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arXiv:quant-ph/0612011 · Journal of Physics A Mathematical and Theoretical

Boundary effect of a partition in a quantum well

Tamás Fülöp, Izumi Tsutsui

Published Apr 11, 2007• 7 citations
arXivHTMLPDF

Abstract

The paper wishes to demonstrate that, in quantum systems with boundaries, different boundary conditions can lead to remarkably different physical behaviour. Our seemingly innocent setting is a one-dimensional potential well that is divided into two halves by a thin separating wall. The two half wells are populated by the same type and number of particles and are kept at the same temperature. The only difference is in the boundary condition imposed at the two sides of the separating wall, which is the Dirichlet condition from the left and the Neumann condition from the right. The resulting different energy spectra cause a difference in the quantum statistically emerging pressure on the two sides. The net force acting on the separating wall proves to be nonzero at any temperature and, after a weak decrease in the low-temperature domain, to increase and diverge with a square-root-of-temperature asymptotics for high temperatures. These observations hold for both bosonic and fermionic type particles, but with quantitative differences. We work out several analytic approximations to explain these differences and the various aspects of the found unexpectedly complex picture.

DOI
10.1088/1751-8113/40/17/013

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The paper wishes to demonstrate that, in quantum systems with boundaries, different boundary conditions can lead to remarkably different physical behaviour. Our seemingly innocent setting is a one-dimensional potential well that is divided into two halves by a thin separating wall. The two half wells are populated by the same type and number of particles and are kept at the same temperature. The only difference is in the boundary condition imposed at the two sides of the separating wall, which is the Dirichlet condition from the left and the Neumann condition from the right. The resulting different energy spectra cause a difference in the quantum statistically emerging pressure on the two sides. The net force acting on the separating wall proves to be nonzero at any temperature and, after a weak decrease in the low-temperature domain, to increase and diverge with a square-root-of-temperature asymptotics for high temperatures. These observations hold for both bosonic and fermionic type particles, but with quantitative differences. We work out several analytic approximations to explain these differences and the various aspects of the found unexpectedly complex picture.
Abstract. We calculate the quantum statistical force acting on a partition wall that divides a one dimensional box into two halves. The two half boxes contain the same (fixed) number of noninteracting bosons, are kept at the same temperature, and admit the same boundary conditions at the outer walls; the only difference is the distinct boundary conditions imposed at the two sides of the partition wall. The net force acting on the partition wall is nonzero at zero temperature and remains almost constant for low temperatures. As the temperature increases, the force starts to decrease considerably, but after reaching a minimum it starts to increase, and tends to infinity with a square-root-of-temperature asympotics. This example demonstrates clearly that distinct boundary conditions cause remarkable physical effects for quantum systems.
We calculate the quantum statistical force acting on a partition wall that divides a one-dimensional box into two halves. The two half boxes containing the same (fixed) number of noninteracting bosons are kept at the same temperature, and admit the same boundary conditions at the outer walls; the only difference is the distinct boundary conditions imposed at the two sides of the partition wall. The net force acting on the partition wall is nonzero at zero temperature and remains almost constant for low temperatures. As the temperature increases, the force starts to decrease considerably, but after reaching a minimum it starts to increase, and tends to infinity with a square-root-of-temperature asympotics. This example demonstrates clearly that distinct boundary conditions cause remarkable physical effects for quantum systems.
Boundary effects in quantum mechanics are examined by considering a partition wall inserted at the centre of a harmonic oscillator system. We put an equal number of particles on both sides of the impenetrable wall keeping the system under finite temperatures. When the wall admits distinct boundary conditions on the two sides, then a net force is induced on the wall. We study the temperature behaviour of the induced force both analytically and numerically under the combination of the Dirichlet and the Neumann conditions, and determine its scaling property for two statistical cases of the particles: fermions and bosons. We find that the force has a nonvanishing limit at zero temperature T = 0 and exhibits scalings characteristic to the statistics of the particles. We also see that for higher temperatures the force decreases according to 1/sqrt{T}, in sharp contrast to the case of the infinite potential well where it diverges according to sqrt{T}. The results suggest that, if such a nontrivial partition wall can be realized, it may be used as a probe to examine the profile of the potentials and the statistics of the particles involved.
Boundary effects in quantum mechanics are examined by considering a partition wall inserted at the centre of a harmonic oscillator system. We put an equal number of particles on both sides of the impenetrable wall keeping the system under finite temperatures. When the wall admits distinct boundary conditions on the two sides, then a net force is induced on the wall. We study the temperature behaviour of the induced force both analytically and numerically under the combination of the Dirichlet and the Neumann conditions, and determine its scaling property for two statistical cases of the particles: fermions and bosons. We find that the force has a nonvanishing limit at zero temperature T = 0 and exhibits scalings characteristic to the statistics of the particles. We also see that for higher temperatures the force decreases according to 1/sqrt{T}, in sharp contrast to the case of the infinite potential well where it diverges according to sqrt{T}. The results suggest that, if such a nontrivial partition wall can be realized, it may be used as a probe to examine the profile of the potentials and the statistics of the particles involved.
In this paper, we show that there exists an essential difference of boundary effects between Bose and Fermi systems both for Dirichlet and Neumann boundary conditions: at low temperatures and high densities the influence of the boundary on the Bose system depends on the temperature but is independent of the density, but for the Fermi case the influence of the boundary is independent of the temperature but depends on the density, after omitting the negligible high-order corrections. We also show that at high temperatures and low densities the difference of the influence of the boundary between Bose and Fermi systems appears in the next-to-leading order boundary contribution, and the leading boundary contribution is independent of the density. Moreover, for calculating the boundary effects at high temperatures and low densities, since the existence of the boundary modification causes the standard virial expansion to be invalid, we introduce a modified virial expansion. PACS numbers: 05.30.-d, 05.70.Ce, 65.80.+n, 82.60.Qr 1
The recent observation by Hasan and Goble of a discrepancy between their numerical results on the specific heat of an ideal Bose gas confined to a thin-film geometry and the earlier analytic calculations of Pathria employing Neumann boundary conditions is shown to be unjustified. In those cases where a true comparison is possible, e.g., under Dirichlet or periodic boundary conditions, the observed deviations turn out to be insignificant if one uses the more recent second-order calculations of Greenspoon and Pathria rather than the original first-order ones of Pathria. For comparison under Neumann boundary conditions, fresh numerical calculations are required. We also present here second-order analytic calculations of the specific heat of this system (i) under mixed boundary conditions (Dirichlet on one wall, Neumann on the other) and (ii) under antiperiodic boundary conditions, and compare them with the relevant numerical calculations of Hasan and Goble.