Double-well Oscillator Research Articles

Analytical results for the classical and semiclassical coordinate probability distribution function of anharmonic oscillators

The classical and Wigner—Kirkwood semiclassical distribution functions for single and double-well anharmonic oscillators with Ĥ = p̂ 2/2 m + k 2 x̂ 2 + k 2n x̂ 2 n are given in closed form. Applications in the theory of electron diffraction are discussed briefly.

Analytical results for the classical and semiclassical coordinate probability distribution function of anharmonic oscillators

The classical and Wigner—Kirkwood semiclassical distribution functions for single and double-well anharmonic oscillators with Ĥ = p̂ 2 /2 m + k 2 x̂ 2 + k 2n x̂ 2 n are given in closed form. Applications in the theory of electron diffraction are discussed briefly.

Exact results for the statistical thermodynamics of the asymmetric single- and double-well anharmonic oscillator

Exact analytical results are presented for the high-temperature behaviour of the anharmonic oscillator with Ĥ = p̂ 2/2m + k 2x̂ 2 + k 3^x 3 + k 4x̂ 4, k 4 > 0. The classical partition function and the exact Golden-Thompson upper bound with and without variation of the harmonic frequency are calculated for both single- and double-well oscillators.

Complex saddle points versus dilute-gas approximation in the double-well anharmonic oscillator

The energy shift between the two lowest eigenstates in the left- and right-hand wells of the double potential is computed by evaluating functional integrals using the saddle point method. The complex saddle points are considered on the same footing as the real ones. They happen to be a perfect substitute for the dilute gas.

Nonlinear contributions to the partition function of a double-well oscillator

Nonlinear contributions to the partition function of a double-well oscillator

Effect of fermions upon tunneling in a one-dimensional system

We study the effects of a fermion with Yukawa-type coupling upon the tunneling of the double-well anharmonic oscillator. These effects prove to be nondramatic, despite the zero-frequency bound-state eigenmode one would encounter if one applied the conventional boundary conditions to the path integral.

Pseudoparticle contributions to the energy spectrum of a one-dimensional system

We show that classical solutions of the Euclidean action can be used to calculate the shift in energy levels due to tunneling through a potential barrier. In particular, we use the path integral to compute the kernel of the double-well anharmonic oscillator for a large, but finite, Euclidean time interval by expanding about pseudoparticle solutions (i.e., the kink). This allows us to determine the ground-state energy plus that of the first excited state (the splitting is due to barrier penetration). We find that not only the classical solution must be expanded about, but also nearly stationary trajectories corresponding to kink plus kink-antikink pairs. The quasitranslational invariance must also be dealt with carefully. We compare with the WKB result and find our result more accurate, because it avoids the errors introduced by the linear (Airy functions) connecting formulas.

Functional integral approach to quantum double-well oscillator

The functional integral method, as developed by Wang, Evenson and Schrieffer and by Hamann is applied to calculate the partition function for a double well oscillator, which is a simple model for a ferroelectric crystal. The free energy and the polarizability of the system are evaluated using the static approximation and the RPA′. The results in the former case resemble those of the classical solution whereas those of the latter agree fairly well with the exact results from high temperatures down to the ground state energy.