Harmonic-oscillator system force constant

A similar expression would be obtained along the same lines for the final system, yet with 8 replaced by 1 — 5. Comparison of Eqs. (69) and (60) shows that the outer shell (solvent) contributions are equivalent to a single harmonic oscillator of force constant given in Eq. (70),... 

A harmonic oscillator of force constant k and mass m has potential energy jkx2 and kinetic energy p2/2m, where x and p are its instantaneous displacement and momentum, and it oscillates with a frequency v0 = l2nyj(klm). Thus the kinetic energy of the system of Fig. 4.1 is... 

Consider a general three dimensional harmonic oscillator with force constants /C , k, and k. Write down the total energy expression for the energy levels of this system. What is the ground state energy ... 

Oscillator system where two particles, each constrained to move along the x-axis, are attached by a harmonic spring with force constant k. The positions of the two particles along the x-axis are designated x and Xj- The equilibrium length of the spring is the constant Xq. 

The problem of the quantum mechanical description of a harmonic oscillator is our first example of applying the postulates of quantum mechanics. It also provides a valuable comparison with the classical description considered earlier in this chapter. The picture of the system is the same as that in Figure 7.1, a mass m connected to an immovable wall by a harmonic spring with force constant k. The steps for quantum mechanical treatment of this problem, as well as any other problem, are the following ... 

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

The function time-dependent force acting on the harmonic oscillator and mo = K/m is the angular frequency of the system (Section 5.3.3). 

In an elastic material medium a deformation (strain) caused by an external stress induces reactive forces that tend to recall the system to its initial state. When the medium is perturbed at a given time and place the perturbation propagates at a constant speed (or celerity) c that is characteristic of the medium. This propagating strain is called an elastic (or acoustic or mechanical) wave and corresponds to energy transport without matter transport. Under a periodic stress the particles of matter undergo a periodic motion around their equilibrium position and may be considered as harmonic oscillators. 

Figure 3.1 Phase-space trajectory (center) for a one-dimensional harmonic oscillator. As described in the text, at time zero the system is represented by the rightmost diagram (q = b, p = 0). The system evolves clockwise until it returns to the original point, with the period depending on the mass of the ball and the force constant of the spring...

While Eq. (9.49) has a well-defined potential energy function, it is quite difficult to solve in the indicated coordinates. However, by a clever transfonnation into a unique set of mass-dependent spatial coordinates q, it is possible to separate the 3 Ai-dirncnsional Eq. (9.49) into 3N one-dimensional Schrodinger equations. These equations are identical to Eq. (9.46) in form, but have force constants and reduced masses that are defined by the action of the transformation process on the original coordinates. Each component of q corresponding to a molecular vibration is referred to as a normal mode for the system, and with each component there is an associated set of harmonic oscillator wave functions and eigenvalues that can be written entirely in terms of square roots of the force constants found in the Hessian matrix and the atomic masses. 

The harmonic oscillator is used as a simple model to represent the vibrations in bonds. It includes two masses that can move on a plane without friction and that are joined by a spring (see Fig. 10.3). If the two masses are displaced by a value x0 relative to the equilibrium distance / , the system will start to oscillate with a period that is a function of the force constant k (N m ) and the masses involved. The frequency, which is independent of the elongation, can be approximated by equation (10.2) where n (kg) represents the reduced mass of the system. The term harmonic oscillator comes from the fact that the elongation is proportional to the exerted force while the frequency i/yib is independent of it. 

Quantitative evaluation of a force-distance curve in the non-contact range represents a serious experimental problem, since most of the SFM systems give deflection of the cantilever versus the displacement of the sample, while the experimentalists wants to obtain the surface stress (force per unit contact area) versus tip-sample separation. A few prerequisites have to be met in order to convert deflection into stress and displacement into tip-sample separation. First, the point of primary tip-sample contact has to be determined to derive the separation from the measured deflection of the cantilever tip and the displacement of the cantilever base [382]. Second, the deflection can be converted into the force under assumption that the cantilever is a harmonic oscillator with a certain spring constant. Several methods have been developed for calibration of the spring constant [383,384]. Third, the shape of the probe apex as well as its chemical structure has to be characterised. Spherical colloidal particles of known radius (ca. 10 pm) and composition can be used as force probes because they provide more reliable and reproducible data compared to poorly defined SFM tips [385]. 

The energy levels of the vibrational modes can be predicted with a reasonable accuracy on the basis of the standard Wilson vibrational analysis (241,244) (called GF analysis). The vibrational motion of atoms in the polyatomic system is approximated by harmonic oscillations in a quadratic force field. Computations of the force constants are the subject of quantum chemistry. 

Because of their importance to nucleation kinetics, there have been a number of attempts to calculate free energies of formation of clusters theoretically. The most important approaches for the current discussion are harmonic models, " Monte Carlo studies, and molecular dynamics calcula-tions. In the harmonic model the cluster is assumed to be composed of constituent atoms with harmonic intermolecular forces. The most recent calculations, which use the harmonic model, have taken the geometries of the clusters to be those determined by the minimum in the two-body additive Lennard-Jones potential surface. The oscillator frequencies have been obtained by diagonalizing the Lennard-Jones force constant matrix. In the harmonic model the translational and rotational modes of the clusters are treated classically, and the vibrational modes are treated quantum mechanically. The harmonic models work best at low temjjeratures where anharmonic-ity effects are least important and the system is dominated by a single structure. 

Another three-dimensional problem which is soluble in Cartesian coordinates is the three-dimensional harmonic oscillator, a special case of which, the isotropic oscillator, we have treated in Section la by the use of classical mechanics. The more general system consists of a particle bound to the origin by a force whose components along the x, y, and z axes are equal to —kzx, —Jtvy, and — k,z, respectively, where kx, kv, kt are the force constants in the three directions and x, y, z are the components of the displacement along the three axes. The potential energy is thus... 

The last reference system we discuss is the lattice of interacting harmonic oscillators. In this system each atom is connected to its neighbors by a Hookean spring. By diagonalizing the quadratic form of the Hamiltonian, the system may be transformed into a collection of independent harmonic oscillators, for which the free energy is easily obtained. This reference system is the basis for lattice-dynamics treatments of the solid phase [67]. If D is the dynamical matrix for the harmonic system (such that element Dy- describes the force constant for atoms i and j), then the free energy is... 

The resonant frequency (v) of any harmonic oscillator is proportional to (fc/A/) where M is the effective mass of the system and / is a restoring force constant for the oscillator. 

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