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Register a new userInstantonic approach to triple well potential
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By using a usual instanton method we obtain the energy splitting due to quantum tunneling through the triple well barrier. It is shown that the term related to the midpoint of the energy splitting in propagator is quite different from that of double well case, in that it is proportional to the algebraic average of the frequencies of the left and central wells.
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In this paper, we introduce a family of sextic potentials that are exactly solvable, and for the first time, a family of triple-well potentials with their whole energy spectrum and wavefunctions using supersymmetry method. It was suggested since three decades ago that all additive or translational shape invariant superpotentials formed by two combination of functions have been found and their list was already exhausted by the well-known exactly solvable potentials that are available in most textbooks and furthermore, there are no others. We have devised a new family of superpotentials formed by a linear combination of three functions (two monomials and one rational) and where the change of parameter function is linear in four parameters. This new family of potentials with superpotential $W(x,A,B,D,G) = Ax^3 + Bx -frac{Dx}{1+Gx^2}$ will extend the list of exactly solvable Schrodinger equations. We have shown that the energy of the bound states is rational in the quantum number. Furthermore, approximating the potential around the central well by a harmonic oscillator, as a usual practice, is not valid. The two outer wells affect noticeably the probability density distribution of the excited states. We have noticed that the populations of the triple-well potentials are localized in the two outer wells. These results have potential applications to explore more physical phenomena such as tunneling effect, and instantons dynamics.
We investigate the effect of anharmonicity on the WKB approximation in a double well potential. By incorporating the anharmonic perturbation into the WKB energy splitting formula we show that the WKB approximation can be greatly improved in the region over which the tunneling is appreciable. We also observe that the usual WKB results can be obtained from our formalism as a limiting case in which the two potential minima are far apart.
We consider an Ehrenfest approximation for a particle in a double-well potential in the presence of an external environment schematized as a finite resource heat bath. This allows us to explore how the limitations in the applicability of Ehrenfest dynamics to nonlinear systems are modified in an open system setting. Within this framework, we have identified an environment-induced spontaneous symmetry breaking mechanism, and we argue that the Ehrenfest approximation becomes increasingly valid in the limit of strong coupling to the external reservoir, either in the form of increasing number of oscillators or increasing temperature. The analysis also suggests a rather intuitive picture for the general phenomenon of quantum tunneling and its interplay with classical thermal activation processes, which may be of relevance in physical chemistry, ultracold atom physics, and fast-switching dynamics such as in superconducting digital electronics.
We propose a new way to implement an inflationary prior to a cosmological dataset that incorporates the inflationary observables at arbitrary order. This approach employs an exponential form for the Hubble parameter $H(phi)$ without taking the slow-roll approximation. At lowest non-trivial order, this $H(phi)$ has the unique property that it is the solution to the brachistochrone problem for inflation.
We investigate the dynamical properties for non-Hermitian triple-well system with a loss in the middle well. When chemical potentials in two end wells are uniform and nonlinear interactions are neglected, there always exists a dark state, whose eigenenergy becomes zero, and the projections onto which do not change over time and the loss factor. The increasing of loss factor only makes the damping form from the oscillating decay to over-damping decay. However, when the nonlinear interaction is introduced, even interactions in the two end wells are also uniform, the projection of the dark state will be obviously diminished. Simultaneously the increasing of loss factor will also aggravate the loss. In this process the interaction in the middle well plays no role. When two chemical potentials or interactions in two end wells are not uniform all disappear with time. In addition, when we extend the triple-well system to a general (2n + 1)-well, the loss is reduced greatly by the factor 1=2n in the absence of the nonlinear interaction.
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