Oscillator angular frequency

Below, therefore, the solution of the Bloch equation in Eq. (6) for the canonical density matrix C(r, Tq, P, F, co) for independent electrons in a constant electric field of strength F, with harmonic restoring force corresponding to an oscillator angular frequency to, will be presented. In Sect. 7.1 below, the electric field is taken as the z axis. Then this solution can readily be generalized to include harmonic restoring forces also in the x and y directions. 

Moment of inertia Period of oscillation Angular frequency ... 

Here E(t) denotes the applied optical field, and-e andm represent, respectively, the electronic charge and mass. The (angular) frequency oIq defines the resonance of the hamionic component of the response, and y represents a phenomenological damping rate for the oscillator. The nonlinear restoring force has been written in a Taylor expansion the temis + ) correspond to tlie corrections to the hamionic... 

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 cycle of oscillation is 0 to 2n, precisely the circumference of a circle. After one cycle of 2n radians is complete, another cycle begins, identical to the one before it. The angular frequency in radians co is related to the frequency expressed in units of complete cycles per second v as co = 2nv, whence... 

A 10,0-g mass connected by a spi itig to a statiotiaiy poitit executes exactly 4 complete cycles of harmonic oscillation in 1,00 s. What are the period of oscillation, the frequency, and the angular frequency What is the force constant of the spring ... 

It turns out that these two equations are valid only when u> has one of two possible values called the normal-mode angular frequencies. In either case, both particles oscillate with the same angular frequency. 

Since nuclear masses are much greater than the electron mass we can treat the nucleus as if it were fixed in space. Taking the mass of the electron charge cloud as m, then k = mu>Q where angular frequency of the oscillator. 

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

The function time-dependent force acting on the harmonic oscillator and mo = K/m is the angular frequency of the system (Section 5.3.3). 

The factor before the square brackets is of course the amplitude of the oscillation It reaches a maximum value when the square of the angular frequency of the forcing function is given by... 

In the Einstein model, all the independent oscillators have the same angular frequency, coe, and the average total internal energy is... 

The time-dependent perturbation operator corresponding to the interaction of a molecule with an external oscillating electric field of an angular frequency... 

Planck showed that the mean energy of a great number of oscillators, each with a characteristic angular frequency w, = 2ttv,, is given by... 

As one may verify by direct quantum mechanical calculation (S), the application of an oscillating magnetic field. Hi, of angular frequency , will cause transitions of the nuclear spin between adjacent magnetic energy... 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

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