Linear harmonic oscillator

If only the first derivatives in tie dipole-moment function and the second derivatives (k=j) in the potential function are retained, the strict harmonic-oscillator-linear-dipole-moment approximation, the selection rules are strict ... 

Figure 9. Schematic of possible types of oscillations for Model 3 (potential (P) as a function of polarization P) (I) harmonic oscillations (linear regime) (II) nonlinear oscillations with a large amplitude, < P " >T = 0 (III) nonlinear oscillations around the stable states P, with < P > t = 0.

Here we apply the LAND-map approach to compute of the time dependent average population difference, A t) = az t)), between the spin states of a spin-boson model. Here az = [ 1)(1 — 2)(2 ]. Within the limits of linear response theory, this model describes the dissipative dynamics of a two level system coupled to an environment [59,63-65]. The environment is represented by an infinite set of harmonic oscillators, linearly coupled to the quantum subsystem. The characteristics of the system-bath coupling are completely described by the spectral density J(w). In the following, we shall restrict ourselves to the case of an Ohmic spectral density... 

The Harmonic Oscillator Linear Differential Equations with Constant Coefficients... 

Metiu, Oxtoby, and Freed applied a viscoelastic hydrodynamic model to the relaxation of a harmonic oscillator linearly coupled to a heat bath (a model for which the population relaxation and dephasing times differ by only a factor of 2 see Section II.B). They considered the two atoms of the diatomic to be half-spheres in contact with a continuum and used frequency-dependent (viscoelastic) shear and bulk viscosities with slip boundary conditions. The result they derived for the population relaxation... 

A general and convenient choice is to model the bath as a set of independent harmonic oscillators linearly coupled to the system. The Hamiltonian for a collection of oscillators of unit mass and frequencies w. is... 

The results from the general theory for the vibrational spectrum of a localized harmonic oscillator, linearly coupled with a noninteracting boson continuum (phonons, photons, electron-hole pairs), can be used to estimate the contribution of different relaxation processes at smfaces. The spectral function of the oscillator obtained by normal-mode analysis at zero temperature is [25]... 

In the above equations coq is the frequency of a noninteraction oscillator, X(o)) is the eoupling function between the oscillator and the external field, and RHg (ro) is the field density of states. With Eqs (67)-(74) it will now be possible to predict the spectral behavior for localized harmonic oscillators linearly coupled with different model boson continua. 

Consider for the nth time step the linear harmonic oscillator... 

Another option is a q,p) = p and b q,p) = VU q). This guarantees that we are discretizing a pure index-2 DAE for which A is well-defined. But for this choice we observed severe difficulties with Newton s method, where a step-size smaller even than what is required by explicit methods is needed to obtain convergence. In fact, it can be shown that when the linear harmonic oscillator is cast into such a projected DAE, the linearized problem can easily become unstable for k > . Another way is to check the conditions of the Newton-Kantorovich Theorem, which guarantees convergence of the Newton method. These conditions are also found to be satisfied only for a very small step size k, if is small. 

Of particular interest is the model of a bath as a set of harmonic oscillators qj with frequencies cOj, which are linearly coupled to the tunneling coordinate... 

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,... 

It is important to note that in all these methods, the first term in the series solution constitutes the so-called approximation of zero order. This is generally the solution of a simple linear problem e.g., the harmonic oscillator the second term appears as the first approximation, and so on. The amount of labor increases very rapidly with the order of approximation, but the additional information obtained from approximations of higher orders (beginning with the second) does not increase our knowledge from the qualitative point of view. It merely adds small quantitative corrections to the first approximation, and in most applied problems, these corrections are scarcely worth the considerable complication in calculations. For that reason the first approximation is generally sufficient in exploring a new problem, or in investigating the qualitative aspect of a phenomenon. 

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

Show that the wave functions A (y) in momentum space corresponding to 0 ( ) in equation (4.40) for a linear harmonic oscillator are... 

The Sehrodinger equation for the linear harmonic oscillator leads to the differential equation (4.17)... 

Because the degrees of freedom decouple in the linear approximation, it is easy to describe the dynamics in detail. There is the motion across a harmonic barrier in one degree of freedom and N — 1 harmonic oscillators. Phase-space plots of the dynamics are shown in Fig. 1. The transition from the reactant region at q <0 to the product region at q >0 is determined solely by the dynamics in (pi,qi), which in the traditional language of reaction dynamics is called the reactive mode. 

In the last equation Hi(x) is the th Hermite polynomial. The reader may readily recognize that the functions look familiar. Indeed, these functions are identical to the wave functions for the different excitation levels of the quantum harmonic oscillator. Using the expansion (2.56), it is possible to express AA as a series, as has been done before for the cumulant expansion. To do so, one takes advantage of the linearization theorem for Hermite polynomials [42] and the fact that exp(-t2 + 2tx) is the generating function for these polynomials. In practice, however, it is easier to carry out the integration in (2.12) numerically, using the representation of Po(AU) given by expressions (2.56) and (2.57). 

Consider an ensemble of harmonic oscillators interacting linearly with an ion of charge number z, so that the potential energy of the system is given by ... 

The statistical treatment of the vibrational degrees of freedom of crystals is far more difficult compared to gases. Let us initially consider a monoatomic crystal. An atom in a crystal vibrates about its equilibrium lattice position. In the simplest approach, three non-interacting superimposed linear harmonic oscillators represent the vibrations of each atom. The total energy, given by the sum of the kinetic and potential energies for the harmonic oscillators, is... 

The decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( ) ... 

Since these equations are general for a system exhibiting linear response, we can illustrate them by the simplest such system, a harmonic oscillator... 

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