltonian but is chosen so as to represent some appropriate average of bound and continuum states. These pseudostates can provide fiill polarization distortion to die target by incident electrons and allows flux to be transferred from the the open channels included in the tmncated set. [Pg.2050]

We now describe the relation between a purely formal calculational device, like a gauge transformation that merely admixes the basis states, and observable effects. [Pg.155]

In the present implementation, the unperturbed functions are not subject to any orthogonality constraint nor are required to diagonalize any model hamiltonian. This freedom yields a faster convergence of the variational expansion with the basis size and allows to obtain the phaseshift of the basis states without the analysis of their asymptotic behaviour. [Pg.368]

Using GTO bases, it cannot be expected that the variational representations of the electron waves are snfficiently accnrate far ontside the so-called molecular region , i.e. the rather limited region of space where the potential clearly deviates from the asymptotic Conlomb form. Therefore the phaseshifts of the pwc basis states cannot be obtained from the analysis of their long-range behaviour, as was done in previous works with the STOCOS bases. In the present approach, this analysis may be avoided since the K-matrix techniqne allows to determine, by equation [3] below, the phase-shift difference between the eigenfunctions of Hp and the auxiliary basis functions... [Pg.369]

Fig.2 p-wave phaseshifts. Full line GTO results, broken line STOCOS results, circles GTO basis states, points STOCOS basis states. [Pg.376]

A different sequence of electromagnetic pulses can be used to directly monitor Rabi oscillations between the two spin qubit basis states (Figure 7.7). As expected,... [Pg.198]

Figure 5. Schematic illustration of the time evolution of a time-displaced basis. Basis states 1, 2, and 3 belong to one seed while 4, 5, and 6 belong to another. The basis set is shown at two time points, and the leading basis functions are shaded in gray. The arrows connecting basis functions indicate required new matrix elements at time t + At. For this specific example, 11 new matrix elements are evaluated at each point in time, compared to 21 if all basis functions had been chosen independently. (Figure adapted from Ref. 40.)...

The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A. The quantum numbers are conveniently arranged in patterns (or tableaux), called Young tableaux. Tensor representations of Lie algebras are characterized by a set of integers... [Pg.23]

Chain (I). Basis states in this chain are characterized by the quantum numbers... [Pg.28]

The basis states N,nx> or IF,FZ > can be written explicitly in terms of boson creation and annihilation operators... [Pg.30]

We use the bracket notation of Dirac, following standard practice. The ket J,M> corresponding to (A.32) is also called a basis state. [Pg.204]

In what follows we introduce the model Hamiltonian using both diabatic and adiabatic representations. Adopting diabatic electronic basis states /j ), the molecular model Hamiltonian can be written as [162, 163]... [Pg.251]

The similar appearance of the quantum and classical Liouville equations has motivated several workers to construct a mixed quantum-classical Liouville (QCL) description [27 4]. Hereby a partial classical limit is performed for the heavy-particle dynamics, while a quantum-mechanical formulation is retained for the light particles. The quantities p(f) and H in the mixed QC formulation are then operators with respect to the electronic degrees of freedom, described by some basis states 4> ), and classical functions with respect to the nuclear degrees of freedom with coordinates x = x, and momenta p = pj — for example. [Pg.287]

Let us next consider an Al-Ievel system with basis states / ) n = I,..., N) and the Hamiltonian... [Pg.304]

In obvious analogy to Schwinger s theory of angular momentum, this N-level system can be represented by N oscillators, whereby the mapping relations for the operator and the basis states read [99]... [Pg.304]

Let us start with a brief review of spin-coherent state theory. For simplicity we focus on a two-level (or spin 1/2) system. The coherent states for a two-level system with basis states /i), /2) can be written as [136, 139]... [Pg.355]

A similar construction yields the expansion of the right basis states r)... [Pg.157]

The exciton model of polyene spectra assumes that each excited state of the polyene may be described by a linear combination of basis states, each having only one (singly) excited ethylenic unit. Only states with neighboring excited ethylenic units can interact. [Pg.56]

The basic quantities in this model are Ey the n-ir excitation energy of ethylene, and r, the interaction energy of two basis states with neighboring excited ethylenic-units. Their values are Ey = 7.60 ev, r = -2.54 ev. [Pg.56]

The essential concept in the definition of the CDF is the use of time-dependent basis states in place of stationary basis states in the representation of the time evolution of a system, with the constraint that both sets of states are orthonormal. Consider a complete set of orthonormal stationary states S and a complete set of orthonormal time-dependent basis states D t) related by the unitary transformation U t) ... [Pg.54]

Then any state vector of a system, (f)) that satisfies the time-dependent Schrodinger equation with Hamiltonian H(t) and is represented in terms of the basis states S can also be represented in terms of the basis states D t) via the transformation... [Pg.54]

In principle, the choice of the basis states (p ) is arbitrary as long as the basis set spans the entire Hilbert space. In the following subsections, we will discuss three choices of the basis set relevant for collision problems in external fields. [Pg.323]

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