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European Journal of Physics

Particle tunnelling between two boxes joined with a long thin tube

Milan Batista, Mitja Lakner, J. Peternelj

Published Nov 28, 2003• 3 citations
Publisher

Abstract

The ground state of a particle confined to two two-dimensional square boxes connected with a narrow channel is calculated. It is shown that the presence of the channel results in the splitting of the otherwise degenerate ground state. When the width of the channel is small compared to the wavelength of the particle, the wavefunction in the channel is exponentially suppressed, very much like in the case of tunnelling through a potential barrier. Moreover, in the limit of long and narrow channels, any small asymmetry in the shape of the boxes results in the localization of the particle.

DOI
10.1088/0143-0807/25/2/002

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The ground state of a particle confined to two two-dimensional square boxes connected with a narrow channel is calculated. It is shown that the presence of the channel results in the splitting of the otherwise degenerate ground state. When the width of the channel is small compared to the wavelength of the particle, the wavefunction in the channel is exponentially suppressed, very much like in the case of tunnelling through a potential barrier. Moreover, in the limit of long and narrow channels, any small asymmetry in the shape of the boxes results in the localization of the particle.
We discuss two different approaches for splitting the wave function of a single-particle box (SPB) into two equal parts. Adiabatic insertion of a barrier in the center of a SPB in order to make two compartments which each have probability 1/2 of finding the particle in it is one of the key steps for a Szilard engine. However, any asymmetry between the volume of the compartments due to an off-center insertion of the barrier results in a particle that is fully localized in the larger compartment, in the adiabatic limit. We show that rather than exactly splitting the eigenfunctions in half by a symmetric barrier, one can use a nonadiabatic insertion of an asymmetric barrier to induce excitations to only the first excited state of the full box. As the barrier strength goes to infinity the excited state of the full box becomes the ground state of one of the new boxes. Thus, we can achieve close to exact splitting of the probability between the two compartments using the more realistic nonadiabatic, not perfectly centered barrier, rather than the idealized adiabatic and central barrier normally assumed.
An in depth look to the quantum behavior of a particle in a unidimensional finite, square box of potential energy is presented.Wave function normalization constants are fully determined and are used to establish an expression to calculate the particle allowed energy eigenstates based in the confinement condition for each of the two possible wave function symmetries.These eigenstates turn out to be degenerate and it becomes possible to configure confined systems so that the particle confinement/near free duality condition, also known as entanglement is present.This condition, obtained by using the particle mean presence probability density function of the two wave function symmetries of the system degenerate states, is illustrated with a couple of calculation examples.
Highly accurate closed-form approximations are given for the ground state and first excited state wavefunctions and energies for a nonrelativistic particle in a one-dimensional double square well potential with a square barrier in between (so that the potential is a sequence of five constant pieces that alternate in value from being above and below the ground state and first excited state energies), under the assumption that the barrier is sufficiently wide that the tunneling across it is very small.
We investigate the problem of a quantum particle confined in a finite one-dimensional square well. The eigenvalues and eigenfunctions are found using a computer algebra system to solve the equations that result from matching the logarithmic derivatives at the well boundaries. The wavefunctions are used to investigate the relation of well depth to tunneling and to the Heisenberg uncertainty relations. We demonstrate how dimensional analysis can be used to locate the minimum uncertainty for a given confining potential.
The usual WKB analysis for quantum tunneling applies when the tunneling action is large, as it is for tall, wide potential barriers. In contrast we analyze tunneling when the action is small, as it is for tunneling across a tall, thin barrier. We develop a perturbative analysis where the control parameter is the inverse of the area under the potential barrier and apply our technique to several examples in 1+11+11+1 dimensions. In resonant situations for bound particles we find that the tunneling probability grows with time as ∝t2\propto t^2∝t2, while in non-resonant situations it grows linearly with time. We evaluate not only the tunneling probability but also the time-dependent tunneling wavefunction for a particle that escapes to infinity, {\it i.e.} from a quasi-bound state to the continuum.
Spatial confinement of quantum systems has been used to model atoms under high pressure, quantum dots, and Bose–Einstein condensates. Spherical boxes are commonly employed, yet they are energy-minimizing, and other box geometries will be more relevant for some applications. We study generic single-particle quantum systems comprising a central physical potential confined within a three-dimensional infinite potential well corresponding to an impenetrable box having prescribed geometric properties; we refer to this as geometric confinement. There is a marked distortion of the wavefunction and energy spectrum, relative to the standard unconfined system, when a confining box is made small enough. We employ asymptotic analysis in combination with numerical simulations to better understand this behaviour, determining lengthscales on which each of the physical potential and confining box respectively dominate the structure of quantum states, with Padé approximants employed to connect asymptotic scalings between these two regimes. This combination of methods permits us to determine, in a fairly general manner, how and to what extent box size and shape modify a given single-particle quantum system held under geometric confinement. We provide a variety of examples of single-particle central potentials and confining geometries to illustrate our methods and motivate the theoretical findings.