From reasonable postulates to generalised Hamiltonian systems

• URL: http://arxiv.org/abs/2403.00038v1
• Date: Thu, 29 Feb 2024 07:50:51 GMT
• Title: From reasonable postulates to generalised Hamiltonian systems
• Authors: Libo Jiang, Daniel R. Terno and Oscar Dahlsten
• Abstract summary: Hamiltonian mechanics describes the evolution of a system through its Hamiltonian. In both quantum and classical mechanics, Hamiltonian mechanics demands a precise relationship between time evolution and observable energy.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Hamiltonian mechanics describes the evolution of a system through its Hamiltonian. The Hamiltonian typically also represents the energy observable, a Noether-conserved quantity associated with the time-invariance of the law of evolution. In both quantum and classical mechanics, Hamiltonian mechanics demands a precise relationship between time evolution and observable energy, albeit using slightly different terminology. We distil basic conditions satisfied in both quantum and classical mechanics, including canonical coordinate symmetries and inner product invariance. We express these conditions in the framework of generalised probabilistic theories, which includes generalizing the definition of energy eigenstates in terms of time-invariant properties of the Hamiltonian system. By postulating these conditions to hold, we derive a unified Hamiltonian system model. This unified framework describes quantum and classical mechanics in a consistent language, facilitating their comparison. We moreover discuss alternative theories: an equation of motion given by a mixture of commutation relations, an information-restricted version of quantum theory, and Spekken's toy theory. The findings give a deeper understanding of the Hamiltonian in quantum and classical theories and point to several potential research topics.

Related papers

• Quantifying non-Hermiticity using single- and many-particle quantum properties [14.37149160708975]
  The non-Hermitian paradigm of quantum systems displays salient features drastically different from Hermitian counterparts. We propose a formalism that quantifies the (dis-)similarity of these right and left ensembles, for single- as well as many-particle quantum properties. Our findings can be instrumental in unveiling new exotic quantum phases of non-Hermitian quantum many-body systems.
  arXiv  Detail & Related papers  (2024-06-19T13:04:47Z)
• Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB [0.0]
  Hamilton-Jacobi theory is a powerful formalism, but its utility is not explored in quantum theory beyond the correspondence principle. We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi theory.
  arXiv  Detail & Related papers  (2024-06-12T02:50:43Z)
• Classifying two-body Hamiltonians for Quantum Darwinism [0.0]
  We consider a generic model of an arbitrary finite-dimensional system interacting with an environment formed from an arbitrary collection of finite-dimensional degrees of freedom. We show that such models support quantum Darwinism if the set of operators acting on the system which enter the Hamiltonian satisfy a set of commutation relations with a pointer observable and with one other.
  arXiv  Detail & Related papers  (2024-05-01T18:42:43Z)
• Unification of energy concepts in generalised phase space theories [0.0]
  We consider how to describe Hamiltonian mechanics in generalised probabilistic theories. We define generalised energy eigenstates as the purest stationary states. This allows for a generalised Liouville time-evolution equation that applies to quantum and classical Hamiltonian mechanics.
  arXiv  Detail & Related papers  (2024-02-29T09:04:13Z)
• Independent-oscillator model and the quantum Langevin equation for an oscillator: A review [19.372542786476803]
  A derivation of the quantum Langevin equation is outlined based on the microscopic model of the heat bath. In the steady state, we analyze the quantum counterpart of energy equipartition theorem. The free energy, entropy, specific heat, and third law of thermodynamics are discussed for one-dimensional quantum Brownian motion.
  arXiv  Detail & Related papers  (2023-06-05T07:59:35Z)
• Entanglement in harmonic systems at coherent states [0.0]
  We study entanglement in the toy model of harmonic systems lying at any classicalmost state. We specify the reduced density matrix and its time-evolution, as well as the modular Hamiltonian.
  arXiv  Detail & Related papers  (2022-06-12T16:25:27Z)
• Correspondence Between the Energy Equipartition Theorem in Classical Mechanics and its Phase-Space Formulation in Quantum Mechanics [62.997667081978825]
  In quantum mechanics, the energy per degree of freedom is not equally distributed. We show that in the high-temperature regime, the classical result is recovered.
  arXiv  Detail & Related papers  (2022-05-24T20:51:03Z)
• Unraveling the topology of dissipative quantum systems [58.720142291102135]
  We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. We show for a broad family of translation-invariant collapse models that the set of dark state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hamiltonians.
  arXiv  Detail & Related papers  (2020-07-12T11:26:02Z)
• Emergence of classical behavior in the early universe [68.8204255655161]
  Three notions are often assumed to be essentially equivalent, representing different facets of the same phenomenon. We analyze them in general Friedmann-Lemaitre- Robertson-Walker space-times through the lens of geometric structures on the classical phase space. The analysis shows that: (i) inflation does not play an essential role; classical behavior can emerge much more generally; (ii) the three notions are conceptually distinct; classicality can emerge in one sense but not in another.
  arXiv  Detail & Related papers  (2020-04-22T16:38:25Z)
• Dynamically encircling an exceptional point in a real quantum system [13.510562179346167]
  The exceptional point, known as the non-Hermitian degeneracy, has special topological structure. Here we experimentally demonstrate dynamically encircling the exceptional point with a single nitrogen-vacancy center in diamond. Our work reveals the topological structure of the exceptional point and paves the way to comprehensively explore the exotic properties of non-Hermitian Hamiltonians in the quantum regime.
  arXiv  Detail & Related papers  (2020-02-17T06:41:17Z)
• The Vertical Logic of Hamiltonian Methods (Part 1) [0.0]
  We discuss the key role that Hamiltonian notions play in physics. Five examples are given that illustrate the versatility and generality of Hamiltonian notions.
  arXiv  Detail & Related papers  (2020-01-10T13:08:17Z)

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.