We unveil the existence of a non-trivial Berry phase associated to the dynamics of a quantum particle in a one dimensional box with moving walls. It is shown that a suitable choice of boundary conditions has to be made in order to preserve unitarity. For these boundary conditions we compute explicitly the geometric phase two-form on the parameter space. The unboundedness of the Hamiltonian describing the system leads to a natural prescription of renormalization for divergent contributions arising from the boundary
Abstract. We establish a bijection between the self-adjoint extensions of the Laplace operator on a bounded regular domain and the unitary operators on the boundary. Each unitary encodes a specific relation between the boundary value of the function and its normal derivative. This bijection sets up a characterization of all physically admissible dynamics of a nonrelativistic quantum particle confined in a cavity. Moreover, this correspondence is discussed also at the level of quadratic forms. Finally, the connection between this parametrization of the extensions and the classical one, in terms of boundary self-adjoint operators on closed subspaces, is shown.
We consider the quantum dynamics of a free nonrelativistic particle moving in a cavity and we analyze the effect of a rapid switching between two different boundary conditions. We show that this procedure induces, in the limit of infinitely frequent switchings, a new effective dynamics in the cavity related to a novel boundary condition. We obtain a dynamical composition law for boundary conditions which gives the emerging boundary condition in terms of the two initial ones.
Starting with a quantum particle on a closed manifold without boundary, we consider the process of generating boundaries by modding out by a group action with fixed points, and we study the emergent quantum dynamics on the quotient manifold.As an illustrative example, we consider a free nonrelativistic quantum particle on the circle and generate the interval via parity reduction. A free particle with Neumann and Dirichlet boundary conditions on the interval is obtained, and, by changing the metric near the boundary, Robin boundary conditions can also be accommodated. We also indicate a possible method of generating non-local boundary conditions. Then, we explore an alternative generation mechanism which makes use of a folding procedure and is applicable to a generic Hamiltonian through the emergence of an ancillary spin degree of freedom.
We present here a set of lecture notes on quantum thermodynamics and canonical typicality. Entanglement can be constructively used in the foundations of statistical mechanics. An alternative version of the postulate of equal a priori probability is derived making use of some techniques of convex geometry.
We present here a set of lecture notes on exact fluctuation relations. We prove the Jarzynski equality and the Crooks fluctuation theorem, two paradigmatic examples of classical fluctuation relations. Finally we consider their quantum versions, and analyze analogies and differences with the classical case.Keywords: quantum thermodynamics; fluctuation relations. IntroductionWe present here the notes of three lectures given by one of us at the "Fifth International Workshop on Mathematical Foundations of Quantum Mechanics and its applications" held in February 2017 in Madrid at the Instituto de Ciencias Matemáticas (ICMAT).We will consider some results about fluctuation theorems both for classical and for quantum systems, a research topic that recently has attracted a great deal of attention. The statistical mechanics of classical and quantum systems driven far from equilibrium has witnessed quite recently a sudden development with the discovery of various exact fluctuation theorems which connect equilibrium thermodynamic quantities to non-equilibrium ones. There are excellent reviews on this topic, which cover both classical [1] and quantum fluctuation relations [2,3]. Here we will follow more closely the exposition by Campisi, Hänggi and Talkner [3], to which we refer the reader for further information.In the first lecture we will recall the derivation of Einstein's fluctuationdissipation relation for a Brownian particle, which is the inception of classical fluctuation relations. Moreover, we will identify the fundamental ingredients which are already present in this early derivation. Then, we will consider the GreenKubo formula, which represents the first general approach to quantum fluctuationdissipation relations. The link between the correlation function of a quantum system and its linear response function will be shown and the classical limit of the Green-Kubo formula will be considered.
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