A Proof of Generalized Kramers-Pasternack Relation using Hyper-Radial Equation

• URL: http://arxiv.org/abs/2502.19462v1
• Date: Wed, 26 Feb 2025 13:06:28 GMT
• Title: A Proof of Generalized Kramers-Pasternack Relation using Hyper-Radial Equation
• Authors: Avoy Jana,
• Abstract summary: We present a proof of the generalized Kramers-Pasternack relation using the hyper-radial equation approach.<n>This approach provides deeper insights into the structure of recurrence relations for diagonal radial matrix elements in quantum mechanics.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present a proof of the generalized Kramers-Pasternack relation using the hyper-radial equation approach. Following Kramers' method, we manipulate the radial equation by multiplying it with an expression closely related to terms in the hyper-virial theorem. Through successive integrations by parts, we systematically derive the second Pasternack formula, extending its validity to arbitrary dimensions. This approach provides deeper insights into the algebraic structure of recurrence relations for diagonal radial matrix elements in quantum mechanics.

Related papers

• Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow [44.500314182711236]
  We derive explicit ordinary equations govern the evolution of mean, covariance, and mass under HK-Boltzmann flow. We demonstrate exponential convergence to equilibrium through constructing a gradient structure associated with a Polyak-Lojasiewicz-type inequalities. We extend our analysis to non-Gaussian targets exhibiting strong log-lambda-concavity.
  arXiv  Detail & Related papers  (2025-04-29T03:54:56Z)
• Leading correction to the relativistic Foldy-Wouthuysen Hamiltonian [55.2480439325792]
  We rigorously derive a leading correction in the weak-field approximation to the known relativistic Foldy-Wouthuysen Hamiltonian. For Dirac particles, the relativistic wave equation of the second order is obtained with the correction similar to that to the Foldy-Wouthuysen Hamiltonian.
  arXiv  Detail & Related papers  (2024-08-03T12:53:41Z)
• On the use of complex GTOs for the evaluation of radial integrals involving oscillating functions [0.0]
  We study two classes of radial integrals involving a product of bound and continuum one-electron states. We investigate the reliability of this scheme for low-energy physical parameters. This study serves as a premise in view of potential applications in molecular scattering processes.
  arXiv  Detail & Related papers  (2024-05-16T12:15:02Z)
• Tensor product random matrix theory [39.58317527488534]
  We introduce a real-time field theory approach to the evolution of correlated quantum systems. We describe the full range of such crossover dynamics, from initial product states to a maximum entropy ergodic state.
  arXiv  Detail & Related papers  (2024-04-16T21:40:57Z)
• Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations [77.34726150561087]
  We reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ''wavefunctions'' of the density matrix.
  arXiv  Detail & Related papers  (2023-02-27T18:56:40Z)
• Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning [24.97120654216651]
  We show how to solve difficult differential equations with momentum on a submanifold. We do so by proposing a generalized version of the Riemannian normal coordinates. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free $2textnd$orders for deep learning with low precision by using only matrix multiplications.
  arXiv  Detail & Related papers  (2023-02-20T03:31:11Z)
• Calculation of the wave functions of a quantum asymmetric top using the noncommutative integration method [0.0]
  We obtain a complete set of solutions to the Schrodinger equation for a quantum asymmetric top in Euler angles. The spectrum of an asymmetric top is obtained from the condition that the solutions are in with respect to a special irreducible $lambda$-representation of the rotation group.
  arXiv  Detail & Related papers  (2022-11-27T12:38:22Z)
• Quantum-based solution of time-dependent complex Riccati equations [0.0]
  We show a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE. As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation.
  arXiv  Detail & Related papers  (2022-09-07T23:52:04Z)
• One bound to rule them all: from Adiabatic to Zeno [0.0]
  We derive a universal nonperturbative bound on the distance between unitary evolutions generated by time-dependent Hamiltonians. We discuss the error of the rotating-wave approximation over long time and in the presence of time-dependent amplitude modulation.
  arXiv  Detail & Related papers  (2021-11-17T08:14:21Z)
• Conformal field theory from lattice fermions [77.34726150561087]
  We provide a rigorous lattice approximation of conformal field theories given in terms of lattice fermions in 1+1-dimensions. We show how these results lead to explicit error estimates pertaining to the quantum simulation of conformal field theories.
  arXiv  Detail & Related papers  (2021-07-29T08:54:07Z)
• Dissipative flow equations [62.997667081978825]
  We generalize the theory of flow equations to open quantum systems focusing on Lindblad master equations. We first test our dissipative flow equations on a generic matrix and on a physical problem with a driven-dissipative single fermionic mode.
  arXiv  Detail & Related papers  (2020-07-23T14:47:17Z)
• Algebraic derivation of Kramers-Pasternack relations based on the Schrodinger factorization method [0.0]
  Kramers-Pasternack relations are used to compute the moments of r for all radial energy eigenfunctions of hydrogenic atoms. Most derivations employ the Feynman-Hellman theorem or a brute-force integration to determine the second inverse moment.
  arXiv  Detail & Related papers  (2020-07-22T01:55:09Z)
• From stochastic spin chains to quantum Kardar-Parisi-Zhang dynamics [68.8204255655161]
  We introduce the asymmetric extension of the Quantum Symmetric Simple Exclusion Process. We show that the time-integrated current of fermions defines a height field which exhibits a quantum non-linear dynamics.
  arXiv  Detail & Related papers  (2020-01-13T14:30:36Z)
• Deep reinforcement learning for complex evaluation of one-loop diagrams in quantum field theory [0.0]
  We present a technique that allows for numerical analytic continuation of integrals encountered in one-loop diagrams in quantum field theory. We train a reinforcement learning agent to perform the required contour deformations. Our study shows great promise for an agent to be deployed in iterative numerical approaches used to compute non-perturbative 2-point functions.
  arXiv  Detail & Related papers  (2019-12-27T19:45:24Z)

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.