Quartic oscillator

We briefiy review results we have obtained on model potentials with the VQRS reference system. The results obtained with the diagonal approximation to the propagator are superior to any previous such approximations that we are aware of. In table 3 classical and quantum results are presented for various moments of the quartic oscillator ... 

Vibrational Spectrum of Quartic Oscillator by Moments Method... 

We consider the same reaction model used in previous studies as a simple model for a proton transfer reaction. [31,57,79] The subsystem consists of a two-level quantum system bilinearly coupled to a quartic oscillator and the bath consists of v — 1 = 300 harmonic oscillators bilinearly coupled to the non-linear oscillator but not directly to the two-level quantum system. In the subsystem representation, the partially Wigner transformed Hamiltonian for this system is,... 

Given the important role of Arnold diffusion in understanding chaotic transport in many-dimensional systems, it is quite surprising that a smdy of the quantization effect on Arnold diffusion was not carried out until very recently [94-96]. In particular, Izrailev and co-workers are the first to carefully examine quantum manifestations of Arnold diffusion in a well-studied model system. The model system is comprised of two coupled quartic oscillators, one of which driven by a two-frequency field. Its Hamiltonian is given by... 

A general comment was offered by fung on studies of the O-H and O-D vibrations of some hydrogen-bonded systems. It was found that die double-minimum asymmetric quartic oscillator potential seemed to work well. The results were not published because of some uncertainties in the assignments. 

A trial function -"x2/2 used to calculate the ground state energy of a quartic oscillator whose potential is given by V (x) = Cx4... 

A one-dimensional quartic oscillator has V = ex, where c is a constant. Devise a variation function with a parameter for this problem, and find the optimum value of the parameter... 

Revise your solution to Problem 8.52 to apply the pib basis functions to the onedimensional quartic oscillator with V = cx. See Problem 8.55 for hints. Take the box to extend from x, = —3.5 to 3.5, where x, is as found in Problem 4.32. Increase the number of pib basis functions until the lowest three energy values remain stable to three decimal places Compare the lowest three energies with those found by the Numerov method in Problem 452. Check the appearance of the lowest three variational functions. Now repeat for the box going from X, = -4.5 to 4.5. For which box length do we get faster convergence to the true energies ... 

Thus, the solutions are analogous to the case of a quartic oscillator of strength —E, which can be either positive or negative, with a small attractive... 

Problem Use the functional form of the ground state wavefunction of the harmonic oscillator as a trial wavefunction and apply variation theory to find an approximate wavefunction and energy for the quartic oscillator, an oscillator with V x) = kx. ... 

Apply the variational method to the quartic oscillator problem in Example 8.4 using the trial wavefunction in Exercise 8.28. 

The performance of the prefactor-free FBSD expressions. Equations (6) or (15), is clearly illustrated in several numerical examples. The first involves the average position in a model of an initially displaced quartic oscillator described by the Hamiltonian (41)... 

Figure L Average position for a one-dimensional quartic oscillator. Solid line exact results obtained via the split propagator method. Solid circles FBSD, Hollow squares full semiclassical calculation.

Figure 2. Average position of a quartic oscillator coupled to a bath of 30 harmonic degrees of freedom at zero temperature. The markers show the results of FBSD. Filled circles = 0.25. Hollow circles = 0.50. The lines show the...

Figure 3. Real and imaginary parts of the position correlation function for the quartic oscillator described in this section at a high temperature, h(op-ypil Q, Solid lines exact quantum mechanical results. Markers FBSD-path integral results with iV = 1 and 10,000 Monte Carlo points per integration variable. Dashed lines classical results.

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