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European Journal of Physics

Capture of a quantum particle by a deep accelerating potential well

Mark R. A. Shegelski, Jamie Sanchez-Fortún Stoker, Tyler Poole

Published Jan 19, 2015• 0 citations
Publisher

Abstract

Abstract We investigate the capture of a quantum particle in free space by a deep, accelerating potential well. The dependence of the probability of capture on the depth of the well is investigated. We take the potential well to move with an initial speed, v 0 , and to decelerate with constant acceleration, a c , until the well stops moving. We obtain and explain some unexpected results. Our work may be of interest to instructors of and students in upper-year undergraduate quantum mechanics courses.

DOI
10.1088/0143-0807/36/2/025007

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Abstract We investigate the capture of a quantum particle in free space by a deep, accelerating potential well. The dependence of the probability of capture on the depth of the well is investigated. We take the potential well to move with an initial speed, v 0 , and to decelerate with constant acceleration, a c , until the well stops moving. We obtain and explain some unexpected results. Our work may be of interest to instructors of and students in upper-year undergraduate quantum mechanics courses.
We investigate the capture of a quantum particle in free space by a moving trapping potential. The capture is investigated for various initial conditions. We examine the dependence of the probability of capture on the shape and depth of the trapping potential, initial speed and mass of the particle, and other parameters. We take the trapping potential to move with an initial speed v 0 and to decelerate with constant acceleration a c until the well stops moving. We study the motion during the time the well is moving and after the well has stopped. We compare the probability of capture to the probability that the particle is in a stationary state (while the well is moving) and the probability the particle is in a bound state (after the well's motion has stopped). Our work could be of interest to instructors and students in upper-year undergraduate quantum mechanics courses.
Quantum mechanical control of the position of a particle by using a trapping potential well is an important problem for the manipulation of a quantum particle. We study the probability of successful conveyance of the particle trapped in a potential well for a given length within a given fixed time, i.e., survival probability after the motion. For the actual motion of conveyance, we need to accelerate the particle to move and then decelerate it to stop at the destination. Acceleration and deceleration cause a dropoff of a particle from the trapping potential well. The relaxation of the survival probability in a process with a constant acceleration rate is studied in detail. First, the process is studied as the relaxation of the survival probability of the trapped particle by direct numerical calculations. The survival probability obeys an exponential decay in a long time, which is analyzed from a viewpoint of eigenvalue problem. The value of survival probability is also estimated by the Wentzel-Kramers-Brillouin method with connection formulas using the Airy function and the Weber function. The value is further estimated by a method of the resonance states. We emphasize the fact that an important source of dropoff comes from a nonanalytic change of velocity at the starting point. When the rested particle begins to move, the ground state of the rest frame is redistributed to eigenstates of the moving frame, and then each eigenstate of the moving frame evolves in time. The dephasing of wave functions of the distributed populations reduces the probability of successful conveyance. In general, a smooth start gives a small initial disturbance but it requires a large acceleration during the process to reach the destination in the fixed time which causes a larger dropoff in the process. Considering these conflicting facts, we study the survival probability in concrete conveyance schemes, i.e., protocols with (1) a sudden change of velocity to a constant velocity, (2) a sudden change of acceleration rate to a constant acceleration, and (3) a smooth change of acceleration by studying the real-time change of populations of the adiabatic (instantaneous) eigenstates. We observe the time evolution of the trapped probability and the population distribution during the conveyance process. In cases that the potential well has several bound states, we propose a method to select the particle trapped at the ground state by making use of the difference of survival probabilities of bound states. Published by the American Physical Society 2024
The results of analytical and numerical investigation of quantum dissipative dynamics a particle moving in a potential well with two equilibrium states is presented. A classical analogue of this problem is the capture into the resonance a particle in a double-well potential, when two states of equilibrium in the phase space are separated by a separatrix. The probability of finding the system in one of states of equilibrium for the quantum particle, depending on initial conditions and noise, is discussed.
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.
We describe the dynamics of a bound state of an attractive δ-well under displacement of the potential. Exact analytical results are presented for the suddenly moved potential. Since this is a quantum system, only a fraction of the initially confined wave function remains confined to the moving potential. However, it is shown that besides the probability to remain confined to the moving barrier and the probability to remain in the initial position, there is also a certain probability for the particle to move at double speed. A quasi-classical interpretation for this effect is suggested. The temporal and spectral dynamics of each one of the scenarios is investigated.