Harmonic oscillator quantum theory

For a harmonic oscillator, quantum theory requires that the transition integral (f Q i) disappears unless the vibrational quantum numbers Vi and Vf are related by v,-- = 0 or 1. This requires that over-... 

Perturbation theory offers another method for finding quantum mechanical wavefunc-tions. It is especially suited to problems that are similar to model or ideal situations differing only in some small way. For example, the potential for an oscillator might be harmonic except for a feature such as the small "bump" depicted in Figure 8.8. Because the bump is a small feature, one expects the system s behavior to be quite similar to that of a harmonic oscillator. Perturbation theory affords a way to correct the description of the system, obtained from treating it as a harmonic oscillator, so as to account for the effects of the bump in the potential. In principle, perturbation theory can yield exact wavefunctions and eigenenergies, but usually it is employed as an approximate approach. 

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

Quantization (the idea of quantums, photons, phonons, gravitons) is postulated in Quantum Mechanics, while the Theory of Relativity does not derive quantization from geometric considerations. In the case of the established phenomenon the quantized nature of portioned energy transfer stems directly from the mechanisms of the process and has a precise mathematical description. The quasi-harmonic oscillator obeys the classical laws to a greater extent than any other system. A number of problems, related to quasi-harmonic oscillators, have the same solution in classical and quantum mechanics. 

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 ( ) ... 

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

Thus, in the high-temperature limit, the mean-square displacement of the harmonic oscillator, and therefore the temperature factor B, is proportional to the temperature, and inversely proportional to the frequency of the oscillator, in agreement with Eq. (2.43). At very low temperatures, the second term in Eq. (2.51a) becomes negligible. The mean-square amplitude of vibrations is then a constant, as required by quantum-mechanical theory, and evident in Fig. 2.5. 

Slater s theory assumes that the normal modes behave as harmonic oscillators, which requires that there be no flow of energy between the normal modes once the molecule is suitably activated, and so the energy distribution remains fixed between collisions. But spectroscopy shows that energy can flow around a molecule, and allowing for such a flow between collisions vastly improves the theory. Like Kassel s theory a fully quantum theory would be superior. 

Quantum-chemical calculations for pyrylium including one, two, or three water molecules using DFT and 6-31 + G(d,p) basis set revealed that the aromaticity (estimated by harmonic oscillator stabilization energy, HOSE natural resonance theory, NRT harmonic oscillator model of aromaticity, HOMA and nucleus-independent chemical shifts, NICS) is not influenced by water molecules [82],... 

Memory effects play an important role for the description of dynamical effects in open quantum systems. As mentioned above, Meier and Tannor [32] developed a time-nonlocal scheme employing the numerical decomposition of the spectral density. The TL approach as discussed above as well as the approaches by Yan and coworkers [33-35] use similar techniques. Few systems exist for which exact solutions are available and can serve as test beds for the various theories. Among them is the damped harmonic oscillator for which a path-integral solution exists [1], In the simple model of an initially excited... 

The theory of rotational and vibrational Raman intensities is discussed in detail elsewhere (e.g.. References 1-6). Relative rotational Raman intensities are proportional to Raman line strength factors (S ). For rigid rotator, harmonic oscillator diatomic molecules S (J.,Jf) = 3(Jj+1)(Jj+2)/(2(2L+3)) where J is a rotational quantum number. However, real molecules are not rigid rotators and S must be... 

In this expression, according to the theory of the quantum harmonic oscillator, the operator q appearing on the right-hand side, may couple two successive eigenstates /c ) of the Hamiltonian of the harmonic oscillator. Consequently, by ignoring the scalar term p(0,0), which does not couple these states, we may write the dipole moment operator according to... 

As usual in the quantum theory of damping [54], the thermal bath may be figured by an infinite set of harmonic oscillators and its coupling with the H-bond bridge by terms that are linear in the position coordinates of the bridge and of the bath oscillators ... 

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