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arXiv:2012.06699 · Journal of Physics A Mathematical and Theoretical

Evolution and invariants of free-particle moments

Mark Andrews

Published Mar 26, 2021• 2 citations
arXivHTMLPDF

Abstract

Abstract Moments are expectation values of products of powers of position and momentum, taken over quantum states (or averages over a set of classical particles). For free particles, the evolution in the quantum case is closely related to that of a set of classical particles. Here we consider the evolution of symmetrized moments for free particles in one dimension, first examining the geometric properties of the evolution for moments up to the fourth order, as determined by their extrema and inflections. These properties are specified by combinations of the moments that are invariant in that they remain constant under free evolution. An inequality constrains the fourth-order moments and shows that some geometric types of evolution are possible for a quantum particle but not possible classically, and some examples are examined. Explicit expressions are found for the moments of any order in terms of their initial values, for the invariant combinations, and for the moments in terms of these invariants.

DOI
10.1088/1751-8121/abf27c

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The classical and quantum evolution of a generic probability distribution is analyzed. To that end, a formalism based on the decomposition of the distribution in terms of its statistical moments is used, which makes explicit the differences between the classical and quantum dynamics. In particular, there are two different sources of quantum effects. Distributional effects, which are also present in the classical evolution of an extended distribution, are due to the fact that all moments cannot be vanishing because of the Heisenberg uncertainty principle. In addition, the noncommutativity of the basic quantum operators add some terms to the quantum equations of motion that explicitly depend on the Planck constant and are not present in the classical setting. These are thus purely quantum effects. Some particular Hamiltonians are analyzed that have very special properties regarding the evolution they generate in the classical and quantum sector. In addition, a large class of inequalities obeyed by high-order statistical moments, and in particular uncertainty relations that bound the information that is possible to obtain from a quantum system, are derived.
Abstract The correspondence principle provides a prescription to connect quantum physics to classical. It asserts that the physical quantities evaluated quantum mechanically approach their respective classical values for large quantum numbers. This has been shown for the pedagogically important cases of the particle in a box and a harmonic oscillator. However, a particle in a constant field has a wave function related to the Airy function and has at best been treated numerically. Employing energy eigenstates we obtain the expectation values of the position, the momentum and their moments upto fourth order, rigorously and without resorting to numerical or graphical techniques. We compare them with the corresponding classical values. We also examine the uncertainty product for the system.
Abstract We present a formulation of measurement-based feedback control of a single quantum particle in one spatial dimension to consider arbitrary linear combinations of the position and momentum of the particle used as observables for monitoring and as generators of unitary feedback with strength proportional to the measured signal. We derive a feedback master equation and discuss a general approach to computing the steady-state solutions for arbitrary potentials. For a quantum harmonic oscillator or a free particle, we show that it is possible to cool and confine the system using feedback that simultaneously damps the measured observable and its conjugate momentum. Our general approach allows to identify a combination of measurement and feedback variables that, under certain circumstances, completely mitigates the noise induced by the measurement. The resulting deterministic evolution of expectation values in each realisation of the measurement resembles the dynamics of a damped classical oscillator, leading to a stationary state centred at the minimum of the potential, which becomes the ground state in the weak measurement limit. Remarkably, this stabilisation can be achieved using a fixed generator of feedback that is determined by the asymptotic values of the second-order moments of the steady state. In addition, we demonstrate that appropriate feedback adds a quadratic term in the measured observable to the Hamiltonian of the system. Moreover, we provide an argument for the possibility to cool systems with arbitrary potentials, provided that the measurement is strong enough to localise the particle on an interval smaller than the characteristic length scale of the potential.
Important information about stationary flows (B I ) I Î I is contained in moments of the corresponding process. We consider the case where we have an order invariance of the moments of the increments, in the sense that such moments do not change if we shift the increments against each other, as long as we do not change the relative position of the intervals. We then show that for large classes of quantum processes one can make precise statements about the time behaviour of their moments. Keywords: Von Neumann algebras, Stationary flows and quantum processes, order invariant, Trace and moments.