Harmonic oscillator motion equations

It should be evident that the expressions for the Laplace transforms of derivatives of functions can facilitate the solution of differential equations. A trivial example is that of the classical harmonic oscillator. Its equation of motion is given by Eq. (5-33), namely,... 

An important example of one-dimensional motion is provided by a simple harmonic oscillator. The equation of motion is... 

Similar to the treatment of the harmonic oscillator, this equation of motion can be... 

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

It is noteworthy that eq. (4.15a) is nothing but the linearized classical upside-down barrier equation of motion (8S/8x = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15a) describes how the trajectory escapes from the instanton solution, when it deviates from it. The parameter X, referred to as the stability angle [Gutzwil-ler 1967 Rajaraman 1975], generalizes the harmonic-oscillator phase co, which would appear in (4.15), if CO, were a constant. The fact that X is real indicates the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det( — -I- co, ) is a function of X only,... 

It is left to reader to verify that, under Lee .s discrete mechanics, both free particles and particles subjected to a constant force, behave in essentially the sa e way as they do under continuous equations of motion. Moreover, the time intervals At = t-i i — ti are all equal. While the spatial behavior for non-constant forces (ex particles in a harmonic oscillator V potential) also remains essentially... 

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

Thus, we have demonstrated that a small motion of a mass is described by the equation of a harmonic oscillator, Equation (3.25), and, as is well known, its solution is... 

The vibrational and rotational motions of the chemically bound constituents of matter have frequencies in the IR region. Industrial IR spectroscopy is concerned primarily with molecular vibrations, as transitions between individual rotational states can be measured only in IR spectra of small molecules in the gas phase. Rotational - vibrational transitions are analysed by quantum mechanics. To a first approximation, the vibrational frequency of a bond in the mid-IR can be treated as a simple harmonic oscillator by the following equation ... 

With the use of the differential operator the equation of motion for the harmonic oscillator [Eq. (29)], can be expressed as... 

To solve this equation, an appropriate basis set ( >.,( / ) is required for the nuclear functions. These could be a set of harmonic oscillator functions if the motion to be described takes place in a potential well. For general problems, a discrete variable representation (DVR) [100,101] is more suited. These functions have mathematical properties that allow both the kinetic and potential energy... 

Equation 3.24, the equation of motion for a harmonic oscillator with a frequency... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

The total vibrational energy is a sum of energies of 3N-6 distinct harmonic oscillators. Indeed, 3N-6 is the final number of coordinates in Equation 8. Namely, the number of 3N+3 coordinates of the initial equation has been reduced by three through the elimination of internal rotations. Furthermore, the equation of nuclear motion (mainly its potential) has to be invariant under rotations and translations of a molecule as a whole (which is equivalent to the momentum and angular momentum preservation laws). The latter requirement leads to a further reduction of the number of coordinates by six (five in the case of linear molecules for which there are only two possible rotations). 

The equation of motion for a single harmonic oscillators of mass m, and force... 

Niunerical algorithms for solving the GLE are readily available. Only recently, Hershkovitz has developed a fast and efficient 4th order Runge-Kutta algorithm. Memory friction does not present any special problem, especially when expanded in terms of exponentials, since then the GLE can be represented as a finite set of memoiy-less coupled Langevin equations. " Alternatively (see also the next subsection), one can represent the GLE in terms of its Hamiltonian equivalent and use a suitable discretization such that the problem becomes equivalent to that of motion of the reaction coordinate coupled to a finite discrete bath of harmonic oscillators. ... 

The j-th harmonic bath mode is characterized by the mass mj, coordinate Xj, momentum pxj and frequency coj. The exact equation of motion for each of the bath oscillators is mjxj + mj(0 Xj = Cj q and has the form of a forced harmonic oscillator equation of motion, ft may be solved in terms of the time dependence of the reaction coordinate and the initial value of the oscillator coordinate and momentum. This solution is then placed into the exact equation of motion for the reaction coordinate and after an integration by parts, one obtains a GLE whose... 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

A further refinement of the harmonic oscillator model is possible, in which the lattice is put into contact with a heat bath at temperature T and remains in contact with the heat bath, so that the initial correlations decay not only through mutual interactions but also through random collisions with an external fluctuating field. Although it might appear that such a case would contain features of both the independent particle case and the harmonic oscillator model just analyzed, the resulting formalism is much closer to that required for the latter, and the results differ only in detail. The model to be discussed is specified by the equations of motion... 

In these last two examples of equations of motion, the objective is to determine functions of the form h = /(/) or x=g(t), respectively, which satisfy the appropriate differential equation. For example, the solution of the classical harmonic motion equation is an oscillatory function, x=g t), where g(f) = cos a>t, and a> defines the frequency of oscillation. This function is represented schematically in Figure 7.1 (see also Worked Problem 4.4). 

As was shown by Bohm and Pines,90 under certain conditions the equations of motion for a Fourier component of the electron density can be reduced to harmonic-oscillation equations with the frequency depending only on the density of electrons ne ... 

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