The notion of a double well potential typically involves two regions of space separated by a repulsive potential barrier. The ground state is a wave function that is suppressed in the barrier region and localized in the two surrounding regions. We illustrate that an attractive potential well (a quantum moat) with a finite nonzero width also acts as a barrier, using a simple square well model. We also show how the pseudopotential method both explains the role of the well as a barrier, and greatly improves the efficiency of constructing wave functions for this system using matrix diagonalization. With this simplified model we provide an introduction to the ideas typically used to simplify calculations in solids, where in place of the double well potential, multiple potentials occur in a periodic array.
The notion of a double well potential typically involves two regions of space separated by a repulsive potential barrier. The ground state is a wave function that is suppressed in the barrier region and localized in the two surrounding regions. We illustrate that an attractive potential well (a quantum moat) with a finite nonzero width also acts as a barrier, using a simple square well model. We also show how the pseudopotential method both explains the role of the well as a barrier, and greatly improves the efficiency of constructing wave functions for this system using matrix diagonalization. With this simplified model we provide an introduction to the ideas typically used to simplify calculations in solids, where in place of the double well potential, multiple potentials occur in a periodic array.
The solution to a problem in quantum mechanics is generally a linear superposition of states. The solutions for double well potentials epitomize this property, and go even further than this: they can often be described by an effective model whose low energy features can be described by two states—one in which the particle is on one side of the barrier, and a second where the particle is on the other side. Then the ground state remains a linear superposition of these two macroscopic-like states. In this paper, we illustrate that this property is achieved similarly with an attractive potential that separates two regions of space, as opposed to the traditionally repulsive one. In explaining how this comes about we revisit the concept of “orthogonalized plane waves,” first discussed in 1940 to understand electronic band structure in solids, along with the accompanying concept of a pseudopotential. We show how these ideas manifest themselves in a simple double well potential, whose “barrier” consists of a moat instead of the conventional wall.
Chapter 5 familiarizes the student with some of the most basic features of quantum systems, by applying the Schrödinger equation to simple, one-dimensional problems. It shows that it is possible to obtain qualitatively correct conclusions about the behavior of quantum particles in the presence of more or less arbitrary potentials by replacing them with piecewise-constant potentials, which greatly simplifies the mathematics. Replacing the real potential with a simpler one that retains its basic characteristics has the great advantage of allowing us to study the essence of the physical situation, as illustrated by the examples of the one-dimensional potential step, the square well, the square potential barrier and the symmetric double well. These examples are used to discuss specific features such as quantum tunneling, and applications such as masers and molecular clocks.
Double-well potentials are important potential models that underlie the study of macroscopic quantum tunneling in Bose–Einstein Condensates (BECs), which are most commonly constructed in the form of a combination of a finite square-well potential and a square potential barrier. We propose another possibility of constructing a double-well potential, which consists of the combination of a square-well potential and a movable Gaussian-type potential barrier, and by adjusting the width, height, symmetry, and position of the movable potential barrier, symmetric and asymmetric double-well potentials with rich structures can be constructed. By numerically solving the Gross–Pitaevskii equation, we give the ground state structures of the BEC in this double-well potential, and then study the tunneling dynamics in this double-well potential. Different from trivial symmetric barrier displacement, the independently tunable two half-Gaussian widths bring an extra control degree of freedom; it is shown that the ground–state properties and tunneling dynamics are far more sensitive to asymmetric segmented barriers than to symmetrically shifted barriers. This feature offers a flexible strategy for atomic manipulation and is not merely a trivial qualitative observation.
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.
Abstract In this paper, the finite square well and its application are investigated, namely the quantum state discrimination. The finite square well is treated in all standard textbooks on introductory quantum mechanics. It is used as a simple ‘model of departure’ in many areas of physics. In atomic and molecular physics, it may be used as a model of an electron moving in the mean field of a linear molecule It also arises as the partial wave radial equation for a spherically symmetric, finite square-well potential. The Schrodinger equation for finite square well is solved. The domain is divided into three regions by the existing potential V 0, so for convenience, those three regions are named Region I, Region II, and Region III, respectively. Specifically, the potential of Region I and III are V 0, and that of Region II is 0. Also, a constant is needed to make this wave function normalized. Then since it is a wave function, it is necessary to make sure the function is continuous and differentiable everywhere within the domain. Besides, the wave function needs to be either odd or even just like infinite square well. After that, the wave functions for the three intervals can be obtained, and the exact quantum state is distinguished from a group of different quantum states. In the end, two wave functions are obtained; one for even form and one for odd form. Then all the energy level of the corresponding function with different wavelength needs to be found and listed. Local discrimination of orthogonal quantum states has attracted much attention during the last twenty years. The results are applied to the quantum-information task of state discrimination, by using the obtained six states in finite square well. It is assumed that all the quantum states are locally distinguishable, and the six states are distinguished using the hypothesis of quantum measurements.
The Gross-Pitaevskii equation is solved by analytic methods for an external double-well potential that is an infinite square well plus a δ-function central barrier. We find solutions that have the symmetry of the non-interacting Hamiltonian as well as asymmetric solutions that bifurcate from the symmetric solutions for attractive interactions and from the antisymmetric solutions for repulsive interactions. We present a variational approximation to the asymmetric state as well as an approximate numerical approach. Stability of the states is briefly considered.