Motion equations quantum

Discrete Fourier transform (DFT), non-adiabatic coupling, Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 153-155 Discrete variable representation (DVR) direct molecular dynamics, nuclear motion Schrodinger equation, 364-373 non-adiabatic coupling, quantum dressed classical mechanics, 177-183 formulation, 181-183... 

Using a cumulant expansion, we have shown how to obtain quantum corrections to purely classical equations of motion. Quantum correction reduces chaos in... 

These methods have been applied to calculate the polarizabilities of atoms,31 and the long-ranged forces between atoms,33 with a typical calculated accuracy of 10 % or less. Thus, we have been able to estimate successfully the significant features of zero-point fluctuations of atomic dipole moments, without actually solving the quantum equations of motion to obtain all the excited state energies and wave functions. 

This expression will be used in the next section to obtain the classical equation of motion as a straighforward formal limit of the quantum equation of motion. 

If the system has an internal angular momentum (associated with rotational states of molecules) there will, in the absence of an external field, be degeneracies in the system that will be practical to display explicitly in the expression for microscopic reversibility in Eq. (B.22). For systems with angular momenta, time reversal of the quantum equations of motion reverses the signs of both the momenta and their projections on a given direction, just like in a classical system. To express this explicitly, Eq. (B.13) is written as... 

The use of the quantum treatment in dealing with processes in polyatomic systems is rather limited (6,7). Nevertheless, the quantum formulation implies the most general features of the problem, so that it is convenient to commence our consideration with the quantum equations of motion. 

There is an uncertainty in pressure of an order of 0.1 MPa depending on whether classical or quantum partition function is used for vibrational motions. Clearly, equation (14) gives too low dissociation pressure. Therefore, the influence of the guest molecule on the host lattice is fairly large and cannot be neglected[22, 24, 33]. Comparison with experiment will be made below. 

The computation of the equilibrium properties of quantum systems is a challenging problem. The simulation of dynamical properties, such as transport coefficients, presents additional problems since the solution of the quantum equations of motion for many-body systems is even more difficult. This fact has prompted the development of approximate methods for dealing with such problems. 

However, another subtle consequence of these relations is that from dynamical quantum equation may be abstracted (beyond the coordinate and momentum operators) the general equation of motion for an arbitrary operator ... 

An elegant method to introduce the quantum particles motion s equation, called - at non-relativistic level - the Schrodinger equation, consists in employing the classical-quantum correspondences in Table 3.1. Thus, for the conservation of the total energy of a particle under the action of an external potential V(x) there is equivalently obtained that the classical form of energy conservation (Putz, 2006)... 

Shenvi N, Cheng HZ, Tully JC (2006) Nonadiabatic dynamics netir metal surfaces decoupling quantum equations of motion in the wide-band limit. Phys Rev A 74 10... 

Once the potential energy surface is obtained, the reaction can be studied by solving for the nuclear motion along the reaction coordinate. Most nuclei other than hydrogen are sufficiently massive that classical mechanics is thought to be an adequate approximation, but quantum calculations are also carried out. Neither the classical nor quantum equations can be solved in closed form, and the motions are numerically simulated using computer programs. If classical mechanics is used, the calculation is carried out for a number of different trajectories. The fraction of the trajectories that... 

It is useful to write the molecular quantum equations of motion explicitly in terms of the time independent field free molecular Hamiltonian and the energy V arising from the interaction with the field in equation (19) ... 

The theory of multiphoton excitation has thus been established, starting from the time dependent quantum equations of motion. It leads to a complete, time dependent quantum description for simple systems, to a statistical mechanical description for complex systems and finally includes the derivation of the simplest experimental quantities such as the rate constants. Its computational implementation exists at all these levels. Table 1 gives a summary of some selected applications of the theory to specific systems at various levels of the theoretical and numerical treatment. Numerous further applications to experimental systems can be found in the review literature and some original papers cited. 

Once a potential surface has been obtained, a variety of dynamics methods can be used to determine reaction cross sections and rate constants. The "exact approach would involve solving the nuclear-motion Schrodinger equation for the appropriate scattering wave function. Because of the complexity associated with doing this, such exact solutions have been confined to the H4-H2 reaction. Much progress in the past year has been reported in the determination of approximate quantum solutions in three dimensions, so some of these applications are described in section II.B. Many more atom-diatom reactions have been treated exactly using one-dimensional models. 

In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle coordinates r. and velocities dr./dt at any given time. These evolve according to Newton s equations of motion. In principle, one can write down equations involving the state variables and forces acting on the particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t, provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the system at time t is instead described by a well behaved mathematical fiinction of the particle coordinates q- rather than a simple list of positions and velocities. 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

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