Two displaced harmonic oscillators

Once a model potential is derived, it is possible to verify and refine this potential, making use of observed perturbation matrix elements and calculated overlap integrals between the vibrational levels of the two interacting electronic states. Although it is usually more convenient to input the analytic form of V(R) into a numerical integration program to calculate overlap integrals (Section 5.1.3), analytic expressions exist for harmonic and Morse (vi vj) factors. 

The vibrational wavefunctions for a Morse oscillator can be expressed analytically in terms of Whittaker functions (Felenbok, 1963) or in terms of polygamma functions (Matsumoto and Iwamoto, 1993). 

The harmonic potential is a model of last resort for diatomic molecules. Its behavior at R = 0 and R = oo is unphysical, as is the sign of ae. Exact diatomic molecule vibrational wavefunctions for levels above v = 0, except for their number of nodes, differ from harmonic oscillator eigenfunctions (Hermite polynomials with an exponential factor) in that they are not symmetric about Re and, increasingly so at high v, are skewed toward the outer turning point. 

The Morse potential displays the qualitative features of realistic molecular 

In addition to the Morse potential there exist other model potentials defined by three adjustable parameters for example, the Varshni potential (Varshni, 1957) 

---

Again we obtain two displaced harmonic oscillators with the same frequency, but the absolute values of their displacements are different in this case. The eigenvalues... 

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

For example, in the case of H tunneling in an asymmetric 0i-H - 02 fragment the O1-O2 vibrations reduce the tunneling distance from 0.8-1.2 A to 0.4-0.7 A, and the tunneling probability increases by several orders. The expression (2.77a) is equally valid for the displacement of a harmonic oscillator and for an arbitrary Gaussian random value q. In a solid the intermolecular displacement may be contributed by various lattice motions, and the above two-mode model may not work, but once q is Gaussian, eq. (2.77a) will still hold, however complex the intermolecular motion be. 

Pj and p2 represent the displacement vectors of the nuclei A and D (the corresponding polar coordinates are p1 cji, and p2, < )2, respectively) p, and pc are the displacement vectors and pT, r and pc, <[)f the corresponding polar coordinates of the terminal nuclei at the (collective) trans-bending and cis-bending vibrations, respectively. As a consequence of the use of these symmetry coordinates the nuclear kinetic energy operator for small-amplitude bending vibrations represents the kinetic energy of two uncoupled 2D harmonic oscillators ... 

The Hamiltonian function for a system of bound harmonic oscillators is, in the most general form, a sum of two positively definite quadratic forms composed of the particle momentum vectors and the Cartesian projections of particle displacements about equilibrium positions ... 

Vibrational energy and transitions As seen in Fig. 3.2a, the bond between the two atoms in a diatomic molecule can be viewed as a vibrating spring in which, as the internuclear distance changes from the equilibrium value rc, the atoms experience a force that tends to restore them to the equilibrium position. The ideal, or harmonic, oscillator is defined as one that obeys Hooke s law that is, the restoring force F on the atoms in a diatomic molecule is proportional to their displacement from the equilibrium position. 

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. 

If the duration of the experiment is measured not from the onset of the UV pulse but from t0, the time spent in boosting the momentum in the lower electronic state should be less than the time saved evolving on V2. For the two harmonic oscillators displaced the distance D, this requirement gives in the limit of T small (a more general case is analyzed in [7]) the following condition for the strength E0 of the IR field (m is the mass and qel is the equilibrium distance of oscillator Vi) ... 

The simple class of models just discussed is of interest because it is possible to characterize the decay of correlations rather completely. However, these models are rather far from reality since they take no account of interparticle forces. A next step in our examination of the decay of initial correlations is to find an interacting system of comparable simplicity whose dynamics permit us to calculate at least some of the quantities that were calculated for the noninteracting systems. One model for which reasonably complete results can be derived is that of an infinite chain of harmonic oscillators in which initial correlations in momentum are imposed. Since the dynamics of the system can be calculated exactly, one can, in principle, study the decay of correlations due solely to internal interactions (as opposed to interactions with an external heat bath). We will not discuss the most general form of initial correlations but restrict our attention to those in which the initial positions and momenta have a Gaussian distribution so that two-particle correlations characterize the initial distribution completely. Let the displacement of oscillator j from its equilibrium position be denoted by qj and let the momentum of oscillator j be pj. On the assumption that the mass of each oscillator is equal to 1, the momentum is related to displacement by pj =. We shall study... 

A schematic view of the nanomechanical GMR device to be considered is presented in Fig. 1. Two fully spin-polarized magnets with fully spin-polarized electrons serve as source and drain electrodes in a tunneling device. In this paper we will consider the situation when the electrodes have exactly opposite polarization. A mechanically movable quantum dot (described by a time-dependent displacement x(t)), where a single energy level is available for electrons, performs forced harmonic oscillations with period T = 2-k/uj between the leads. The external magnetic field is perpendicular to the orientation of the magnetization in both leads. 

Let us take two spherically symmetrical systems, each with a polaris-ability a, say two three-dimensional isotropic harmonic oscillators with no permanent moment in their rest position. If the charges e of these oscillators are artificially displaced from their rest positions by the displacements... 

A molecule with N atoms has a total of 37V degrees of freedom for its nuclear motions, since each nucleus can be independently displaced in three perpendicular directions. Three of these degrees of freedom correspond to translational motion of the center of mass. For a nonlinear molecule, three more degrees of freedom determine the orientation of the molecule in space and thus its rotational motion. This leaves 37V - 6 vibrational modes. For a linear molecule, there are just two rotational degrees of freedom, which leaves 3N -5 vibrational modes. For example, the nonlinear molecule H2O has three vibrational modes, while the linear molecule CO2 has four vibrational modes. The vibrations consist of coordinated motions of several atoms in such a way as to keep the center of mass stationary and nonrotating. These are called the normal modes. Each normal mode has a characteristic resonance frequency Vj (expressed in cm ), which is usually determined experimentally. To a reasonable approximation, each normal mode behaves as an independent harmonic oscillator of frequency u . The normal modes of H2O and CO2 are shown in Figs. 14.2 and 14.3. A normal mode will be infrared active only if it involves an oscillation of the dipole moment. All three modes of H2O are... 

In the second part of this work, we addressed the problem of how to use the above described effects of a space-dependent interaction to steer molecular transition. We proposed a model that allows us to induce a space-dependent coupling between two molecular potentials via a steady-state coupling to a third potential surface. By changing the frequency and intensity of the steady state laser, we can shape the space-dependence of the coupling. We illustrated the method with an example of three coupled harmonic oscillators and showed how displacement and width of the excited wave packet can be controlled. 

Figure 12. Schematic representation of two coupled linear harmonic oscillators. Equilibrium positions are at q 0. cri and Q2 are relative displacements.

This work is intended as an attempt to present two essentially different constructions of harmonic oscillator states in a FD Hilbert space. We propose some new definitions of the states and find their explicit forms in the Fock representation. For the convenience of the reader, we also bring together several known FD quantum-optical states, thus making our exposition more self-contained. We shall discuss FD coherent states, FD phase coherent states, FD displaced number states, FD Schrodinger cats, and FD squeezed vacuum. We shall show some intriguing properties of the states with the help of the discrete Wigner function. 

---