arXiv subjects

Bill Poirier

Publications and source records attributed to Bill Poirier.

At least 19 recordsLinked to original sources

arXiv2026

On Hydrodynamic Formulations of Quantum Mechanics and the Problem of Sparse Ontology

Hydrodynamic reformulations of the Schr\"odinger equation suggest an interpretation of quantum mechanics in terms of a fluid flowing on configuration space. In the discrete hydrodynamic view, this fluid is not fundamental but emerges from many underlying microscopic fluid components whose collective behavior reproduces quantum phenomena. The most developed realization of this idea is the discrete many interacting worlds (MIW) framework, in which discrete particle-like worlds interact via inter-world forces and quantum probabilities are grounded in direct world counting. But there is also an older, continuous version of MIW. After reviewing the hydrodynamic and MIW formalisms, and emphasizing some of their interpretational advantages over the Everettian Many Worlds and Bohmian approaches, we argue that all discrete hydrodynamic models face a generic structural difficulty, which we call the problem of sparse ontology. Because wavefunctions typically branch under decoherence, the discrete components of the fluid are repeatedly partitioned into sub-ensembles, thereby thinning their density in configuration space and driving the dynamics away from the quantum regime once the components become sufficiently sparse. We conclude that successful hydrodynamic completions of quantum mechanics plausibly require an essentially continuous ontology.

quant-ph↗

arXiv2023

ExoMol line lists -- L: High-resolution line lists of H$_3^+$, H$_2$D$^+$, D$_2$H$^+$ and D$_3^+$

New MiZo line lists are presented for the D$_2$H$^+$ and D$_3^+$ isotopologues of H$_3^+$. These line lists plus the existing H$_3^+$ MiZATeP and the Sochi H$_2$D$^+$ line lists are updated using empirical energy levels generated using the MARVEL procedure for H$_3^+$, H$_2$D$^+$ and D$_2$H$^+$, and effective Hamiltonian energies for D$_3^+$ for which there is significantly less laboratory data available. These updates allow accurate frequencies for far infrared lines for these species to be predicted. Assignments of the energy levels of H$_3^+$ and D$_3^+$ are extended using a combination of high accuracy variational calculations and analysis of transition intensities. All line lists are made available via www.exomol.com.

astro-ph.SR↗

arXiv2022

The Differentiation Lemma and the Reynolds Transport Theorem for Submanifolds with Corners

The Reynolds Transport Theorem, colloquially known as 'differentiation under the integral sign', is a central tool of applied mathematics, finding application in a variety of disciplines such as fluid dynamics, quantum mechanics, and statistical physics. In this work we state and prove generalizations thereof to submanifolds with corners evolving in a manifold via the flow of a smooth time-independent or time-dependent vector field. Thereby we close a practically important gap in the mathematical literature, as related works require various 'boundedness conditions' on domain or integrand that are cumbersome to satisfy in common modeling situations. By considering manifolds with corners, a generalization of manifolds and manifolds with boundary, this work constitutes a step towards a unified treatment of classical integral theorems for the 'unbounded case' for which the boundary of the evolving set can exhibit some irregularity.

math-ph↗

arXiv2022

Towards a mathematical Theory of the Madelung Equations

Even though the Madelung equations are central to many 'classical' approaches to the foundations of quantum mechanics such as Bohmian and stochastic mechanics, no coherent mathematical theory has been developed so far for this system of partial differential equations. Wallstrom prominently raised objections against the Madelung equations, aiming to show that no such theory exists in which the system is well-posed and in which the Schrödinger equation is recovered without the imposition of an additional 'ad hoc quantization condition'--like the one proposed by Takabayasi. The primary objective of our work is to clarify in which sense Wallstrom's objections are justified and in which sense they are not, with a view on the existing literature. We find that it may be possible to construct a mathematical theory of the Madelung equations which is satisfactory in the aforementioned sense, though more mathematical research is required.

math-ph↗

arXiv2021

Efficient Evaluation of Exponential and Gaussian Functions on a Quantum Computer

The exponential and Gaussian functions are among the most fundamental and important operations, appearing ubiquitously throughout all areas of science, engineering, and mathematics. Whereas formally, it is well-known that any function may in principle be realized on a quantum computer, in practice present-day algorithms tend to be very expensive. In this work, we present algorithms for evaluating exponential and Gaussian functions efficiently on quantum computers. The implementations require a (generally) small number of multiplications, which represent the overall computational bottleneck. For a specific, realistic NISQ application, the Toffoli count of the exponential function is found to be reduced from 15,690 down to 912, when compared against a state-of-the art competing method by Häner and coworkers [arXiv:1805.12445], under the most favorable conditions for each method. For the corresponding Gaussian function comparison, the Toffoli count is reduced from 19,090 down to 704. Space requirements are also quite modest, to the extent that the aforementioned NISQ application can be implemented with as few as 71 logical qubits. More generally, the methods presented here could also be equally well applied in a fault-tolerant context, using error-corrected multiplications, etc.

quant-ph↗

arXiv2021

Full-Dimensional Schrödinger Wavefunction Calculations using Tensors and Quantum Computers: the Cartesian component-separated approach

Traditional methods in quantum chemistry rely on Hartree-Fock-based Slater-determinant (SD) representations, whose underlying zeroth-order picture assumes separability by particle. Here, we explore a radically different approach, based on separability by Cartesian component, rather than by particle [J. Chem. Phys., 2018, 148, 104101]. The approach appears to be very well suited for 3D grid-based methods in quantum chemistry, and thereby also for so-called "first-quantized" quantum computing. We first present an overview of the approach as implemented on classical computers, including numerical results that justify performance claims. In particular, we perform numerical calculations with four explicit electrons that are equivalent to full-CI matrix diagonalization with nearly $10^{15}$ SDs. We then present an implementation for quantum computers, for which both the number of qubits, and the number of quantum gates, may be substantially reduced in comparison with other quantum circuitry that has been envisioned for implementing first-quantized "quantum computational chemistry" (QCC).

quant-ph↗

arXiv2021

Viewing quantum mechanics through the prism of electromagnetism

In this paper, we demonstrate novel relationships between quantum mechanics and the electromagnetic wave equation. In our approach, an invariant interference-dependent electromagnetic quantity, which we call "quantum rest mass", replaces the conventional role of the inertial rest mass. In the ensuing results, photons, during interference, move slower than the speed of light in vacuum, and possess de Broglie wavelength. Further, we use our electromagnetic approach to examine double-slit photon trajectories, and to arrive at the Schrodinger equation's results for a particle in an infinite square well potential.

quant-ph↗

arXiv2020

Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime

In this work we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of quantum field theory (QFT) such as the `problem of infinities' (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work we consider the $1$-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the $1$-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the `problem of time' in this approach and therefore serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved (example given for the latter).

math-ph↗

arXiv2019

Exact Schrodinger wave-function solutions of the fully-correlated few-electron gas and the uniform electron gas

We present numerically exact solutions to the full-dimensional Schrodinger Equation for the few-electron gas (few-EG) model of electronic structure theory. Our core methodology uses a Sum-of-Products (SOP) representation of singular potentials to make the fully correlated multi-particle quantum mechanical problem computationally tractable. With full access to periodic wave-functions, we analyze the exchange/correlation, spin symmetry, and thermodynamic ($T = 0$K) phases for few-EG systems with up to four explicit electrons per unit cell. We also present pure-spin exchange/correlation functionals for the same few-EG solutions. Finally, we consider the family of few-EG calculations in juxtaposition with the closely related uniform electron gas (UEG) system, for which previous calculations have been performed using density functional theory and quantum Monte Carlo methods. The exact few-EG solutions all display a similar oscillatory behavior at lower densities, which is not predicted by previous UEG simulations.

cond-mat.str-el↗

arXiv2017

Final State Resolved Quantum Predissociation Dynamics of SO2(C1B2) and Its Isotopomers via a Crossing with a Singlet Repulsive State

The fragmentation dynamics of predissociative SO2(C1B2) is investigated on an accurate adiabatic potential energy surface (PES) determined from high level ab initio data. This singlet PES features non-C2v equilibrium geometries for SO2, which are separated from the SO + O dissociation limit by a barrier resulting from a conical intersection with a repulsive singlet state. The ro-vibrational state distribution of the SO fragment is determined quantum mechanically for many predissociative states of several sulfur isotopomers of SO2. Significant rotational and vibrational excitations are found in the SO fragment. It is shown that these fragment internal state distributions are strongly dependent on the predissociative vibronic states, and the excitation typically increases with the photon energy.

physics.chem-ph↗

arXiv2016

Exploring the propagation of relativistic quantum wavepackets in the trajectory-based formulation

In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schrödinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [Phys. Rev. X, 4, 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature---suggesting a "many interacting worlds" interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time---in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.

quant-ph↗

arXiv2012

Trajectory-based Theory of Relativistic Quantum Particles

Recently, a self-contained trajectory-based formulation of non-relativistic quantum mechanics was developed [Ann. Phys. 315, 505 (2005); Chem. Phys. 370, 4 (2010); J. Chem. Phys. 136, 031102 (2012)], that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories that extremize a suitable action. Here, the trajectory-based approach is developed into a viable, generally covariant, relativistic quantum theory for single (spin-zero, massive) particles. Central to this development is the introduction of a new notion of global simultaneity for accelerated particles--together with basic postulates concerning probability conservation and causality. The latter postulate is found to be violated by the Klein-Gordon equation, leading to its well-known problems as a single-particle theory. Various examples are considered, including the time evolution of a relativistic Gaussian wavepacket.

quant-ph↗

arXiv2012

Quantum Mechanics Without Wavefunctions

We present a self-contained formulation of spin-free nonrelativistic quantum mechanics that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories, obtained by extremizing an action and satisfying energy conservation. The theory applies for arbitrary configuration spaces and system dimensionalities. Various beneficial ramifications - theoretical, computational, and interpretational - are discussed.

quant-ph↗

arXiv2008

Development and Numerical Analysis of "Black-box" Counterpropagating Wave Algorithm for Exact Quantum Scattering Calculations

In a recent series of papers [J. Chem. Phys. 121 4501 (2004), J. Chem. Phys. 124 034115 (2006), J. Chem. Phys. 124 034116 (2006)] a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary bound states Psi of the one-dimensional Shrodinger equation, such that the components Psi+- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. In this paper, the earlier results are used to construct a universal ``black-box'' algorithm, numerically robust, stable and efficient, for computing accurate scattering quantities of any quantum dynamical system in one degree of freedom.

quant-ph↗

arXiv2008

Flux Continuity and Probability Conservation in Complexified Bohmian Mechanics

Recent years have seen increased interest in complexified Bohmian mechanical trajectory calculations for quantum systems, both as a pedagogical and computational tool. In the latter context, it is essential that trajectories satisfy probability conservation, to ensure they are always guided to where they are most needed. In this paper, probability conservation for complexified Bohmian trajectories is considered. The analysis relies on time-reversal symmetry considerations, leading to a generalized expression for the conjugation of wavefunctions of complexified variables. This in turn enables meaningful discussion of complexified flux continuity, which turns out not to be satisfied in general, though a related property is found to be true. The main conclusion, though, is that even under a weak interpretation, probability is not conserved along complex Bohmian trajectories.

quant-ph↗

arXiv2008

Reconciling Semiclassical and Bohmian Mechanics: V. Wavepacket Dynamics

In previous articles [J. Chem. Phys. 121 4501 (2004), J. Chem. Phys. 124 034115 (2006), J. Chem. Phys. 124 034116 (2006), J. Phys. Chem. A 111 10400 (2007)] a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary states Psi of the one-dimensional Schrodinger equation, such that the components Psi+- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. In this paper, the method is generalized for time-dependent wavepacket dynamics applications, and applied to several benchmark problems, including multisurface systems with nonadiabatic coupling.

quant-ph↗

arXiv2008

Reconciling Semiclassical and Bohmian Mechanics: IV. Multisurface Dynamics

In previous articles [J. Chem. Phys. 121 4501 (2004), J. Chem. Phys. 124 034115 (2006), J. Chem. Phys. 124 034116 (2006)] a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary states Psi of the one-dimensional Schrodinger equation, such that the components Psi+- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. In this paper, the method is generalized for multisurface scattering applications, and applied to several benchmark problems. A natural connection is established between intersurface transitions and (+/-) transitions.

quant-ph↗

arXiv2008

Reconciling Semiclassical and Bohmian Mechanics: II. Scattering states for discontinuous potentials

In a previous paper [J. Chem. Phys. 121 4501 (2004)] a unique bipolar decomposition, Psi = Psi1 + Psi2 was presented for stationary bound states Psi of the one-dimensional Schroedinger equation, such that the components Psi1 and Psi2 approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step, square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a future publication.

quant-ph↗