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International Journal of Modern Physics B

QUANTUM EFFECTS OF A PARTICLE IN AN INFINITE SQUARE WELL WITH A MOVING WALL

Jian Zou, Bin Shao

Published Apr 20, 2000• 2 citations
Publisher

Abstract

The quantum behavior of a particle in a one-dimensional infinite square well potential with a moving wall is studied. The particle is assumed to be initially prepared in the coherent state (Gaussian wave packet) and although the boundary is far from the particle, it is shown that the changing of the boundary conditions can instantaneously affect the dynamical behavior of the particle. It is also shown that the initial state can evolve into a squeezed state, and in some cases the spreading of the wavepacket could be suppressed. Finally the Pancharatnam phase is also discussed.

DOI
10.1142/s0217979200001308

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The quantum behavior of a particle in a one-dimensional infinite square well potential with a moving wall is studied. The particle is assumed to be initially prepared in the coherent state (Gaussian wave packet) and although the boundary is far from the particle, it is shown that the changing of the boundary conditions can instantaneously affect the dynamical behavior of the particle. It is also shown that the initial state can evolve into a squeezed state, and in some cases the spreading of the wavepacket could be suppressed. Finally the Pancharatnam phase is also discussed.
Abstract We investigate the issue of single particle nonlocality in a quantum system subjected to time-dependent boundary conditions. We discuss earlier claims according to which the quantum state of a particle remaining localized at the center of an infinite well with moving walls would be specifically modified by the change in boundary conditions due to the wall’s motion. We first prove that the evolution of an initially localized Gaussian state is not affected nonlocally by a linearly moving wall: as long as the quantum state has negligible amplitude near the wall, the boundary motion has no effect. This result is further extended to related confined time-dependent oscillators in which the boundary’s motion is known to give rise to geometric phases: for a Gaussian state remaining localized far from the boundaries, the effect of the geometric phases is washed out and the particle dynamics shows no traces of a nonlocal influence that would be induced by the moving boundaries.
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.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.
As a useful tool to manipulate the feature of quantum systems, periodic drive has attracted widespread attention. Various exotic phases and rich dynamics have been found in the periodically driven quantum systems, such as adiabatic evolution and time crystals. Displaying these strange properties in a simple physical system can not only deepen the understanding of quantum mechanics, but also cultivate students' interest in scientific research. Here, we are interested in a specific case that the well-geometry changing is caused by boundary contraction or expansion. The simplest one-dimensional square well with periodically moving boundary is examined. We explore the influence of the frequency and the intensity of the drive on the time evolution of the system. By numerical simulations, we calculate the evolution of the ground state occupation taking two cases under different boundary parameters and moving paths in both adiabatic and non adiabatic cases. By solving the expected value and deviation value of the energy of the system, the influence of the vibration frequency and intensity of the boundary on the probability of the particles at different energy levels is given. The results show that the expansion and contraction of the potential well with the same velocity have different effects on the time evolution of the particles in the well; Different well wall periodic motion modes have different effects on the time evolution of particles. The results obtained by the numerical method are also explained in this paper.