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Computers & Mathematics with Applications

The quantum square well with moving boundaries: A numerical analysis

O A Fojón, Manuel Gadella, Luis Pedro Lara

Published Dec 17, 2009• 20 citations
Publisher
DOI
10.1016/j.camwa.2009.09.011

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In this paper a systematic formalism for dealing with non-relativistic time-dependent quantum Hamiltonians is presented. The starting point is the well-known Lewis and Riesenfeld idea which involves the construction of an invariant operator I ( x , t ) which defines both the dynamics of the physical system and the canonical formalism that has to be used in order to obtain a consistent theoretical framework. In order to exhibit the full power of the formalism we discuss two examples: the generalized harmonic oscillator and the infinite square well with a moving boundary. As the first example has already been analyzed by the present authors from other different points of view, we are able to compare the results of the canonical formalism with these other approaches and, as it was expected, we obtain identical descriptions of this physical system. After this, we turn to the case of the square well with a moving boundary. The main surprise is that in order to obtain consistency with the formalism an effective interaction appears which seems to be due to the time dependence of the boundary. Also consistency with the principle of minimal coupling and gauge invariance is obtained just by using this canonical operator formalism. Finally some interesting physical applications are suggested and discussed.
Abstract The phase of a quantum state changes rapidly as parameters X = (X1, X2,...) are varied near a degeneracy X*, reflecting the monopole singularity of the underlying phase 2-form V(X) at X*. The singularities may be sources or sinks of V. We study them numerically and display them graphically for two families of hamiltonians whose degeneracy structure is typical. First is a particle moving along a line segment with kinetic energy quartic in the momentum (‘quartic-momentum square well’); the X are incorporated into the boundary conditions. Second is a charged particle moving in a domain D of the plane which is threaded by a magnetic flux line of strength α, with wavefunction vanishing on the boundary ∂D (‘Aharonov-Bohm billiards’); the X are α and parameters specifying ∂D; V is not invariant under gauge transformations of the vector potential generating the flux. For Aharonov-Bohm billiards we study how the spatial patterns of phase of wavefunctions change round circuits near degeneracies; these patterns also have singularities (wavefront dislocations) that appear and disappear by colliding with each other and with ∂D.
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.