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.
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.
This thesis deals with the theoretical and experimental details of trying to understand \nand infer the quantum potential from Bohmian mechanics using Bohmian trajectories. In \nthis work, some of the key components of the history of Bohmian mechanics is given, a \ntheoretical explanation of how to measure the Bohmian potential, followed by an overview \nof the experimental apparatus and its components. Some tests of the experimental apparatus \nare then presented followed by a discussion of the results. Lastly, some interesting \nfollow up experiments are de ned and discussed. \n \nBohmian mechanics, while completely agreeing with quantum mechanics could not be \nmore di erent from the standard notions such as the fact that particles retain their classical \nidentity of a point like object that have de nite and causal trajectories. In this realm, rst \ndeveloped by de Broglie, these particles move and are guided by a wave, which turns out \nto be the wave function itself. Typically, the velocity of the particles anywhere in space \nis given by spatial derivatives of the phase of this wave and the energy content of the \nparticles is given by the temporal derivative. The trajectories can bend even when there \nare no outside (classical) forces that are acting on the objects. This is due to the fact that \nthese observations of Bohmian mechanics must conform to the standard measurements in \nquantum mechanics. \n \nOne detail that has gone unmeasured (in the sense of inference) in a laboratory setting, \nis the quantum potential (the mysterious force that moves particle to make sure they \nreproduce the measurements of quantum mechanics). Using the technique of measuring \nthe trajectories of photons, the question was asked, can the quantum potential for single \nphotons in the double slit be inferred? A setup has been built in order to try and answer this \nquestion using a single photon detecting camera and a heralding spontaneous parametric \ndown conversion source. Unfortunately, the quantum potential still goes unmeasured due \nto noise in the camera pictures destroying the integrity of the signal and an interesting \nproblem of trying to stitch together the quantum potential at the varying z-planes.
Abstract For complex scalar waves, a convenient measure of the local oscillations and (‘faster than Fourier’) superoscillations is the phase gradient vector: the local wavevector, or weak value of the momentum operator. This vanishes for standing waves, described by real functions ψ( r ); for such waves, an alternative descriptor of oscillations is the local weak value of the square of one of the momentum components, i.e. K 2 r = − ∂ 2 ψ r / ∂ x 2 / ψ r , here called the ‘weak curvature’. Superoscillations correspond to places where K 2 lies outside the interval 0 ⩽ K 2 ⩽ 1. Two illustrations are given. First is an explicit family of real waves in dimension d = 2, with arbitrarily strong superoscillations; this could represent Neumann standing modes in a strip waveguide. Second is an exact calculation of the probability distribution of K 2 for Gaussian random real waves in d dimensions. This decays as 1 / K 2 2 , as a consequence of the codimension 1 of nodes (e.g. nodal lines for d = 2). The superoscillation probability varies from 0.3918 (d = 2) to 0.3041 (d = ∞).
The origin of quantum interference characteristic of bound nonlinear systems is investigated within the Bohmian formulation of time-dependent quantum mechanics. By contrast to time-dependent semiclassical theory, whereby interference is a consequence of phase mismatch between distinct classical trajectories, the Bohmian, fully quantum mechanical expression for expectation values has a quasiclassical appearance that does not involve phase factors or cross terms. Numerical calculations reveal that quantum interference in the Bohmian formulation manifests itself directly as sharp spatial/temporal variations of the density surrounding kinky trajectories. These effects are most dramatic in regions where the underlying classical motion exhibits focal points or caustics, and crossing of the Bohmian trajectories is prevented through extremely strong and rapidly varying quantum mechanical forces. These features of Bohmian dynamics, which constitute the hallmark of quantum interference and are ubiquitous in bound nonlinear systems, represent a major source of instability, making the integration of the Bohmian equations extremely demanding in such situations.
Bohmian mechanics is a theory that provides a consistent explanation of quantum phenomena in terms of point particles whose motion is guided by the wave function. In this theory, the state of a system of particles is defined by the actual positions of the particles and the wave function of the system; and the state of the system evolves deterministically. Thus, the Bohmian state can be compared with the state in classical mechanics, which is given by the positions and momenta of all the particles, and which also evolves deterministically. However, while in classical mechanics it is usually taken for granted and considered unproblematic that the state is, at least in principle, measurable, this is not the case in Bohmian mechanics. Due to the linearity of the quantum dynamical laws, one essential component of the Bohmian state, the wave function, is not directly measurable. Moreover, it turns out that the measurement of the other component of the state — the positions of the particles — must be mediated by the wave function; a fact that in turn implies that the positions of the particles, though measurable, are constrained by absolute uncertainty. This is the key to understanding how Bohmian mechanics, despite being deterministic, can account for all quantum predictions, including quantum randomness and uncertainty.
The concept of superoscillations is extended beyond bandlimited functions, to include monochromatic waves in space-varying media, such as wavefunctions representing quantum particles in non-constant potentials. 'Semiclassical superoscillations' are defined as places where the phase gradient (local wavevector, or weak momentum) exceeds the classically allowed momentum. The phenomenon is essentially quantum (or, in optics, essentially beyond geometrical optics), and can arise in three different ways: from interference, evanescence (e.g. barrier penetration), and WKB corrections. Illustrations are given, including superpositions of few and many states in a linear potential, angular-momentum circular harmonic-oscillator states, eigenstates of degenerate harmonic-oscillator modes, and above-barrier scattering.