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arXiv:1001.0342

Quantum Fluctuations Contribution to the Random Walk of a Single Molecule and New Estimate of the Planck Constant

J. P. Mbelek

Published Jan 3, 2010• 0 citations
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Abstract

It is shown, by considering the case of the harmonic oscillator, that quantum fluctuations may be the most significant contribution to the random walk of a single molecule. From this point, the controversy on the existence of a standard quantum limit (SQL) is addressed and settled on the experimental ground. Comparisons to the experimental data yet avalaible in the literature provide a new estimate of the reduced Planck constant yielding \hbar = (1.1 \pm 0.2) ~ 10^{-34} J.s.

DOI
10.48550/arxiv.1001.0342

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It is shown, by considering the case of the harmonic oscillator, that quantum fluctuations may be the most significant contribution to the random walk of a single molecule. From this point, the controversy on the existence of a standard quantum limit (SQL) is addressed and settled on the experimental ground. Comparisons to the experimental data yet avalaible in the literature provide a new estimate of the reduced Planck constant yielding \hbar = (1.1 \pm 0.2) ~ 10^{-34} J.s.
Motivated by the Dirac idea that fundamental constants are dynamical variables and by conjectures on quantum structure of space–time at small distances, we consider the possibility that Planck constant [Formula: see text] is a time depending quantity, undergoing random Gaussian fluctuations around its measured constant mean value, with variance [Formula: see text] and a typical correlation timescale [Formula: see text]. We consider the case of propagation of a free particle and a one-dimensional harmonic oscillator coherent state, and show that the time evolution in both cases is different from the standard behavior. Finally, we discuss how interferometric experiments or exploiting coherent electromagnetic fields in a cavity may put effective bounds on the value of [Formula: see text].
The one-dimensional quantum harmonic oscillator is considered. The wave solution of the Schrödinger equation determines the model of motion of particles in the nnn-quantum state as orbits of a stationary random walk at the moment of jump from the n−1n-1n−1-qu
Surprisingly, the natural looking random walk leading to Brownian motion occurs to be often biased in a very subtle way: emphasizing some possibilities by only approximating maximal uncertainty principle. A new philosophy of stochastic modelling has been recently introduced, in which we use the only maximizing entropy choice of transition probabilities instead. Local behaviour of both approaches is similar, but they usually lead to completely different global situations. In contrast to Brownian motion leading to nearly uniform stationary density, this recent approach turns out in agreement with having strong localization properties, thermodynamical predictions of quantum mechanics, like thermalizing to dynamical equilibrium state of probability density as the quantum ground state: squares of coordinates of the lowest energy eigenvector of the Bose-Hubbard Hamiltonian for single particle in discrete case, or of the standard Schrödinger operator while including potential and making infinitesimal limit.
In the world of computational probing, random walks and quantum walks are two separate but tightly linked paradigms. Classical stochastic processes like random walks are based on the random movement of particles in mathematical space and have been extensively used in data mining, network analysis, computer vision, etc. Quantum walks, by contrast generalize random walks into the quantum realm, taking the benefit of the quantum superposition and interference in order to investigate multiple paths at the same time. Quantum walks provide significant potential for algorithmic development: in quantum computing and communication, and in Scientific modelling and simulation. This paper provides an in-depth survey of random walks and quantum walks, outlining historical context, fundamental principles, applications, challenges, and future prospects. In this paper, motivated by the abstractness of both random walks and quantum walks, we survey the field of computational exploration at large scales, coalescing the myriad notions and difficult challenges addressed by existing research efforts on random and quantum walks, inferring a nuanced and multifaceted view on how both random walks and quantum walks are involved in paving the way to future development in many fields.
The nonzero ground-state energy of the quantum mechanical harmonic oscillator implies quantum fluctuations around the minimum of the potential with the mean-square value proportional to Planck's constant. On the other hand, thermal fluctuations occur when the oscillator is coupled to a heat bath. Using quantum statistical mechanics, this paper describes the transition from pure quantum fluctuations at zero temperature to classical thermal fluctuations in the high-temperature limit. It was early pointed out by Peierls that the classical mean-square thermal fluctuations in a harmonic chain—a linear chain of coupled harmonic oscillators—increase linearly with the distance of the masses from the fixed end of the chain, destroying long-range crystalline order. As shown here, the corresponding quantum fluctuations lead to a much slower logarithmic increase with the distance from the end. It is also shown that this behavior implies, for example, the absence of sharp Bragg peaks in x-ray scattering in an infinite chain at zero temperature, which instead shows power-law behavior typical for one-dimensional quantum liquids (called Luttinger liquids).
The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial; pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e. when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk owing to the interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point than a classical walker on average, and this forms the basis of a quantum speed up, which can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, even with a small quantum computer available, the development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems.