The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.
This chapter considers a particle with mass m and total energy E moving under the influence of the one-dimensional finite square well. An alternative method which turns out to be much more flexible when arrays of two or more wells are considered. The starting point of this alternative method is the boundary condition that the gradient of the wave function must be continuous at both edges of the well, and this implies that the gradient change across the well must be the same whether one uses the internal or external wave functions. Consider the array of three one-dimensional delta wells. This potential configuration can have up to three allowed states, depending on the well separation r. Since the potential is symmetrical about r = 0, the wave functions must have even or odd parity. Two of them have even parity while the third has odd parity.
Methods previously developed for the case of a symmetric double-well potential are here extended to study tunneling in an asymmetric, piecewise harmonic, double-well potential. The approach involves the decomposition of the real stationary states into complex right- and left-moving states and the determination of exact transmission coefficients for the latter. Numerical solutions for a selection of representative parameter values are obtained for the time evolution of states which correspond initially to a Gaussian wave packet localized in the upper well. Both resonant and nonresonant cases are studied and the significance of the transmission coefficient is established for each.
Double-well potentials are widely used to model phenomena in physics and chemistry. The system is assumed to be formed in a metastable state, its dynamical evolution providing the clues for the interpretation of the experimental data. Quantum mechanics predicts coherent oscillations of probability between wells if the double-well potential is nearly symmetric and irreversible exponential decay if the final well has an infinite width. For very asymmetric double-well potentials, these two extreme behaviors are expected to coexist. The purpose of the present paper is to investigate this coexistence and its evolution as a function of the width of (or density of states in) the second well. In this sense, increasing the density of states can be regarded as a mechanism for coherence breakdown. The dynamical evolution of the metastable state can be simulated by solving numerically the time-dependent Schroedinger equation (TDSE). This approach is general, intuitive, and gives access to time scales and dynamical effects. It also allows the inclusion of phenomenological dissipation.
The motion of the center of a wave packet localized in one of the wells of an asymmetric double-well potential is studied. The asymmetric potential is obtained from a solvable symmetric potential using the Gel'fand–Levitan formalism, by keeping the energy eigenvalues unchanged, but changing the normalization of the ground-state wave function. While in a symmetric double well the wave packet tunnels back and forth between the two wells, for an asymmetric well, in general, tunneling does not take place. This is independent of the initial location of the wave packet, i.e., whether it is centered at the minimum of the shallower or of the deeper well. However, when such a system is coupled to a dissipative force, then the tunneling becomes possible. For instance it is shown that if, as a model of dissipative coupling, one chooses Gisin's nonlinear evolution equation, then the center of the wave packet ends up in the deeper well (decay of false vacuum). This result depends on the particular model of dissipation, for instance, an optical potential model yields a different result.
The analytic nature of the transmission coefficient for a δ-function barrier makes it a useful tool to examine a variety of technologically important applications, such as photoemission from semiconductors with an alkali coating, the examination of tunneling times for wave packets incident on a barrier, and for parameterizing tunneling through the narrow barrier of a normal-superconducting point contact. The analytic model of a δ-function barrier inside a confining well is extended to the finite height and width rectangular barrier (a delta-function sequence). Methods to exactly evaluate the eigenstates are given and their dependencies are examined. The time evolution of a superposition of the lowest eigenstates is considered for barriers having comparable Gamow tunneling factors so as to quantify the impact of barrier height and shape on time evolution in a simple and exact system and, therefore, serve as a proxy for tunneling time. Last, density profiles and associated quantum potentials are examined for coupled wells to show changes induced by weaker and wider barriers.
An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of �ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1