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Quantum Force due to Distinct Boundary Conditions

Tamás Fülöp, Hitoshi Miyazaki, Izumi Tsutsui

Published Jan 1, 2003• 3 citations
OpenAlex

Abstract

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.

Related papers

QUANTUM FORCE DUE TO DISTINCT BOUNDARY CONDITIONS

TAMÁS FÜLÖP, HITOSHI MIYAZAKI, Izumi Tsutsui

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.

arXiv:quant-ph/0309095•Modern Physics Letters A•Dec 28, 2003•4 citations

Boundary effect of a partition in a quantum well

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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.

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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.

•Jan 1, 2009•4 citations

Quantum force induced on a partition wall in a harmonic potential

Tamás Fülöp, I Tsutsui

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.

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Quantum effective force in an expanding infinite square-well potential and Bohmian perspective

S V Mousavi

The Schrödinger equation is solved for the case of a particle confined to a small region of a box with infinite walls. If walls of the well are moved, then, due to an effective quantum nonlocal interaction with the boundary, even though the particle is nowhere near the walls, it will be affected. It is shown that this force apart from a minus sign is equal to the expectation value of the gradient of the quantum potential for vanishing at the walls boundary condition. Variation of this force with time is studied. A selection of Bohmian trajectories of the confined particle is also computed.

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Quantum effective force and Bohmian approach to time-dependent traps

S V Mousavi

Trajectories of a Bohmian particle confined in time-dependent cylindrical and spherical traps are computed for both contracting and expanding boxes. Quantum effective force is considered in arbitrary directions. It is seen that in contrast to the problem of a particle in an infinite rectangular box with one wall in motion, if particle initially is in an energy eigenstate of a tiny box the force is zero in all directions. Trajectories of a two-body system confined in the spherical trap are also computed for different statistics. Computations show that there are situations for which distance between bosons is greater than the fermions. However, results of average separation of the particles confirm our expectation about the statistics.

arXiv:1309.0993•Physica Scripta•Apr 23, 2014•5 citations
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.
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.
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.
The Schrödinger equation is solved for the case of a particle confined to a small region of a box with infinite walls. If walls of the well are moved, then, due to an effective quantum nonlocal interaction with the boundary, even though the particle is nowhere near the walls, it will be affected. It is shown that this force apart from a minus sign is equal to the expectation value of the gradient of the quantum potential for vanishing at the walls boundary condition. Variation of this force with time is studied. A selection of Bohmian trajectories of the confined particle is also computed.
Trajectories of a Bohmian particle confined in time-dependent cylindrical and spherical traps are computed for both contracting and expanding boxes. Quantum effective force is considered in arbitrary directions. It is seen that in contrast to the problem of a particle in an infinite rectangular box with one wall in motion, if particle initially is in an energy eigenstate of a tiny box the force is zero in all directions. Trajectories of a two-body system confined in the spherical trap are also computed for different statistics. Computations show that there are situations for which distance between bosons is greater than the fermions. However, results of average separation of the particles confirm our expectation about the statistics.