Oscillators, 3-dimensional harmonic results

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]. 

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

We have considered only one particular degenerate vibrational level. The general case is dealt with by solving the isotropic two-dimensional harmonic oscillator in plane polar coordinates (Problem 6.19). The result is... 

Similarly, during their effort to understand the thermal energy of solids, Einstein and Debye quantized the lattice waves and the resulting quantum was named phonon. Consequently, it is possible to consider the lattice waves as a gas of noninteracting quasiparticles named phonons, which carries energy, E=U co, and momentum, p = Uk. That is, each normal mode of oscillation, which is a one-dimensional harmonic oscillator, can be considered as a one-phonon state. 

In 34 the eigenvalue problem of the one-dimensional time-independent Schrodinger equation is studied. Exponentially fitted and trigonometrically fitted symplectic integrators are developed, by modification of the first and second order Yoshida symplectic methods. Numerical results are presented for the one-dimensional harmonic oscillator and Morse potential. 

In 35 the numerical solution of the two-dimensional time-independent Schrodinger equation is studied using the method of partial discretization. The discretized problem is treated as a problem of the numerical solution of a system of ordinary differential equations and Numerov type methods are used to solve it. More specifically the classical Numerov method, the exponentially and trigonometrically fitting modified Numerov methods of Vanden Berghe el al. and the minimum phase-lag method of Rao et al. are applied to this problem. The methods are applied for the calculation of the eigenvalues of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heils potential. The results are compared with the results produced by full discretization. Conclusions are presented. 

Figure 2 shows the results for simulations of the same one-dimensional harmonic oscillator with Nose-Hoover chains of var3ung lengths. The first column shows the results for a chain of length M = 1, which is equivalent to the NosAHoover algorithm. Notice that neither the momentum (b) nor position (c) distributions are canonical in nature. This is further confirmed by the presence of a Hoover hole in the Poincare section depicted in (a). 

Let us now consider a system whose Schrodinger time functions corresponding to the stationary states of the system are k0, i, , kn, . Suppose that we carry out an experiment (the measurement of the values of some dynamical quantities) such as to determine the wave function uniquely. Such an experiment is called a maximal measurement. A maximal measurement for a system with one degree of freedom, such as the one-dimensional harmonic oscillator, might consist in the accurate measurement of the energy the result of the measurement would be one of the characteristic energy values W and the corresponding wave function would then represent the... 

Before studying the hydrogen atom, we shall consider the more general problem of a single particle moving under a central force. The results of this section will apply to any central-force problem, for example, the hydrogen atom (Section 6.5), the isotropic three-dimensional harmonic oscillator (Problem 6.1). 

For the ground state of the one-dimensional harmonic oscillator, compute the standard deviations Ax and IXpy and check that the uncertainty principle is obeyed. Use the results of Prob. 4.9 to save time. 

Use the generalized Helhnann-Feynman theorem to find pD for the one-dimensional harmonic-oscillator stationary states. Check that the result obtained agrees with the virial theorem. 

Which one of the following statements conflicts with the quantum mechanical results for a one-dimensional harmonic oscillator ... 

There is great similarity between the mathematical techniques used in solving the R and 0 equations and those used to solve the one-dimensional harmonic oscillator problem of Chapter 3. Hence, we will only suimnarize the steps involved in these solutions and make a few remarks about the results. More detailed treatments are presented in many texts. ... 

Besides the mechanical renormalization considered above, which leads to an increase in the adsorbate vibrational frequency, there may be also a frequency shift due to the interaction of the molecular dipole with its image in the substrate. Such a shift has the same origin as that discussed for the electronic states in physisorption (Section 2.2.1). Applying that theory to the model of a one-dimensional harmonic oscillator, one finds that the polarizability a (Eq. (2.102)) is independent of the state a), and thus the van der Waals shift (Eq. (2.100)) is the same for each state. As a result, the fre-... 

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. 

The wavefunction of the parent molecule in the electronic ground state is assumed to be a product of two harmonic oscillator wavefunctions with m and n quanta of excitation along R and r, respectively. In Figure 13.2(b) only the vibrational mode of BC is excited, n = 3, while the dissociation mode is in its lowest state, m = 0. The corresponding spectrum is smooth without any reflection structures. Conversely, the wavefunction in Figure 13.2(a) shows excitation in the dissociation mode, m = 3, while the vibrational mode of BC is unexcited. The resulting spectrum displays very clear reflection structures in the same way as in the one-dimensional case. Thus, we conclude that, in general ... 

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