State in classical mechanics

The word state in classical mechanics means a specification of the position and velocity of each particle of the system at some instant of time, plus specification of the forces acting on the particles. According to Newton s second law, given the state of a system at any time, its future state and future motions are exactly determined, as shown by Eqs. (1.9)-(1.11). The impressive success of Newton s laws in explaining planetary motions led many philosophers to use Newton s laws as an argument for philosophical determinism. The mathematician and astronomer Laplace (1749-1827) assumed that the universe consisted of nothing but particles that obeyed Newton s laws. Therefore, given the state of the universe at some instant, the future motion of... 

The time-dependent Schrodinger equation is a first-order difierential equation in the time, so that, just as in classical mechanics, the present state of an undisturbed system determines the future state. However, unlike knowledge of the state in classical mechanics, knowledge of the state in quantum mechanics involves a knowledge of only... 

The word state in classical mechanics means a specification of the position and velocity of each particle of the system at some instant of time, plus specification of the forces... 

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. 

As in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrence of IVR in polyatomic molecules, depends both on the Flamiltonian and the initial conditions, i.e. the initial quantum mechanical state I /(tQ)). We focus here on the time-dependent aspects of IVR, and in this case such initial conditions always correspond to the preparation, at a time of superposition states of molecular (spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occurs if these levels involve at least two distinct... 

Problems in classical mechanics can be solved by the application of Newton s (hree laws, which can be stated as follows. 

In classical mechanisms for C—H bond activation either C—H rr-bond donation or cyclic. 4-centered transition states are important, but these are precluded in the porphyrin systems and the mechanism proposed for activation of CH4, toluene, and Hy by the Rh porphyrin radicals is a new mechanistic possibility. 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

Spin is a quantum mechanical property that does not appear in classical mechanics. An electron can have one of two distinct spins, spin up or spin down. The full specification of an electron s state must include both its location and its spin. The Pauli exclusion principle only applies to electrons with the same spin state. 

Boltzmann distribution law in classical mechanics involves the Boltzmann factor e wlkT in the same way as that for quantum mechanics. In classical mechanics the state of a system can be described by giving the values of the coordinates and the momenta, for example, for a single particle the values of the three coordinates x, y, and z, and of the three linear momenta px, py, and pt which are equal to the mass of the particle multiplied by the components of velocity in the x, y, and z di-... 

This type of energy exchange in an autoionization process may correspond with the behavior of a kicked rotator in classical mechanics, which is known to exhibit chaos. It would be worthwhile to consider an autoionization process of a simple diatomic molecule in its Rydberg states to understand experimentally the essential dynamics of a quantum system, whose classical counterpart exhibits chaos. 

Classical Path. Another approach to scattering calculations uses a quantum-mechanical description of the internal states, but classical mechanics for the translational motion. This "classical path" method has been popular in line-shape calculations (37,38). It is almost always feasible to carry out such calculations in the perturbation approximation for the internal states (37). Only recently have practical methods been developed to perform non-perturbative calculations in this approach (39). 

The final vibrational state distribution of the photofragment manifests the change of its bond length along the dissociation path [see Simons and Yarwood (1963) and Mitchell and Simons (1967) for early references]. Let us consider the dissociation of a linear triatomic molecule ABC described by Jacobi coordinates R and r as defined in Figure 2.1. In classical mechanics, it is the force... 

These equations have a close resemblance to Hamilton s equations of classical mechanics, Eq. (4.63). An identity is, however, only obtained for potentials with terms of no more than second order (note, e.g., that (x2) A (x 2)- One simple application is to the dynamics of a free particle (e.g., the motion of the center of mass). The expectation values behave like in classical mechanics—the expectation value of the momentum is constant as is the associated momentum uncertainty. That is, except for the inherent momentum uncertainty of the initial state, the free particle behaves as in classical mechanics. 

In classical mechanics, if a system is in a given state (q,p), the measurement of the dynamical variable uj will yield a value u> q,p). The state of the system will remain unaffected. 

In classical mechanics, the state variables change with time according to Hamilton s equations of motion... 

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