The variational method applied to the harmonic oscillator in presence of
a delta function potential
- URL: http://arxiv.org/abs/2012.00559v2
- Date: Tue, 17 Aug 2021 17:42:44 GMT
- Title: The variational method applied to the harmonic oscillator in presence of
a delta function potential
- Authors: Indrajit Ghose, Parongama Sen
- Abstract summary: 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.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: The problem of the harmonic oscillator with a centrally located delta
function potential can be exactly solved in one dimension where the
eigenfunctions are expressed as superpositions of the Hermite polynomials or as
confluent hypergeometric functions in general. The eigenfunctions obtained
exactly are difficult to visualise and hence to gain more insight, one can
attempt using model wave functions which are explicitly and simply expressed.
Here we apply the variational method to verify how close one can approach the
exact ground state eigenvalues using such trial wave functions. 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.
Related papers
- A mathematical analysis of the adiabatic Dyson equation from
time-dependent density functional theory [0.0]
We analyze the Dyson equation for the density-density response function (DDRF) that plays a central role in linear response time-dependent density functional theory.
We derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix.
We show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial density-density response function and the solution of the Dyson equation are the same.
arXiv Detail & Related papers (2023-05-15T15:46:33Z) - Non-relativistic quantum particles interacting with pseudoharmonic-type
potential under flux field in a topological defect geometry [0.0]
We investigate the quantum motions of non-relativistic particles interacting with a potential in the presence of the Aharonov-Bohm flux field.
Our findings reveal that the eigenvalue solutions are significantly influenced by the topological defect characterized by the parameter $beta$.
This influence manifests as a shift in the energy spectrum, drawing parallels to the gravitational analog of the Aharonov-Bohm effect.
arXiv Detail & Related papers (2023-02-01T17:45:02Z) - Structural aspects of FRG in quantum tunnelling computations [68.8204255655161]
We probe both the unidimensional quartic harmonic oscillator and the double well potential.
Two partial differential equations for the potential V_k(varphi) and the wave function renormalization Z_k(varphi) are studied.
arXiv Detail & Related papers (2022-06-14T15:23:25Z) - Counting Phases and Faces Using Bayesian Thermodynamic Integration [77.34726150561087]
We introduce a new approach to reconstruction of the thermodynamic functions and phase boundaries in two-parametric statistical mechanics systems.
We use the proposed approach to accurately reconstruct the partition functions and phase diagrams of the Ising model and the exactly solvable non-equilibrium TASEP.
arXiv Detail & Related papers (2022-05-18T17:11:23Z) - On the exactly-solvable semi-infinite quantum well of the
non-rectangular step-harmonic profile [0.0]
The model behaves itself as a semi-infinite quantum well of the non-rectangular profile.
We show that wavefunctions of the discrete spectrum recover wavefunctions in terms of the Hermites.
We also present a new limit relation that reduces Bessels directly to Hermites.
arXiv Detail & Related papers (2021-11-07T12:23:17Z) - Deformed Explicitly Correlated Gaussians [58.720142291102135]
Deformed correlated Gaussian basis functions are introduced and their matrix elements are calculated.
These basis functions can be used to solve problems with nonspherical potentials.
arXiv Detail & Related papers (2021-08-10T18:23:06Z) - Uhlmann Fidelity and Fidelity Susceptibility for Integrable Spin Chains
at Finite Temperature: Exact Results [68.8204255655161]
We show that the proper inclusion of the odd parity subspace leads to the enhancement of maximal fidelity susceptibility in the intermediate range of temperatures.
The correct low-temperature behavior is captured by an approximation involving the two lowest many-body energy eigenstates.
arXiv Detail & Related papers (2021-05-11T14:08:02Z) - Leveraging Global Parameters for Flow-based Neural Posterior Estimation [90.21090932619695]
Inferring the parameters of a model based on experimental observations is central to the scientific method.
A particularly challenging setting is when the model is strongly indeterminate, i.e., when distinct sets of parameters yield identical observations.
We present a method for cracking such indeterminacy by exploiting additional information conveyed by an auxiliary set of observations sharing global parameters.
arXiv Detail & Related papers (2021-02-12T12:23:13Z) - Alternative quantisation condition for wavepacket dynamics in a
hyperbolic double well [0.0]
We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width.
Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and quasiprobability distributions.
arXiv Detail & Related papers (2020-09-18T10:29:04Z) - Probing eigenstate thermalization in quantum simulators via
fluctuation-dissipation relations [77.34726150561087]
The eigenstate thermalization hypothesis (ETH) offers a universal mechanism for the approach to equilibrium of closed quantum many-body systems.
Here, we propose a theory-independent route to probe the full ETH in quantum simulators by observing the emergence of fluctuation-dissipation relations.
Our work presents a theory-independent way to characterize thermalization in quantum simulators and paves the way to quantum simulate condensed matter pump-probe experiments.
arXiv Detail & Related papers (2020-07-20T18:00:02Z) - Exponentially confining potential well [0.0]
We introduce an exponentially confining potential well that could be used as a model to describe the structure of a strongly localized system.
We obtain an approximate partial solution of the Schr"odinger equation with this potential well where we find the lowest energy spectrum and corresponding wavefunctions.
arXiv Detail & Related papers (2020-05-15T17:10:35Z)
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.