Harmonic-oscillator system

We first expand the full 3N-dof potential energy surface about a chosen stationary point, that is, minimum, saddle, or higher rank saddle. By taking the zeroth-order Hamiltonian as a harmonic oscillator system, which might include some negatively curved modes, that is, reactive modes, we establish the higher order perturbation terms to consist of nonlinear couplings expressed in arbitrary combinations of coordinates. 

By introducing a coordinate transformation, we may reduce the harmonic oscillator system to a pair of decoupled complex scalar ODEs of the form... 

FIGURE 9 Classical (dashed) and quantum (solid) probability of a coupled Morse-harmonic oscillator system remaining in its initial state. The energy is sufficient to allow dissociation. [From Kay, K. (1984). J. Chem. Phys. 80, 4973.]... 

This is the Lagrangian hmction for the harmonic oscillator system. 

Now we need to determine the values of the constants c . Recall that this equation was determined by substituting a trial wavefunction into the Schrodinger equation, so that if the harmonic oscillator system has wavefunctions that are eigenfunctions of the Schrodinger equation, those wavefunctions would be of the form given in equation 11.8 [that is, R = f(x)]. By identifying the constants, we... 

Changes in vibrational energy should be exact multiples of the Av = 1 transition. However, real normal vibrations are not ideal (which is why such transitions are observed occasionally in the first place), so absorptions due to overtone transitions are usually less than an integral number of hv. This deviation is a measure of anhar-monicity, which we will consider in the next section. Table 14.3 lists the absorptions due to the fundamental and overtone vibrational transitions for HCl (g). Also listed are the various multiples of the fundamental vibrational frequency, and the variance from the multiple as shown by experiment. Note how the overtone absorptions get farther and farther from ideal. The fact that Av > 1 is possible (although to a much lesser extent than Av = 1) and the variance from exact multiples of the fundamental vibrational frequency are both reminders that molecules are not true harmonic oscillators. They are anharmonic oscillators. The use of the ideal harmonic oscillator system in describing molecular vibrations is an approximation—but a good approximation. 

Spreadsheet problem] (a) Evaluate cj imitonai ir Equation 11.46 for a quantum mechanical harmonic oscillator system for which the energies are E (kJ moh ) = 23.0 (n +1/2) at 20 temperatures from 10 to 1,000K. (b) Evaluate q it,raHonai the energy expression includes an anharmonidty term -0.1(n + 1/2). Plot the harmonic and anharmonic values of cimbranonai 3 function of temperature. 

Now consider a system of N one-dimensional harmonic oscillators with the Hamiltonian... 

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]. 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

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 hamionic oscillator (Fig. 4-1) is an idealized model of the simple mechanical system of a moving mass connected to a wall by a spring. Oirr interest is in ver y small masses (atoms). The harmonic oscillator might be used to model a hydrogen atom connected to a large molecule by a single bond. The large molecule is so... 

Using MMd. calculate A H and. V leading to ATT and t his reaction has been the subject of computational studies (Kar, Len/ and Vaughan, 1994) and experimental studies by Akimoto et al, (Akimoto, Sprung, and Pitts. 1972) and by Kapej n et al, (Kapeijn, van der Steen, and Mol, 198.V), Quantum mechanical systems, including the quantum harmonic oscillator, will be treated in more detail in later chapters. 

This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /iLorentzian distribution of the system s normal modes. In the absence of friction (2.27) describes thermally activated as well as tunneling processes when < 1, or fhcoo > 1, respectively. At first glance it may seem surprising... 

Harmonic Oscillator The simplest example of an integrable system is the har-... 

The deep philosophical significance of the new theory lies precisely at this point, and consists in replacing a somewhat metaphysical concept of the harmonic oscillator (which could never be produced experimentally) by the new concept of a physical oscillator of the limit cycle type, with which we are dealing in the form of electron tube circuits and similar self-excited systems. 

Problem of Poincard (Nonresonance Case).—Now consider Eqs. (6-47) or (6-48), which are sufficiently general to furnish a basis for further discussion of these systems. If p = 0, one has the differential equation of the harmonic oscillator + x = 0 whose solutions we know. As we assume that p is small, Eq. (6-50) differs but little from that of the harmonic oscillator one often says that the two differential equations are in the neighborhood of each other. But from this fact one cannot conclude that their solutions (trajectories) are also in the neighborhood of each other. Let us take a simple example F(t,x,x) — x and compare the two equations x + x = 0 and x + px + x = 0. For the first the trajectories are circles, whereas for the second they are spirals, so that for a sufficiently large t the solutions certainly are not in the neighborhood of each other, although the differential equations are. 

In NMR theory the analogue of the relation (1.57) connects the times of longitudinal (Ti) and transverse (T2) relaxation [39]. In the case of weak non-adiabatic interaction with a medium it turns out that T = Ti/2. This also happens in a harmonic oscillator [40, 41] and in any two-level system. However, if the system is perturbed by strong collisions then Ti = T2 as for y=0 [42], Thus in non-adiabatic theory these times differ by not more than a factor 2 regardless of the type of system, or the type of perturbation, which may be either impact or a continuous process. 

The most simple way to accomplish this objective is to correct the external field operator post factum, as was repeatedly done in magnetic resonance theory, e.g. in [39]. Unfortunately this method is inapplicable to systems with an unrestricted energy spectrum. Neither can one use the method utilizing the Landau-Teller formula for an equidistant energy spectrum of the harmonic oscillator. In this simplest case one need correct... 

Visco-elastic fluids like pectin gels, behave like elastic solids and viscous liquids, and can only be clearly characterized by means of an oscillation test. In these tests the substance of interest is subjected to a harmonically oscillating shear deformation. This deformation y is given by a sine function, [ y = Yo sin ( t) ] by yo the deformation amplitude, and the angular velocity. The response of the system is an oscillating shear stress x with the same angular velocity . 

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

The harmonic oscillator is an idealized one-dimensional physical system in which a single particle of mass m is attracted to the origin by a force F proportional to the displacement of the particle Ifom the origin... 

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