Harmonic Oscillator Trajectories

Let us consider the phase space trajectory traced out by diis behavior beginning with the position vector. Over any arbitrary time interval, die relationship between two positions is 

These equations map out the oval phase space trajectory depicted in the figure. 

Certain aspects of this phase space trajectory merit attention. We noted above that a 

as required, these (periodic) trajectories do not cross one another. 

Certain aspects of this phase space trajectory merit attention. We noted above that a phase space trajectory cannot cross itself. However, it can be periodic, which is to say it can trace out the same path again and again the harmonic oscillator example is periodic. Note that the complete set of all harmonic oscillator trajectories, which would completely fill the corresponding two-dimensional phase space, is composed of concentric ovals (concentric circles if we were to choose the momentum metric to be (mk) 1/2 times the position metric). Thus, as required, these (periodic) trajectories do not cross one another. 

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Figure 1. Mathcad worksheet calculating the harmonic oscillator trajectory with parameters appropriate for the iodine molecule.

Fig. 5. Langevin trajectories for a harmonic oscillator of angular frequency u = 1 and unit mass simulated by a Verlet-like method (extended to Langevin dynamics) at a timestep of 0.1 (about 1/60 the period) for various 7. Shown for each 7 are plots for position versus time and phase-space diagrams.

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]. 

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,... 

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. 

As an example, consider the differential equation x + x => 0 of the harmonic oscillator, whose trajectories are circles. Choose one of these circles (corresponding to given initial conditions) and on this circle take a point A for t = 0. The transformation effected by this differential equation after the time Ztt will result in a return to the same point, which can be written as... 

One can interpret this physically as follows suppose that the trajectory of the harmonic oscillator be represented by a point on a rotating wheel. The eye observes a circle (the path of the point) if the wheel rotates rapidly this corresponds to continuous illumination. On the other hand, if one illuminates the rotating wheel with stroboscopic flashes separated by a period 2n, a given mark on the wheel appears as a fixed point. Thus, under continuous illumination one sees ... 

Because T -> V energy transfer does not lead to complex formation and complexes are only formed by unoriented collisions, the Cl" + CH3C1 -4 Cl"—CH3C1 association rate constant calculated from the trajectories is less than that given by an ion-molecule capture model. This is shown in Table 8, where the trajectory association rate constant is compared with the predictions of various capture models.9 The microcanonical variational transition state theory (pCVTST) rate constants calculated for PES1, with the transitional modes treated as harmonic oscillators (ho) are nearly the same as the statistical adiabatic channel model (SACM),13 pCVTST,40 and trajectory capture14 rate constants based on the ion-di-pole/ion-induced dipole potential,... 

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, harmonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the trajectory, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

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

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

The initial exploration in this unit requires the students to compare the trajectories calculated for several different energies for both Morse oscillator and harmonic oscillator approximations of a specific diatomic molecule. Each pair of students is given parameters for a different molecule. The students explore the influence of initial conditions and of the parameters of the potential on the vibrational motion. The differences are visualized in several ways. The velocity and position as a function of time are plotted in Figure 2 for an energy approximately 50% of the Morse Oscillator dissociation energy. The potential, kinetic and total energy as a function of time are plotted for the same parameters in Figure 3. 

For e = 0, the quantities (10) and (17) become first integrals for the harmonic oscillator [141]. It is obvious from (15)—(16) that a trajectory in phase space (p, q) for the Kerr oscillator is analytically the same ellipse as for the harmonic oscillator... 

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...

For systems more complicated than the harmonic oscillator, it is almost never possible to write down analytical expressions for the position and momentum components of the phase space trajectory as a function of time. However, if we approximate Eqs. (3.10) and (3.12) as... 

One might suspect that the discrepancy between the thermal rigidity factors of Eqs. (24) and (36) is due to inadequacies of the SACM treatment in general or of the pure harmonic oscillator model. However, after having corrected the analytical representation of the trajectory results through Eqs. (28)—(31), for x 11, Eq. (28) leads to... 

Figure 3.2. Time contour (a) and real part (b) of the tunneling trajectory for a separable system of a parabolic barrier-harmonic oscillator (schematic). Curves E, E and E" are equipotentials. Vertical and horizontal dashed lines show the loci of vibrational and translational turning points. Points A, B, and C indicate the corresponding times and positions along trajectory. (From Altcorn and Schatz [1980]).

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