Abstract For quantum or other waves that are band-limited, the quantum potential in the Madelung–Bohm representation vanishes on the boundaries of regions where the waves are superoscillatory (i.e. where they vary faster than any of their Fourier components). This connection is illustrated by calculations of the quantum potential zeros for a superoscillatory superposition of plane waves, and by Aharonov–Bohm waves.
Abstract For quantum or other waves that are band-limited, the quantum potential in the Madelung–Bohm representation vanishes on the boundaries of regions where the waves are superoscillatory (i.e. where they vary faster than any of their Fourier components). This connection is illustrated by calculations of the quantum potential zeros for a superoscillatory superposition of plane waves, and by Aharonov–Bohm waves.
Abstract. Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter-intuitive physical effects. Following up on this work, we here present more general methods which allow the calculation of superoscillatory wave functions with custom-designed physical properties. We give concrete examples and we prove results about the limits to superoscillatory behavior. We also give a simple and intuitive new explanation for the exponential computational cost of superoscillations. 1 2 1.
Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter-intuitive physical effects. Following up on this work, we here present more general methods which allow the calculation of superoscillatory wave functions with custom-designed physical properties. We give concrete examples and we prove results about the limits to superoscillatory behavior. We also give a simple and intuitive new explanation for the exponential computational cost of superoscillations.
Abstract Bohmian mechanics is a deterministic account of quantum phenomena in terms of particle trajectories. Guided by a quantum force, the gradient of a quantum potential, trajectories bend to account for quantum interference. The quantum potential is expressible in terms of weak values, complex numbers that characterize quantum systems for which both the initial and final states are predetermined. In Berry and Shukla (2010 J. Phys. A: Math. Theor. 43 354024), the typical probabilities of weak values were calculated. In this paper, expressing the quantum potential in terms of weak values, we find the typical probability distribution of the quantum potential for randomly selected states. The latter allows the calculation of the typical probability of semiclassical superoscillation, a generalization of superoscillations (functions that oscillate faster than their fastest Fourier component) to functions that are not band-limited.
In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations . Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.
A superoscillatory function—that is, a band-limited function f(x) oscillating faster than its fastest Fourier component—is taken to be the initial state of a freely-evolving quantum wavefunction ψ. The superoscillations persist for unexpectedly long times, but eventually disappear through the interaction of contributions to ψ with complex momenta that are exponentially disparate in magnitude; this is established by applying the asymptotics of integrals, supported by numerics. f(x) can alternatively be regarded as the wave generated by a diffraction grating, propagating paraxially and without evanescence as ψ in the space beyond. The persistence of superoscillations is then interpreted as the propagation of sub-wavelength structure farther into the field than the more familiar evanescent waves.
Abstract In single-particle Madelung mechanics, the single-particle quantum state Ψ ( x → , t ) = R ( x → , t ) e i S ( x → , t ) / ℏ is interpreted as comprising an entire conserved fluid of classical point particles, with local density R ( x → , t ) 2 and local momentum ∇ → S ( x → , t ) (where R and S are real). The Schrödinger equation gives rise to the continuity equation for the fluid, and the Hamilton–Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term Q ( x → , t ) = − ℏ 2 2 m ∇ → R ( x → , t ) R ( x → , t ) , which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of Ψ exceeds its global band limit. Berry showed that for states of definite energy E, the regions of superoscillation are exactly the regions where Q ( x → , t ) < 0 . For energy superposition states with band-limit