Bohmian Trajectories Within Hilbert Space Based Quantum Mechanics. Solution of the Measurement Problem

• URL: http://arxiv.org/abs/2512.07007v1
• Date: Sun, 07 Dec 2025 21:16:14 GMT
• Title: Bohmian Trajectories Within Hilbert Space Based Quantum Mechanics. Solution of the Measurement Problem
• Authors: Tulsi Dass,
• Abstract summary: dBBT treats quantum particles as point objectsmoving along well defined (Bohmian) trajectories.<n>It has problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework.<n>We present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: de Broglie-Bohm theory (dBBT), treating quantum particles as point objectsmoving along well defined (Bohmian) trajectories, offers an appealingsolution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function. Given a Schrodinger wave function, we use its value u at some fixed time (say, t = 0) to define the probability measure |u|sqdx on the system configuration space. On the resulting probability space M0, we introduce a stochastic process xi(t) corresponding to the Heisenberg position operator XH(t) such that, in the Heisenberg state phi corresponding to u, the expectation value of XH(t) equals that of xi(t) in M0. This condition leads to the de Broglie-Bohmguidance equation for the sample paths of the process xi(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the u-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger - Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.

Related papers

• Exact Solutions to the Quantum Schrödinger Bridge Problem [3.2480327833037226]
  The Quantum Schr"odinger Bridge Problem (QSBP) describes the evolution of a process between two arbitrary probability distributions.<n>We show that the resulting evolution equations involve the so-called Bohm (quantum) potential, representing a notion of non-locality in the process.<n>We present a modified algorithm based on a Gaussian Mixture Model framework, and demonstrate its effectiveness across several experimental settings.
  arXiv  Detail & Related papers  (2025-09-30T09:11:24Z)
• A pseudo-random and non-point Nelson-style process [49.1574468325115]
  We take up the idea of Nelson's processes, the aim of which was to deduce Schr"odinger's equation.<n>We consider deterministic processes which are pseudo-random but which have the same characteristics as Nelson's processes.
  arXiv  Detail & Related papers  (2025-04-29T16:18:51Z)
• What Bohmian mechanic says about arrival times of 1D vacuum squeezed states [0.0]
  We calculate the time of arrival probability distribution of a quantum particle using the Bohmian formalism.<n>The pilot-wave is given by the wave function of the one dimensional vacuum squeezed state but written in the Schr"odinger representation.
  arXiv  Detail & Related papers  (2025-02-09T00:58:00Z)
• Phase-space gaussian ensemble quantum camouflage [0.0]
  We extend the phase-space description of the Weyl-Wigner quantum mechanics to a subset of non-linear Hamiltonians in position and momentum.<n>For gaussian statistical ensembles, the exact phase-space profile of the quantum fluctuations over the classical trajectories are found.
  arXiv  Detail & Related papers  (2024-09-24T18:14:07Z)
• Quantum Mechanics in Curved Space(time) with a Noncommutative Geometric Perspective [0.0]
  We take seriously the noncommutative symplectic geometry corresponding to the quantum observable algebra.<n>The work points to a very different approach to quantum gravity, plausibly with a quantum Einstein equation suggested.
  arXiv  Detail & Related papers  (2024-06-20T10:44:06Z)
• Double-scale theory [77.34726150561087]
  We present a new interpretation of quantum mechanics, called the double-scale theory. It is based on the simultaneous existence of two wave functions in the laboratory reference frame. The external wave function corresponds to a field that pilots the center-of-mass of the quantum system. The internal wave function corresponds to the interpretation proposed by Edwin Schr"odinger.
  arXiv  Detail & Related papers  (2023-05-29T14:28:31Z)
• Describing the Wave Function Collapse Process with a State-dependent Hamiltonian [3.8326963933937885]
  We show how the continuous collapse of the wave function can be described by the Schr"odinger equation with a time-dependent Hamiltonian. We then discuss how the above formalism can also be applied to describe the collapse of the wave function of mixed quantum states.
  arXiv  Detail & Related papers  (2023-01-23T05:08:35Z)
• Quantum dynamics corresponding to chaotic BKL scenario [62.997667081978825]
  Quantization smears the gravitational singularity avoiding its localization in the configuration space. Results suggest that the generic singularity of general relativity can be avoided at quantum level.
  arXiv  Detail & Related papers  (2022-04-24T13:32:45Z)
• Probabilistic deconstruction of a theory of gravity, Part I: flat space [0.0]
  We show that Einstein's equations of the theory arise in the semi-classical limit of the quantum evolution of probability under the process. In particular, in flat Jackiw-Teitelboim gravity, the area of compactified space solved for by Einstein's equations can be identified as a probability density evolving under the Markovian process.
  arXiv  Detail & Related papers  (2021-08-24T18:52:31Z)
• Breakdown of quantum-classical correspondence and dynamical generation of entanglement [6.167267225728292]
  We study the generation of quantum entanglement induced by an ideal Fermi gas confined in a chaotic cavity. We find that the breakdown of the quantum-classical correspondence of particle motion, via dramatically changing the spatial structure of many-body wavefunction, leads to profound changes of the entanglement structure.
  arXiv  Detail & Related papers  (2021-04-14T03:09:24Z)
• Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
  The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
  arXiv  Detail & Related papers  (2021-02-18T20:55:04Z)
• Evolution of a Non-Hermitian Quantum Single-Molecule Junction at Constant Temperature [62.997667081978825]
  We present a theory for describing non-Hermitian quantum systems embedded in constant-temperature environments. We find that the combined action of probability losses and thermal fluctuations assists quantum transport through the molecular junction.
  arXiv  Detail & Related papers  (2021-01-21T14:33:34Z)
• Standard Quantum Mechanics without observers [0.0]
  The Projection Postulate from Standard Quantum Mechanics relies fundamentally on measurements. measurements implicitly suggest the existence of anthropocentric notions like measuring devices. This article proposes an alternative formulation of the Standard Quantum Mechanics.
  arXiv  Detail & Related papers  (2020-08-11T18:01:15Z)
• External and internal wave functions: de Broglie's double-solution theory? [77.34726150561087]
  We propose an interpretative framework for quantum mechanics corresponding to the specifications of Louis de Broglie's double-solution theory. The principle is to decompose the evolution of a quantum system into two wave functions. For Schr"odinger, the particles are extended and the square of the module of the (internal) wave function of an electron corresponds to the density of its charge in space.
  arXiv  Detail & Related papers  (2020-01-13T13:41:24Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.