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arXiv:1501.06490 · International Journal of Geometric Methods in Modern Physics

Quantum systems with time-dependent boundaries

Sara Di Martino, Paolo Facchi A

Published Mar 2, 2015• 14 citations
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Abstract

We present here a set of lecture notes on quantum systems with time-dependent boundaries. In particular, we analyze the dynamics of a non-relativistic particle in a bounded domain of physical space, when the boundaries are moving or changing. In all cases, unitarity is preserved and the change of boundaries does not introduce any decoherence in the system.

DOI
10.1142/s0219887815600038

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We present here a set of lecture notes on quantum systems with time-dependent boundaries. In particular, we analyze the dynamics of a non-relativistic particle in a bounded domain of physical space, when the boundaries are moving or changing. In all cases, unitarity is preserved and the change of boundaries does not introduce any decoherence in the system.
We analyze the quantum dynamics of a non-relativistic particle moving in a bounded domain of physical space, when the boundary conditions are rapidly changed. In general, this yields new boundary conditions, via a dynamical composition law that is a very simple instance of the superposition of different topologies. In all cases, unitarity is preserved and the quick change of boundary conditions does not introduce any decoherence in the system. Among the emerging boundary conditions, the Dirichlet case (vanishing wavefunction at the boundary) plays the role of an attractor. Possible experimental implementations with superconducting quantum interference devices are explored and analyzed.
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.
We show how to use boundary conditions to drive the evolution on a quantum mechanical system. We will see how this problem can be expressed in terms of a time-dependent Schrödinger equation. In particular, we will need the theory of self-adjoint extensions of differential operators in manifolds with boundary. An introduction of the latter as well as meaningful examples will be given. It is known that different boundary conditions can be used to describe different topologies of the associated quantum systems. We will use the previous results to study the topology change and to obtain necessary conditions to accomplish it in a dynamical way.
Some recent results concerning a particle confined in a one-dimensional box with moving walls are briefly reviewed. By exploiting the same techniques used for the 1D problem, we investigate the behavior of a quantum particle confined in a two-dimensional box (a 2D billiard) whose walls are moving by recasting the relevant mathematical problem with moving boundaries in the form of a problem with fixed boundaries and time-dependent Hamiltonian. Changes of the shape of the box are shown to be important, as it clearly emerges from the comparison between the ‘pantographic’ case (same shape of the box through the entire process) and the case with deformation.