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arXiv:1111.3962 · Physica Scripta

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

S V Mousavi

Published Aug 24, 2012• 4 citations
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Abstract

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.

DOI
10.1088/0031-8949/86/03/035004

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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.
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. Results of average separation of the particles confirm our expectation about the statistics.
The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.
In this paper we have solved analytically the Schrodinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. Based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square of the linear momentum are equal to the classical time average of particle position and its square of the linear momentum, respectively.
The effect of a δδδ distribution potential on a quantum mechanical particle in a box
We study the effect of a δδδ distribution potential placed at x0≥0x_0\geq 0x0​≥0 and multiplied by a parameter ααα on a quantum mechanical particle in an infinite square well over the segment [− L2,L2]\left[-\,\frac{L}{2},\frac{L}{2}\right][−2L​,2L​]. We obtain the limit of the eigenfunctions of the time independent Schrödinger equation as α↗+∞α\nearrow+\inftyα↗+∞ and as α↘−∞α\searrow-\inftyα↘−∞. We see how each solution of the Schrödinger equation corresponding to α=0α=0α=0 changes as ααα runs through the real line. When x0x_0x0​ is a rational multiple of LLL, there exist solutions of the Schrödinger equation which vanish at x0x_0x0​ and are unaffected by the value of ααα. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking ∣α∣→∞|α|\to\infty∣α∣→∞. The expectation value of the position of a particle with wave function equal to the limiting eigenfunction is x0x_0x0​.
Ehrenfest’s theorem states that as a quantum state evolves in time, the rate of change of the expectation value of momentum is equal to the expectation value of the force. In the familiar ‘‘particle-in-a-box,’’ however, the probability of finding the particle at a wall—the only place where forces act—is zero, so at first glance it appears that the expectation value of the force should vanish for any state, which would violate Ehrenfest‘s theorem. This argument is flawed, however, since the forces at the walls of the box are infinite. We consider the box as a limit of a very deep square well and confirm that Ehrenfest’s theorem emerges safe and sound.