Classical variable

When we wish to replace the quantum mechanical operators with the corresponding classical variables, the well-known expression for the kinetic energy in hyperspherical coordinates [73] is... 

Huller and Baetz [1988] have undertaken a numerical study of the role played by shaking vibrations. The vibration was supposed to change the phase of the rotational potential V (p — a(t)). The phase a(t) was a stochastic classical variable subject to the Langevin equation... 

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. 

The variables cd, without a ket, represent classical variables belonging to Z. ... 

The basic idea underlying AIMD is to compute the forces acting on the nuclei by use of quantum mechanical DFT-based calculations. In the Car-Parrinello method [10], the electronic degrees of freedom (as described by the Kohn-Sham orbitals y/i(r)) are treated as dynamic classical variables. In this way, electronic-structure calculations are performed on-the-fly as the molecular dynamics trajectory is generated. Car and Parrinello specified system dynamics by postulating a classical Lagrangian ... 

If one is interested in spectroscopy involving only the ground Born Oppenheimer surface of the liquid (which would correspond to IR and far-IR spectra), the simplest approximation involves replacing the quantum TCF by its classical counterpart. Thus pp becomes a classical variable, the trace becomes a phase-space integral, and the density operator becomes the phase-space distribution function. For light frequency co with ho > kT, this classical approximation will lead to substantial errors, and so it is important to multiply the result by a quantum correction factor the usual choice for this application is the harmonic quantum correction factor [79 84]. Thus we have... 

Following Ziesche [35, 55], in order to develop the theory of cumulants for noncommuting creation and annihilation operators (as opposed to classical variables), we introduce held operators /(x) and / (x) satisfying the anticommutation relations for a Grassmann held. 

This solves the problem. The fact that a0, an are operators rather than classical variables made no difference thanks to the linearity. 

Often we must compute values of quantities that are not simple functions of the space coordinates, such as the y component of the momentum, py, where Equation (E.4) is not applicable. To get around this, we say that corresponding to every classical variable, there is a quantum mechanical operator. An operator is a symbol that directs us to do some mathematical operation. For example, the momentum operators are... 

Sources of external potential can be produced in a number of ways in which there is no need for special massive nuclei. As they identify external sources they are classical variables, namely, position coordinates for the sources. There is no quantum dynamics related to them yet. Symmetry constraints can be naturally defined. We formally write He(p a) to distinguish this situation from the standard approach. Since the primacy is given to the electronic wave function, and no Schrodinger equation is available at this point, its existence has to be taken as workinghypotheses. 

Deforming the geometry of Coulomb sources adiabatically is an allowed process, since these are classical variables the electronic energy, eq.(4), can only increase, but there cannot be a change of electronic state. The energy of two electronic states with the same space symmetry for a given a can be accidentally equal. Still, they cannot produce an "avoided crossing" in the sense of the BO picture. If each state represents, for instance, the reactants ( 1R>) and products ( 1P>), respectively, and the system is prepared as reactants, the matrix element... 

The short-time propagators can be solved through application of the Dyson identity truncated to first order. The subsequent dynamics of the quantity of interest are obtained by propagating the classical variables along a surface that corresponds to the quantum state (aa1) followed by Monte Carlo sampling of the nonadiabatic transition events ... 

We can summarize the procedure followed in the present section to achieve the result obtained above. In Ref. [15] the new momentum operator was correctly introduced, together with the new commutation relations of eq.(70), but Pjt(X was not used in the formal construction of the theory. The expressions for the quantum-classical variables (position and momentum) are those shown in eqs.(67), insted of eq.(72), because XjiCC and are represented using Phi,h2 9i, 92) in which the operators haDjia, no more generators of the corresponding Lie group, are present11. 

The DC motor is the classic variable speed drive. It is utilized as a universal main drive for capacities of up to several hundred kW and for servo drives down to the Watt range. 

The function AB(f) should be evaluated using a rigorous quantum mechanical calculation. In earlier work [108] we noted that decoherence theory simplifies this task by assuming that the time-dependent quantum operator ,(t) is a stochastic classical variable, so that the function becomes a characteristic... 

In the matrix model (Jongschaap, 1990), the global thermodynamic system is composed of two separate physical parts, which are called the environment and the internal variables. For the polymer solutions, for example, the pressure tensor Pv may be the internal variables, and the classical variables density, velocity, and internal energy are the environment variables. 

The next approximation is to clamp the nuclei at classically variable coordinates. This approximation still allows freedom to study the electron density quantum-mechanically. However, in view of the nature of the valence state developed here there is precious little to gain by attempting all-electron calculations. 

The central concept of AIMD as introduced by Car and Parrinello [1] lies in the idea to treat the electronic degrees of freedom, as described by e.g. one-electron wavefunctions ipi, as dynamical classical variables. The mixed system of nuclei and electrons is then described in terms of the extended classical Lagrangian Cex. ... 

Car-Parrinello techniques have been used to describe classical variables whose behavior, like quantum electrons in the Born-Oppenheimer approximation, is nearly adiabatic with respect to other variables. In simulations of a colloidal system consisting of macroions of charge Ze, each associated with Z counterions of charge —e, Lowen et al. [192] eliminated explicit treatment of the many counterions using classical density functional theory. Assuming that the counterions relax instantaneously on the time-scale of macroion motion, simulations of the macroion were performed by optimizing the counterion density at each time step by simulated annealing. 

Statistical mechanics is the branch of physical science that studies properties of macroscopic systems from the microscopic starting point. For definiteness we focus on the dynamics ofan A-particle system as our underlying microscopic description. In classical mechanics the set of coordinates and momenta, (r, p ) represents a state of the system, and the microscopic representation of observables is provided by the dynamical variables, v4(r, p, Z). The equivalent quantum mechanical objects are the quantum state [/ ofthe system and the associated expectation value Aj = of the operator that corresponds to the classical variable A. The corresponding observables can be thought of as time averages... 

Fig. 2.2 A schematic description of the LZ problem Two quanmm states and the coupling between them depend parametrically on a classical variable ff. The energies of the zero-order states a and b cross at R = R. The energies obtained by diagonalizing the Hamiltonian at any point R (adiabatic states) are ] (R) aDdE2(R).

Having the Hamiltonian treatment, it is easy to proceed to a quantum mechanical formulation of the Fermi resonance problem. The classical variables (9.15) and their complex conjugates correspond to the annihilation and creation operators of a quantum oscillator ... 

There is no evidence that any classical attribute of a molecule has quantum-mechanical meaning. The quantum molecule is a partially holistic unit, fully characterized by means of a molecular wave function, that allows a projection of derived properties such as electron density, quanmm potential and quantum torque. There is no operator to define those properties that feature in molecular mechanics. Manual introduction of these classical variables into a quantum system is an unwarranted abstraction that distorts the non-classical picture irretrievably. Operations such as orbital hybridization, LCAO and Bom-Oppenheimer separation of electrons and nuclei break the quantum symmetry to yield a purely classical picture. No amount of computation can repair the damage. 

Thus the asymptotic quantum states are labeled by the vibrational and rotational quantum numbers, whereas the projection quantum number is treated classically. We have in the above Hamiltonian indicated that it depends upon time through the classical variables. Thus the quantum mechanical problem consists of propagating the solution to the TDSE... 

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