Representation coordinate

In the above discussion of relaxation to equilibrium, the density matrix was implicitly cast in the energy representation. However, the density operator can be cast in a variety of representations other than the energy representation. Two of the most connnonly used are the coordinate representation and the Wigner phase space representation. In addition, there is the diagonal representation of the density operator in this representation, the most general fomi of p takes the fomi... 

The coordinate representation of the density matrix, in the canonical ensemble, may be written... 

Now we do one of the standard quantum mechanical tricks, inserting the identity operator as a complete sum of states in the coordinate representation ... 

The potential energy part is diagonal in the coordinate representation, and we drop the hat indicating an operator henceforth. The kinetic energy part may be evaluated by transfonning to the momentum representation and carrying out a Fourier transform. The result is... 

This section attempts a brief review of several areas of research on the significance of phases, mainly for quantum phenomena in molecular systems. Evidently, due to limitation of space, one cannot do justice to the breadth of the subject and numerous important works will go unmentioned. It is hoped that the several cited papers (some of which have been chosen from quite recent publications) will lead the reader to other, related and earlier, publications. It is essential to state at the outset that the overall phase of the wave function is arbitrary and only the relative phases of its components are observable in any meaningful sense. Throughout, we concentrate on the relative phases of the components. (In a coordinate representation of the state function, the phases of the components are none other than the coordinate-dependent parts of the phase, so it is also true that this part is susceptible to measurement. Similar statements can be made in momentum, energy, etc., representations.)... 

We can state the form of the conjugate relationship in a setting more general than 4 (a, t), which is just a particular, the coordinate representation of the evolving state. For this purpose, we write the state function in a more general way, through... 

A time-varying wave function is also obtained with a time-independent Hamiltonian by placing the system initially into a superposition of energy eigenstates ( n)), or forming a wavepacket. Frequently, a coordinate representation is used for the wave function, which then may be written as... 

FIGURE 8.2 Cartesian coordinate representation of ethane. The first column is the element, the other columns are the x, y, and z Cartesian coordinates. 

The Schrddinger Equation.—We return to Eq. (8-50), the coordinate representation of the operator Pfc. Multiply that equation by and integrate over all coordinates ... 

We interpret this equation as an historical development of the state t> from the initial state 0> at the earlier time t0 < t. Taking the coordinate representation of this operator relation we have... 

The analog of Eq. (8-59), giving the coordinate representation of the momentum operator is... 

We first write down the coordinate representation of the effect of operating by (8-147) on the general state ,jV)> ... 

We shall now express the projection operator as a matrix in coordinate representation, the typical element being... 

The coordinate representation matrix of this operator has the typical element ... 

This expression is the trace, in coordinate representation, of the operator WXHB — WXRH. But the trace of a product of matrices is independent of the order of the product, so the two parts of (8-253) cancel.29... 

We have carried out tins discussion in occupation number representation or coordinate representation each with a definite number N of particles. Similar results follow for the Fock space representation and the properties of grand ensembles. Averages over grand ensembles are also independent of time when the probabilities > are independent of time, whether the observable commutes with H or not. 

Separating the Hamiltonian into parts describing a free rotator and the orienting effect of a liquid cage, we have in a coordinate representation... 

In this form, which is analogous to Eq. (26) in the photon absorption case, the rate is expressed as a sum over the neutral molecule s vibration-rotation states to which the specific initial state having energy , can decay of (a) a translational state density p multiplied by (b) the average value of an integral operator A whose coordinate representation is... 

The density operator in the coordinate representation is given by the functions T q, Q, q Q, <), and can be expressed for Hamiltonian systems in terms of its amplitudes, which become the wavefunctions Q, <) I convenient to introduce the formally exact eikonal representation. 

In the nuclear coordinate representation, the expectation values of an operator A = A Q,Q, t) follow from... 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

In coordinate representation, there exists alternative base representations, adiabatic and diabatic. Both representations are equivalent if the basis are complete. For a thorough discussion on adiabatic-diabatic electronic state transformations the reader is referred to the work by Baer [49, 50], see also the work by Chapuisat et al. [51] In this... 

The usefulness of potential energy hypersurfaces in describing reaction dynamics and chemical reactivity is well illustrated by Levine and Bernstein [84] and Shaik et al. [85] books. See also the fundamental paper of Hase [86]. This success does not assure that the coordinate representation of quantum system is necessarily truthful. It goes without saying, the coordinate representation is an extremely useful mathematical model. However, from recent inelastic neutron scattering experiments on hydrogen bonded system, the idea that the BO approximation may be inadequate has been advanced by Kearley and coworkers[87]. 

The theoretical view advocated here focus attention on the quantum states relevant to the description of particular phenomena. The concept of stationary Hamiltonians follows from the coordinate representation and leads to a numerical determination of relevant quantum states. He will denote this class of Hamiltonians. Thus, given a system that can be decomposed into n-electrons and m-nuclei in different dispositions, the set of He =Ea(l/tna)[pa]2+Vcoui are written in terms of fluctuation coordinates around particular... 

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