Non-harmonic oscillator

The Non-harmonic Oscillator.—We shall consider the case of a linear oscillator of slightly non-harmonic character, i.e. with a... 

In the case of radiation from an atomic system which may be represented approximately by a non-harmonic oscillator it becomes of importance to determine which transitions between the energy steps given by (9) are permissible accordingto the correspondence principle. In order to find this, we calculate q as a function of the angle variable w. The latter is given by... 

In the case of the non-harmonic oscillator, therefore, the co-ordinate oscillates about a mean position differing from the position of equilibrium. The oscillation is not harmonic, for overtones occur, the first of which has an amplitude of the order a. 

The theory of the non-harmonic oscillator finds a further application in the explanation of the increase in the specific heat of solid bodies at high temperatures above Dulong and Petit s value,1 and also in the explanation of band spectra (see 20). 

This reduces the problem to that of the non-harmonic oscillator, which we have discussed in 12. 

As an example of a more complicated case, we may indicate the method of calculation applicable to a spatial non-harmonic oscillator consisting of any number / of coupled linear non-harmonic oscillators.1 Its Hamiltonian function is... 

Perturbations of a Non-degenerate System 42. Application to the Non-harmonic Oscillator 43. Perturbations of an Intrinsically Degenerate System 44. An Example of Accidental Degeneration. 45. Phase Relations in the Case of Bohr Atoms and Molecules... 

Xvf (Ra - Ra,e) Xvi> will be non-zero and probably quite substantial (because, for harmonic oscillator functions these "fundamental" transition integrals are dominant- see earlier) ... 

It is left to reader to verify that, under Lee .s discrete mechanics, both free particles and particles subjected to a constant force, behave in essentially the sa e way as they do under continuous equations of motion. Moreover, the time intervals At = t-i i — ti are all equal. While the spatial behavior for non-constant forces (ex particles in a harmonic oscillator V potential) also remains essentially... 

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. 

According to Bartell (1961a), the relative motion of the interacting non-bonded atoms is described by means of a harmonic oscillator when the two atoms are bonded to the same atom, and by means of two superimposed harmonic oscillators when the atoms are linked to each other via more than one intervening atom. It is the second case which is of interest in connection with the biphenyl inversion transition state. The non-bonded interaction will of course introduce anharmonicity, but since a first-order perturbation calculation of the energy only implies an... 

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... 

Two of the most severe limitations of the harmonic oscillator model, the lack of anharmonicity (i.e., non-uniform energy level spacings) and lack of bond dissociation, result from the quadratic nature of its potential. By introducing model potentials that allow for proper bond dissociation (i.e., that do not increase without bound as x=>°°), the major shortcomings of the harmonic oscillator picture can be overcome. The so-called Morse potential (see the figure below)... 

If the vibrational functions are described within the harmonic oscillator approximation, it can be shown that the <%vf I (Ra - Ra e) I %vj> integrals vanish unless vf = vi +1, vi -1 (and that these integrals are proportional to (vi +1)1/2 and (vi)1/2 in the respective cases). Even when %vf and %vi are rather non-harmonic, it turns out that such Av = 1 transitions have the largest <%vf I (Ra - Ra e) I %vj> integrals and therefore the highest infrared intensities. For these reasons, transitions that correspond to Av = 1 are called "fundamental" those with Av = 2 are called "first overtone" transitions. 

The fundamental frequencies 9t (t = 1, 2,... 3tf—6) are related to and since Xt are the roots of det B—XE) — 0, r, are related to the matrix B and to the molecular force constants Bif. Hence the vibrational energy levels for a non-linear polyatomic molecule in the harmonic oscillator approximation are given by... 

The first term on the right-hand side corresponds to Eq. (2), whereas the second term describes dissipative effects that are induced in the system due to its coupling to the environment. The latter is modeled, as usual [32, 33], as the thermal (temperature T) ensemble of harmonic oscillators, with nonlinear coupling A Qiq) F( thermal bath, expressed in terms of nonlinear molecular and linear environment coupling operators Q(q) and F( qk )- As shown in Ref. 15, it is important to describe the dissipative term in Eq. (10) by making use of the non-Markovian expression... 

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