Quantum dynamics Hamiltonian equation

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. 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

Further Analysis of Solutions to the Time-Independent Wave Packet Equations of Quantum Dynamics II. Scattering as a Continuous Function of Energy Using Finite, Discrete Approximate Hamiltonians. 

The general relations among the coefficients - and Dy are presented elsewhere [179]. The quantities yj and y2 are the damping constants for the fundamental and second- harmonic modes, respectively. In Eq.(82) we shall restrict ourselves to the case of zero-frequency mismatch between the cavity and the external forces (ff>i — ff> = 0). In this way we exclude the rapidly oscillating terms. Moreover, the time x and the external amplitude have been redefined as follows x = Kf and 8F =. The s ordering in Eq.(80) which is responsible for the operator structure of the Hamiltonian allows us to contrast the classical and quantum dynamics of our system. If the Hamiltonian (77)-(79) is classical (i.e., if it is a c number), then the equation for the probability density has the form of Eq.(80) without the s terms ... 

The dynamics associated with the Hamiltonian Eq. (8) or its variants Eq. (11) and Eq. (14) can be treated at different levels, ranging from the explicit quantum dynamics to non-Markovian master equations and kinetic equations. In the present context, we will focus on the first aspect - an explicit quantum dynamical treatment - which is especially suited for the earliest, ultrafast events at the polymer heterojunction. Here, the coherent vibronic coupling dynamics dominates over thermally activated events. On longer time scales, the latter aspect becomes important, and kinetic approaches could be more appropriate. 

Quantum mechanics involves two distinct sets of hypotheses—the general mathematical scheme of linear operators and state vectors with its associated probability interpretation and the commutation relations and equations of motion for specific dynamical systems. It is the latter aspect that we wish to develop, by substituting a single quantum dynamical principle for the conventional array of assumptions based on classical Hamiltonian dynamics and the correspondence principle. 

Huang. Y.. Iyengar. S.S.. Kouri, D.J. and Hoffman. D.K. (1996) Further analysis of solutions to the time-independent wnve packet equations of quantum dynamics. 2. Scattering as a continuous function of energt- using finite, discrete approximate Hamiltonians, J. Chem. Phys. 105, 927-939. 

This chapter is intended to present an integrated description of this general approach to quantum dynamics. Applications of the equations and strategies both to scattering and bound state problems will be discussed. In the next section, we begin with a detailed summary of the salient features of the DAFs as they are used to represent the Hamiltonian operator. Then in Sec. Ill, we discuss the TIWSE and some of the choices that can be made in solving for bound states and scattering information. Included in this is a discussion of the polynomial representations of various operators involved in the TIW form of quantum mechanics. Finally, in Sec. IV we briefly summarize some of the applications made to date of this overall approach. 

The random lifetime assumption is perhaps most easily tested by classical trajectory calculations (Bunker, 1962 1964 Bunker and Hase, 1973). Initial momenta and coordinates for the Hamiltonian of an excited molecule can be selected randomly, so that a microcanonical ensemble of states is selected. Solving Hamilton s equations of motion, Eq. (2.9), for an initial condition gives the time required for the system to reach the transition state. If the unimolecular dynamics of the molecule are in accord with RRKM theory, the decomposition probability of the molecule versus time, determined on the basis of many initial conditions, will be exponential with the RRKM rate constant. That is, the decay is proportional to exp[-k( )t]. The observation of such an exponential distribution of lifetimes has been identified as intrinsic RRKM behavior. If a microcanonical ensemble is not maintained during the unimolecular decomposition (i.e., IVR is slower than decomposition), the decomposition probability will be nonexponential, or exponential with a rate constant that differs from that predicted by RRKM theory. The implication of such trajectory studies to experiments and their relationship to quantum dynamics is discussed in detail in chapter 8. 

The quantum dynamics of these phenomena is described by the time-independent Schrodinger equation with scattering boundary conditions, or alternatively by the operator equations of scattering theory. The Hamiltonian for the system is... 

The methodology of molecular quantum dynamics applied to non-adiabatic systems is presented from a time-dependent perspective in Chap. 4. The representation of the molecular Hamiltonian is first discussed, with a focus on the choice of the coordinates to parametrize the nuclear motion and on the discrete variable representation. The multi-configuration time-dependent Hartree (MCTDH) method for the solution of the time-dependent Schrddinger equation is then presented. The chapter ends with a presentation of the vibronic coupling model of Kdppel, Domcke and Cederbaum and the methodology used in the calculation of absorption spectra. 

Provided the Hamiltonian operator can be written in a suitable form and an efficient integration scheme is used to solve the equations of motion (see Sects. 4.2.3 and 4.2.4), the MCTDH method allows one to perform accurate quantum dynamics calculations for realistic molecular systems with more than twenty degrees of freedom [33-45], and on simple model systems with up to roughly eighty degrees of freedom [46-48]. 

In the transparent interface method, the forces on each atom are derived from the gradient of the total potential energy in Eq. [71]. However, the constraint forces acting on quantum atoms at the boundary, due to the constaints imposed on the pseudoatoms, are neglected in this approach, leading to an approximation of the Hamiltonian equations that provides correct forces and dynamics (within the limit of a classical force field) but violates energy conservation. 

The solution of the dynamical problem for the gas and surface atoms requires in principle solution of the quantum mechanical equations of motion for the system. Since this problem has been solved only for 3-4 atomic systems we need to incorporate some approximations. One obvious suggestion is to treat the dynamics of the heavy solid atoms by classical rather than quantum dynamical equations. As far as the lattice is concerned we may furthermore take advantage of the periodicity of the atom positions. At the surface this periodicity is, however, broken in one direction and special techniques for handling this situation are needed. Lattice dynamics deals with the solution of the equations of motion for the atoms in the crystal. As a simple example we consider first a one-dimensional crystal of atoms with identical masses. If we include only the nearest neighbor interaction, the hamiltonian is given by ... 

The appearance of the Hamiltonian operator in equation (3.55) as stipulated by postulate 5 gives that operator a special status in quantum mechanics. Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator for a given system is sufficient to determine the stationary states of the system and the expectation values of any other dynamical variables. 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

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