Harmonic oscillator equation of motion

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... 

These equations have the form of harmonic oscillator equations of motion for a coordinate q and momentum /). Indeed, Eqs (3.11) can be derived from the Hamiltonian... 

Such correlation functions are often encountered in treatments of systems coupled to their thennal environment, where the mode 1 for the system-bath interaction is taken as a product of A or B with a system variable. In such treatments the coefficients Cj reflect the distribution of the system-bath coupling among the different modes. In classical mechanics these functions can be easily evaluated explicitly from the definition (6.6) by using the general solution of the harmonic oscillator equations of motion... 

To obtain the standard form of Onsager s theory [37,38], we next linearize the thermodynamic forces in eqs. (A. 15) and (A.28). This linearization reduces these equations to coupled damped harmonic oscillator equations of motion. 

In this section we consider the classical equations of motion of particles in cases where the highest-frequency oscillations are nearly harmonic The positions y t) = j/i (t) evolve according to the second-order system of differential equations... 

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... 

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... 

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,... 

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

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

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

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

There are two main ingredients that go into the semiclassical tiunover theory, which differ from the classical limit. In the latter case, a particle which has energy E > 0 crosses the barrier while if the energy is lower it is reflected. In a semiclassical theory, at any energy E there is a trarrsmission probabihty T(E) for the particle to be transmitted through the barrier. The second difference is that the bath, which is harmonic, may be treated as a qrrantum mechanical bath. Within first order perturbation theory, the equations of motion for the bath are those of a forced oscillator, and so their formally exact quantum solution is known. 

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 ... 

An approximate solution of the equation of motion can be found assuming harmonic oscillations even though an intermittent and non-linear force dependence is introduced [102,140-143]. For high oscillation frequencies in a MHz range, where most of the polymer samples behave elastically (low damping, high stiffness), the relation between the cantilever response and the material properties is given by... 

This amplitude is found from the equation of motion of a harmonic oscillator affected by an a.c. field E(t). This approach yields the Lorentz and Van Vleck-Weisskopf lines, respectively, for a homogeneous and Boltzmann distributions of the initial a.c. displacements x(l0) established after instant to of a strong collision. The susceptibility corresponding to the Van Vleck-Weisskopf line in terms of our parameters is given by [66]... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

The free BC oscillator is assumed to be harmonic with force constant k and equilibrium separation r the parameter e controls the coupling between the dissociation coordinate R and the vibrational coordinate r. For e = 0 (elastic limit) the equations of motion for (R, P) and (r, p) decouple and energy cannot flow from one degree of freedom to the other. As a consequence, the vibrational energy of the oscillator remains constant throughout the dissociation and the corresponding vibrational excitation function, which for zero initial momentum po is given by... 

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