Variational approach to the Schr\"odinger equation with a delta-function potential

• URL: http://arxiv.org/abs/2106.10231v1
• Date: Thu, 17 Jun 2021 13:05:33 GMT
• Title: Variational approach to the Schr\"odinger equation with a delta-function potential
• Authors: Francisco M. Fern\'andez
• Abstract summary: We obtain accurate eigenvalues of the one-dimensional Schr"odinger equation with a Hamiltonian of the form $H_g=H+gdelta (x)$. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set account takes into the effect of the Dirac delta on the wavefunction.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We obtain accurate eigenvalues of the one-dimensional Schr\"odinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction.

Related papers

• Quantum simulation of the Fokker-Planck equation via Schrodingerization [33.76659022113328]
  This paper studies a quantum simulation technique for solving the Fokker-Planck equation. We employ the Schrodingerization method-it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrodinger-type equations.
  arXiv  Detail & Related papers  (2024-04-21T08:53:27Z)
• Particle-based Variational Inference with Generalized Wasserstein Gradient Flow [32.37056212527921]
  We propose a ParVI framework, called generalized Wasserstein gradient descent (GWG) We show that GWG exhibits strong convergence guarantees. We also provide an adaptive version that automatically chooses Wasserstein metric to accelerate convergence.
  arXiv  Detail & Related papers  (2023-10-25T10:05:42Z)
• Quantum simulation of partial differential equations via Schrodingerisation: technical details [31.986350313948435]
  We study a new method - called Schrodingerisation introduced in [Jin, Liu, Yu, arXiv: 2212.13969] - for solving general linear partial differential equations with quantum simulation. This method converts linear partial differential equations into a Schrodingerised' or Hamiltonian system, using a new and simple transformation called the warped phase transformation. We apply this to more examples of partial differential equations, including heat, convection, Fokker-Planck, linear Boltzmann and Black-Scholes equations.
  arXiv  Detail & Related papers  (2022-12-30T13:47:35Z)
• Few-cycle excitation of atomic coherence: A closed-form analytical solution beyond the rotating-wave approximation [21.82044445509816]
  We present an approximate, closed-form solution to the Schrodinger equation that describes a two-level atom under the excitation of a far-off-resonance, few-cycle pulse of arbitrary shape without invoking the rotating wave approximation (RWA) We outline an alternative approach that can lead to a more accurate solution by capturing the nonlinear behaviors of the system.
  arXiv  Detail & Related papers  (2021-11-27T18:55:57Z)
• Out-of-equilibrium dynamics of the Kitaev model on the Bethe lattice via coupled Heisenberg equations [23.87373187143897]
  We study the isotropic Kitaev spin-$1/2$ model on the Bethe lattice. We take a straightforward approach of solving Heisenberg equations for a tailored subset of spin operators. As an example, we calculate the time-dependent expectation value of this observable for a factorized translation-invariant.
  arXiv  Detail & Related papers  (2021-10-25T17:37:33Z)
• Scattering solution of Schr\"odinger equation with $\delta$-potential in deformed space with minimal length [0.0]
  We consider the Dirac $delta$-function potential problem in general case of deformed Heisenberg algebra leading to the minimal length. For some resonance energy the incident wave is completely reflected.
  arXiv  Detail & Related papers  (2021-10-24T17:27:56Z)
• Regularization of $\delta'$ potential in general case of deformed space with minimal length [0.0]
  We find exactly the energy level and corresponding eigenfunction for $delta'(x)$ and $delta(x)-delta'(x)$ potentials in deformed space with arbitrary function of deformation.
  arXiv  Detail & Related papers  (2021-08-25T05:27:58Z)
• Machine Learning S-Wave Scattering Phase Shifts Bypassing the Radial Schr\"odinger Equation [77.34726150561087]
  We present a proof of concept machine learning model resting on a convolutional neural network capable to yield accurate scattering s-wave phase shifts. We discuss how the Hamiltonian can serve as a guiding principle in the construction of a physically-motivated descriptor.
  arXiv  Detail & Related papers  (2021-06-25T17:25:38Z)
• The Energy Eigenvalue for the Singular Wave Function of the Three Dimensional Dirac Delta Schrodinger Potential via Distributionally Generalized Quantum Mechanics [0.0]
  Indeterminacy originates from a lack of scale. Wave function not being well defined at the support of the generalized function. Problem is solved here in a completely mathematically rigorous manner.
  arXiv  Detail & Related papers  (2021-01-18T09:07:31Z)
• The variational method applied to the harmonic oscillator in presence of a delta function potential [0.0]
  We show that the eigenfunctions obtained exactly are difficult to visualise and hence to gain more insight. We apply the variational method to verify how close one can approach the exact ground state eigenvalues. We obtain the estimates of the ground state energies which are closer to the exact values in comparison to earlier approximate results for both the repulsive and attractive delta potentials.
  arXiv  Detail & Related papers  (2020-12-01T15:05:01Z)
• Physics-inspired forms of the Bayesian Cram\'er-Rao bound [0.0]
  I find the optimal and naturally invariant bound among the Gill-Levit family of bounds. The problem of finding an unfavorable prior to tighten the bound for minimax estimation is shown. Two quantum estimation problems, namely, optomechanical waveform estimation and subdiffraction incoherent optical imaging, are discussed.
  arXiv  Detail & Related papers  (2020-07-09T14:53:27Z)

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.