Perturbed harmonic oscillator

It may be observed that the two Hamiltonians (269) are those of quantum harmonic oscillators, whereas Hamiltonian (270) is that of a driven damped quantum harmonic oscillator, and Hamiltonians (271) are those of driven undamped quantum harmonic oscillators perturbed by the Davydov coupling... 

The problem considered is the one-dimensional harmonic oscillator perturbed by cubic and quartic potential terms. Thus, the unperturbed Hamiltonian operator is... 

The Morse potential provides a useful model for the anharmonic stretching vibrations of a polyatomic molecule. It is superior to a harmonic oscillator perturbed by cubic and quartic anharmonicity terms in terms of both convergence (a V(r) that includes an r3 term cannot be bound, thus cannot have any rigorously bound vibrational levels) and the need for a smaller number of adjustable parameters to describe both the potential energy curve (a and De for Morse frr, frrri and frrrr for the cubic plus quartic perturbed harmonic oscillator) and the energy levels. 

This treatment of the motion as that of a harmonic oscillator perturbed bj the presence of several positions of minimum potential energy is quantitatively useful only when the splitting is small compared with the spacing of the groups of levels. Beyond this point, other approaches are desirable. For example, it may be possible to express the Eulerian angles from one set of coordinates in terms of those of another and nuth the aid of this relation the rotational parts of both ip a and ipa may be factored out of the combination, leaving... 

Fig. 8.3. Splitting pattern of the /V = 3 multiplet for a harmonic oscillator perturbed by a linear two-body potential. In the first column, the whole N = 3 multiplet is shifted by from its unperturbed value (not shown). The effect of ij induces a splitting which is further modified by the term y, defined in eq. (8.19).

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. 

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. 

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]. 

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... 

As illustrations of the application of perturbation theory we consider two examples of a perturbed harmonic oscillator. In the first example, we suppose that the potential energy V of the oscillator is... 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

Since equation (10.43) with F = 0 is already solved, we may treat V as a perturbation and solve equation (10.43) using perturbation theory. The unperturbed eigenfunctions S H q) are the eigenkets n) for the harmonic oscillator. The first-order perturbation correction to the energy as given by equation (9.24) is... 

Treating vibrational excitations in lattice systems of adsorbed molecules in terms of bound harmonic oscillators (as presented in Chapter III and also in Appendix 1) provides only a general notion of basic spectroscopic characteristics of an adsorbate, viz. spectral line frequencies and integral intensities. This approach, however, fails to account for line shapes and manipulates spectral lines as shapeless infinitely narrow and infinitely high images described by the Dirac -functions. In simplest cases, the shape of symmetric spectral lines can be characterized by their maximum positions and full width at half maximum (FWHM). These parameters are very sensitive to various perturbations and changes in temperature and can therefore provide additional evidence on the state of an adsorbate and its binding to a surface. 

In an elastic material medium a deformation (strain) caused by an external stress induces reactive forces that tend to recall the system to its initial state. When the medium is perturbed at a given time and place the perturbation propagates at a constant speed (or celerity) c that is characteristic of the medium. This propagating strain is called an elastic (or acoustic or mechanical) wave and corresponds to energy transport without matter transport. Under a periodic stress the particles of matter undergo a periodic motion around their equilibrium position and may be considered as harmonic oscillators. 

The first attempts (G. Klein and I. Prigogine, 1953, MSN.5,6,7) were very timid and not very conclusive. They were devoted to a chain of harmonic oscillators. In spite of a tendency to homogenization of the phases, there was no intrinsic irreversibility here, because an essential ingredient is lacking in this model the interaction among normal modes. The latter were introduced as a small perturbation in the fourth paper of the series (MSN.8). 

Or is the frequency of the harmonic oscilator and b) are boson (phonon) creation (annihilation) operators. In order to use the perturbation theory we have to split the Hamiltonian (16) onto the unperturbed part Hq and the perturbation H ... 

It should be noted that (4.28) is only an approximation for the nuclear wave function. The perturbation terms (4.36) will mix into the nuclear wave function small contributions from harmonic-oscillator functions with quantum numbers other than v. These anharmonicity corrections to the vibrational wave function will add further to the probability of transitions with At) > 1. 

This result shows that for slow perturbations, i.e., y a, the average behaves as a superposition of two harmonic oscillators, with frequencies close to co0 a, each being slightly damped. In Fourier language that amounts to two separate Lorentzians. When y grows, both peaks broaden and merge into a single broad line. On the other hand, for rapid perturbations, y > a, one obtains from (6.9) to first order in y 1,... 

The eigenvalue problem for the simple cos y potential of Eq. (4) can be solved easily by matrix diagonalization using a basis of free-rotor wave functions. For practical purposes, however, it is also useful to have approximate analytical expressions for the channel potentials V,(r). The latter can be constructed by suitable interpolation between perturbed free-rotor and perturbed harmonic oscillator eigenvalues in the anisotropic potential for large and small distances r, respectively. Analogous to the weak-field limit of the Stark effect, for linear closed-shell dipoles at large r, one has [7]... 

As a second example we analyze the anisotropic charge-permanent dipole potential where SACM and PST differ from each other. Here, for demonstration, we only consider the low-energy perturbation and the high-energy harmonic oscillator limits. In the former limit, the adiabatic channel potential curve for the lowest channel j = m = 0 has the form... 

The formulas for the susceptibility of a harmonic oscillator, presented above, were first derived in Ref. 18 with neglect of correlation between the particles orientations and velocities. This derivation was based on an early version of the ACF method, in which the average perturbation theorem was not employed, so that the expression equivalent to Eq. (14c) was used. (The integrand of the latter involves the quantities perturbed by an a.c. field.) For a specific case of the parabolic potential, the above-mentioned theory is simple however, it becomes extremely cumbersome for more realistic forms of the potential well. 

---