Oscillator, simple harmonic

Tlris is the Schrodinger equation for a simple harmonic oscillator. The energies of the system are given by E = (i + ) x liw and the zero-point energy is Hlj. 

The TEOM sampler draws air through a hollow tapered tube, the wide end of the tube being fixed, while the narrow end oscillates in response to an applied electric field. The narrow end of the tube contains the filter cartridge. The sampled air flows from the sampling inlet, through the filter and tube, to a flow controller. The tube-filter unit acts as a simple harmonic oscillator with ... 

At low temperatures nearly all bonds will be in their lowest vibrational level, n = 0, and will, therefore, possess the zero-point vibrational energy, Eq = hvl2. Presuming the molecule behaves as a simple harmonic oscillator, the vibrational frequency is given by... 

Discuss how to compute vibrational frequencies using a simple harmonic oscillator model of nuclear motion. 

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

The simple harmonic oscillator picture of a vibrating molecule has important implications. First, knowing the frequency, one can immediately calculate the force constant of the bond. Note from Eq. (11) that k, as coefficient of r, corresponds to the curvature of the interatomic potential and not primarily to its depth, the bond energy. However, as the depth and the curvature of a potential usually change hand in hand, the infrared frequency is often taken as an indicator of the strength of the bond. Second, isotopic substitution can be useful in the assignment of frequencies to bonds in adsorbed species, because frequency shifts due to isotopic substitution (of for example D for H in adsorbed ethylene, or OD for OH in methanol) can be predicted directly. 

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 ... 

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

Figure 5 Trajectory of an element representing a simple harmonic oscillator in phase space.

For elements that have three or more isotopes, isotopic fractionations may be defined using two or more isotopic ratios. Assuming that isotopic fractionation occurs through a mass-dependent process, the extent of fractionation will be a function of the relative mass differences of the two isotope ratios. For example, assuming a simple harmonic oscillator for molecular motion, the isotopic fractionation of may be related to as ... 

Epsilon notation is defined similarly, although the deviations are in parts per 10,000. Comparison between these numbers is straightforward as shown in Figure 2, which plots the co-variations in Fe/ Fe and Te/ Fe of layers from BIFs in both 5 and e notation. The mass-dependent fractionation line, based on a simple harmonic oscillator approximation (Criss 1999), lies close to a line of 1.5 1 for 5 Fe-5 Te variations. For example, point A in Figure 2 has an 5 Te value of+15.0, which would be approximately equal to a 5 Fe value of+1.00, as defined here, assuming normalization to an identical reference reservoir. 

Presented below are three examples designed to give the reader some idea of what one can expect from the theoretical analysis of vibrational spectra based on the simple harmonic oscillator model. Systems have been chosen whose structures have been know for many years and, in fact, were known prior to the availability of IR spectroscopy. Hence their spectra have previously been well characterized and these serve as a test of the method . 

The zero point energy of a simple harmonic oscillator is ... 

Note that if j = 1, (9.12) is formally identical with the classical expression (9.7) the classical multiple oscillator model, which will be discussed in Section 9.2, is even more closely analogous to (9.12). However, the interpretations of the terms in the quantum and classical expressions are quite different. Classically, o30 is the resonance frequency of the simple harmonic oscillator quantum mechanically 03 is the energy difference (divided by h) between the initial or ground state / and excited state j. Classically, y is a damping factor such as that caused by drag on an object moving in a viscous fluid quantum mechanically, y/... 

We start our discussion of laser-controlled electron dynamics in an intuitive classical picture. Reminiscent of the Lorentz model [90, 91], which describes the electron dynamics with respect to the nuclei of a molecule as simple harmonic oscillations, we consider the electron system bound to the nuclei as a classical harmonic oscillator of resonance frequency co. Because the energies ha>r of electronic resonances in molecules are typically of the order 1-10 eV, the natural timescale of the electron dynamics is a few femtoseconds to several hundred attoseconds. The oscillator is driven by a linearly polarized shaped femtosecond... 

These potential energy terms and their attendant empirical parameters define the force field (FF). More complicated FFs which use different and/or more complex functional forms are also possible. For example, the simple harmonic oscillator expression for bond stretching can be replaced by a Morse function, Euorse (3), or additional FF terms may be added such as the stretch-bend cross terms, Estb, (4) used in the Merck molecular force field (MMFF) (7-10) which may be useful for better describing vibrations and conformational energies. 

The attenuation of die dipole of the repeat unit owing to thermal oscillations was modeled by treating the dipole moment as a simple harmonic oscillator tied to the motion of the repeat unit and characterized by the excitation of a single lattice mode, the mode, which describes the in-phase rotation of the repeat unit as a whole about the chain axis. This mode was shown to capture accurately the oscillatory dynamics of the net dipole moment itself, by comparison with short molecular dynamics simulations. The average amplitude is determined from the frequency of this single mode, which comes directly out of the CLD calculation ... 

M for a simple harmonic oscillator where v, the vibrational quantum i AUmber has values 0, 1.2, 3, etc. The potential function F(r) for simple < harmonic motion as derived from Hooke s law is given by... 

The central potential can be a simple harmonic oscillator potential/(r) kr2 or more complicated such as a Yukawa function f(r) (e a,/r) 1 or the Woods-Saxon function that has a flat bottom and goes smoothly to zero at the nuclear surface. The Woods-Saxon potential has the form... 

One of the classic problems of quantum mechanics that is very important for our study of nuclei is the harmonic oscillator. For a simple harmonic oscillator, the restoring force is proportional to the distance from the center, that is, F = — kx, so that V x) = kx2/2. The Schrodinger equation is... 

The potential energy of such an oscillator can be plotted as a function of the separation r, or, for a normal mode in a polyatomic molecule, as a function of a parameter characterizing the phase of the oscillation. For a simple harmonic oscillator, the potential energy function is parabolic, but for a molecule its shape is that indicated in Figure 2.6. The true curve is close to a parabola at the bottom, and it is for this reason that the assumption of simple harmonic motion is justified for vibrations of low amplitude. 

Z is proportional to the gas pressure, and, since Z1>0, the collision number for energy transfer, is constant for a particular transition, the actual value of fi is inversely proportional to the pressure. For convenience relaxation times are usually referred to a pressure of 1 atm. Equation (1) is an approximation, and requires modification to take into account the reversibility between quantum states 0 and 1. For example, the correct equation for vibrational relaxation of a simple harmonic oscillator of fundamental frequency, v, is... 

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