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TIME EVOLUTION OF A PARTICLE IN A ONE-DIMENSIONAL INFINITE SQUARE WELL POTENTIAL WITH MOVING BOUNDARY

Zhuocheng GU, Haodi LIU, X. X. Yi

Published Feb 1, 2023• 0 citations

Abstract

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.

DOI: 10.26599/phys.2023.9320105

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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.

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Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases, non-equilibrium phase transition, and Majorana mode. Recently, the study of effective Hamiltonian using various methods has gained great interest. We consider a vector differential equation of motion to derive the effective Hamiltonian for any periodically driven two-level system, and the dynamics of the spin vector are an evolution under the Bloch sphere. Here, we investigate the properties of this equation and show that a sudden change of the effective Hamiltonian is expected. Furthermore, we present several exact relations, whose expressions are independent of the different starting points. Moreover, we deduce the effective Hamiltonian from the high-frequency limit, which approximately equals the results in previous studies. Our results show that the vector differential equation of motion is not affected by a convergence problem, and thus, can be used to numerically investigate the effective models in any periodic modulating system. Finally, we anticipate that the proposed method can be applied to experimental platforms that require time-periodic modulation, such as ultracold atoms and optical lattices.

Under a periodically driven external field or a temporally oscillating boundary, quantum few-body systems whose corresponding classical systems show chaos are modeled by the random matrices with an oscillating parameter. In such systems, time evolution of the system from the ground state shows the quantum diffusion. However, due to periodic oscillation of the parameter, quantum interference effect prevents the energy of the system from diverging and the energy saturates after some transient time. We show an example where the energy eventually fluctuates around a constant saturated energy. Since the value of this saturated energy depends on the frequency of the driving, we also investigate the dependence of the saturated energy on the frequency in the small frequency region.