Harmonic Oscillator Results

What about the actual form of the wave function The mathematics requires us to backtrack through all the variable changes that were made. Here we present the final results, which are normalized. Other more advanced texts in quantum mechanics will give further details, and if a student is interested in further work, it would be a good idea to purchase the text by Pauling and Wilson [7] in its original form or the reprinted version from Dover Press. 

Note there are odd and even solutions (Table 12.2). Note that the normalization constant N = 

Most textbooks illustrate the lowest wave functions of the harmonic oscillator by plotting the 

Note that N2 and CO have very high force constants and N2 has the highest in Table 12.3. Consider that N=N clearly has a triple bond and so does C=0 when you draw the Lewis electron stmcture  

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The harmonic-oscillator results from the exact calculation have been compared with an approximate formula for AE obtained by Rapp [20] ... 

We now examine actual instances. In Table I we have listed the parameters pertinent to the preceding discussion for a number of solids the lowest two librational frequencies are considered. We see that COj, NjO, Clg, C2N3, OCS, and benzene all have k larger than 10(X), and many of them several times that value. We conclude therefore that the harmonic oscillator result is probably quite accurate for the librations in these solids. On the other hand, a-Ng, CO, CgH, and the hydrogen halides (HCl, HBr) have low values of k and therefore the harmonic oscillator results may be inaccurate. It is difficult, on the other hand, to assess the errors... 

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

A number of empirical tunneling paths have been proposed in order to simplify the two-dimensional problem. Among those are MEP [Kato et al. 1977], sudden straight line [Makri and Miller 1989], and the so-called expectation-value path [Shida et al. 1989]. The results of these papers are hard to compare because slightly different PES were used. As to the expectation-value path, it was constructed as a parametric line q(Q) on which the vibration coordinate q takes its expectation value when Q is fixed. Clearly, for the PES at hand this path coincides with MEP, since is a harmonic oscillator. 

As an example, consider the differential equation x + x => 0 of the harmonic oscillator, whose trajectories are circles. Choose one of these circles (corresponding to given initial conditions) and on this circle take a point A for t = 0. The transformation effected by this differential equation after the time Ztt will result in a return to the same point, which can be written as... 

Such a construction is not a result of perturbation theory in <5 , rather it appears from accounting for all relaxation channels in rotational spectra. Even at large <5 the factor j8 = B/kT < 1 makes 1/te substantially lower than a collision frequency in gas. This factor is of the same origin as the factor hco/kT < 1 in the energy relaxation rate of a harmonic oscillator, and contributes to the trend for increasing xE and zj with increasing temperature, which has been observed experimentally [81, 196]. 

The change in the inner-sphere structure of the reacting partners usually leads to a decrease in the transition probability. If the intramolecular degrees of freedom behave classically, their reorganization results in an increase in the activation barrier. In the simplest case where the intramolecular vibrations are described as harmonic oscillators with unchanged frequencies, this leads to an increase in the reorganization energy ... 

Using the results of Section 4.4, we may easily verify for the harmonic oscillator the Heisenberg uncertainty relation as discussed in Section 3.11. Specifically, we wish to show for the harmonic oscillator that... 

Derive the result that the degeneracy of the energy level E for an isotropic three-dimensional harmonic oscillator is (n + l)(n + 2)/2. 

The matrix elements n x n) for the unperturbed harmonic oscillator are given by equations (4.50). The first-order correction term is obtained by substituting equations (9.50) and (4.50e) into (9.24), giving the result... 

Thus, we obtain a result analogous to the harmonic oscillator solution. The two independent solutions are infinite series, one in odd powers of fi and the other in even powers of fi. The case 5 = 0 gives both solutions, while the case s = 1 merely reproduces the odd series. These solutions are... 

Finally, it is a weU-known result of quantum mechanics" that the wavefunctions of harmonic oscillators extend outside of the bounds dictated by classical energy barriers, as shown schematically in Figure 10.1. Thus, in situations with narrow barriers it can... 

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. 

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