Quantum matrix formulation

Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational... 

In die matrix formulation of quantum mechanics Eq. <121) is transformed. If P and Q are matrices, the Hamiltonian becomes by analogy... 

The fact that quantum observables are represented by matrices immediately suggests problems of non-commutation. For instance, the observables can be measured at the same time only if they have a complete orthogonal set of eigenvectors in common. This happens only when they commute, i.e. XY = YX, or the commutator [X, Y] = XY — YX = 0. This is a central feature of the matrix formulation of quantum theory discovered by Heisenberg, Born and Jordan while trying to explain the observed spectral transitions of the hydrogen atom in a more fundamental way than the quantization... 

The algebraic (or matrix) formulation of quantum mechanics1 is less familiar than the differential (or wave) formulation. This is a disadvantage, and one purpose of the present volume is to show, by explicit examples, the benefits of the algebraic approach. The interested reader will have to judge if the benefits are sufficient to overcome the potential barrier to the understanding of a new approach. We intend to demonstrate that the algebraic formulation is indeed a viable alternative. 

White, S.R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 1992, 69(19), 2863. 

H.-J. Werner, Matrix-Formulated Direct Multiconfiguration Self-Consistent Field and Multiconfiguration Reference Configuration-Interaction Methods, in Ab Initio Methods in Quantum Chemistry - II (K.P. Lawley, ed.), John Wiley Sons Ltd, Chichester (1987). 

The solution of single-particle quantum problems, formulated with the help of a matrix Hamiltonian, is possible along the usual line of finding the wave-functions on a lattice, solving the Schrodinger equation (6). The other method, namely matrix Green functions, considered in this section, was found to be more convenient for transport calculations, especially when interactions are included. 

Another advantage of IET (and MET) is the matrix formulation of the theory making it applicable to reactions of almost arbitrary complexity. A subject of special attention here will be photochemical reactions composed from sequential geminate and bimolecular stages and accompanied by spin conversion, thermal decay, and light saturation of the excited reactants. The quantum yields of fluorescence as well as the yields of charged and excited... 

Schrodinger discovered the equation that bears his name in 1926, and it has provided the foundation for the wave-mechanical formulation of quantum mechanics. Heisenberg had independently, and somewhat earlier, proposed a matrix formulation of the problem, which Schrodinger later showed was an equivalent alternative to his approach. We choose to present Schrodinger s version because its physical interpretation is much easier to understand. 

The classical formulation of the ASC problem presented by Drummond (1988) has some aspects of interest. The supertensor formalism he introduces is similar, although not equal, to the BEM matrix formulation shown in eqs. (51-54). The set of linear equations there which was introduced there may be used in classical, or quantum iterative solutions of the problem (for the quantum use, see Grant et al., 1990, and Coitino et al., 1995a), or, alternatively, in a direct calculation, via the inversion of the D matrix (see eqs. 52-53). Drummond s formulation makes easier the handling of the equivalent supermatrix he defines. This approach has not been tested in quantum calculations. 

Equation (8.3.2) is the quantum-mechanical formulation of (5.4.8). The prefactor —i is missing in (8.3.2), because the equation has been derived from the pulse response and not from a perturbation expansion (cf. Section 5.4.1). The similarity of both equations becomes more obvious by writing (8.3.2) in Liouville space, where the density matrix... 

As another application of classical S-matrix theory it is interesting to see how the scattering of atoms from a solid surface is described. (The extension to scattering of molecules should also be clear.) This has been worked out by Doll49 and closely parallels Wolken s50 quantum mechanical formulation of the problem. 

An advantage of the matrix formulation is that computer subroutines are available which will allow a direct solution of the matrix equation. The subject of matrix manipulation by computer is well advanced and reliable subroutines, which have been developed by mathematicians, are available and can be applied to problems in quantum chemistry and spectroscopy. The solution of eqns. (2)-(5) for a single electrode is fairly straightforward however, the matrix formulation, as in eqn. (16), is a convenient shorthand for the equations and the method comes into its own when more complicated reaction schemes are considered. 

This same result can be derived from the quantum mechanical T matrix formulation of the scat-... 

Werner, H. -J. (1987). Matrix-formulated direct multiconfiguration self-consistent field and multiconfiguration reference configuration-interaction methods. In K.P. Lawley ed., Ab initio Methods in Quantum Chemistry II, Adv. Chem. Phys. Vol 69, pp. 1-62. Wiley, New York. 

Moreover, the presented simple formalism leads towards a diagonal matrix formulation of the energy and other expectation values in computational Quantum Mechanics. The usual quantum mechanical formalism is not at all lost, but... 

The vehicle for the analysis of the scattering problem will be the Wigner-Eisenbud R-matrix formulation. We consider a quantum system with two degrees of freedom represented by coordinates q for the nuclear motion and x for the electronic part. The domain for these variables will be such that the coupling between the two coordinates is considered only when q belongs to the interval [r,p]. The domain for the electronic coordinate is arbitrary, the space will instead be limited by a choice of two basis functions. 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

The main purpose of the present notes was to elucidate in a systematic way the basic concepts, underlying most evolved theories of the electronic structure, dynamics and quantum structural properties of solids. Quite some attention has been paid to the one-electron approximation, which is one of the basic assumptions in actual calculations. Also the dielectric matrix formulation was treated rather in detail, since it enters in many studies of both the static and dynamical properties of solids. 

In this chapter we present the time-dependent quantum wave packet approaches that can be used to compute rate constants for both nonadiabatic and adiabatic chemical reactions. The emphasis is placed on our recently developed time-dependent quantum wave packet methods for dealing with nonadiabatic processes in tri-atomic and tetra-atomic reaction systems. Quantum wave packet studies and rate constants computations of nonadiabatic reaction processes have been dynamically achieved by implementing nuclear wave packet propagation on multiple electronic states, in combination with the coupled diabatic PESs constructed from ab initio calculations. To this end, newly developed propagators are incorporated into the solution of the time-dependent Schrodinger equation in matrix formulism. Applications of the nonadiabatic time-dependent wave packet approaches and the adiabatic ones to the rate constant computations of the nonadiabatic tri-atomic F (P3/2, P1/2) + D2 (v = 0,... 

In Chapter 7 we developed a method for performing linear variational calculations. The method requires solving a determinantal equation for its roots, and then solving a set of simultaneous homogeneous equations for coefficients. This procedure is not the most efficient for programmed solution by computer. In this chapter we describe the matrix formulation for the linear variation procedure. Not only is this the basis for many quantum-chemical computer programs, but it also provides a convenient framework for formulating the various quantum-chemical methods we shall encounter in future chapters. 

S. R. White, Phys. Rev. Lett., 69, 2863 (1992). Density Matrix Formulation for Quantum... 

White SR. Density Matrix Formulation for Quantum Renormalization Groups. Phys Rev Lett. 1992 69(19) 2863. 

Tanner D J and Weeks D E 1993 Wave packet correlation function formulation of scattering theory—the quantum analog of classical S-matrix theory J. Chem. Phys. 98 3884... 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

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