The envelope theory as a pedagogical tool
- URL: http://arxiv.org/abs/2211.06306v2
- Date: Mon, 24 Apr 2023 09:16:42 GMT
- Title: The envelope theory as a pedagogical tool
- Authors: Claude Semay, Maud Balcaen
- Abstract summary: The envelope theory is a reliable and easy to implement method to solve time independent Schr"odinger-like equations.
It is particularly useful to solve many-body systems since the computational cost is independent from the number of particles.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: The envelope theory is a reliable and easy to implement method to solve time
independent Schr\"odinger-like equations (eigenvalues and eigenvectors). It is
particularly useful to solve many-body systems since the computational cost is
independent from the number of particles. The purpose of this paper is twofold.
First, we want to make known a method that is probably too little used. Second,
we also want to show that this method can be used as a pedagogical tool, thanks
to its simplicity and the reliable results that can be obtained. To reach these
goals, the envelope theory is applied to a simple problem in one dimension, the
soft-Coulomb potential $-k/\sqrt{x^2+d^2}$, characterised by a bias distance
$d$. Such interaction is used for the study of excitons, electron-hole bound
pairs where the two charges are kept separated in two different one-dimensional
regions (quantum wires). In addition to its physical interest, this system has
never been treated with the envelope theory.
Related papers
- Confinement of $N$-body systems and non-integer dimensions [0.0]
The squeezing process of a three-dimensional quantum system can be described by the $d$-method, without external field and where the dimension can take non-integer values.
We first generalize both methods to $N$ particles and any transition between dimensions below $3$.
We have in all the cases found that the derived analytic relations between the two methods work very well.
arXiv Detail & Related papers (2024-03-11T08:52:48Z) - 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) - A single $T$-gate makes distribution learning hard [56.045224655472865]
This work provides an extensive characterization of the learnability of the output distributions of local quantum circuits.
We show that for a wide variety of the most practically relevant learning algorithms -- including hybrid-quantum classical algorithms -- even the generative modelling problem associated with depth $d=omega(log(n))$ Clifford circuits is hard.
arXiv Detail & Related papers (2022-07-07T08:04:15Z) - An Introduction to Scattering Theory [0.0]
Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time domain.
Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain.
Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism)
arXiv Detail & Related papers (2022-04-08T11:41:24Z) - Compact equations for the envelope theory [0.0]
The envelope theory is a method to obtain approximate, but reliable, solutions for some quantum many-body problems.
Quite general Hamiltonians can be considered for systems composed of an arbitrary number of different particles in $D$ dimensions.
arXiv Detail & Related papers (2021-08-12T13:04:23Z) - Manifold Hypothesis in Data Analysis: Double Geometrically-Probabilistic
Approach to Manifold Dimension Estimation [92.81218653234669]
We present new approach to manifold hypothesis checking and underlying manifold dimension estimation.
Our geometrical method is a modification for sparse data of a well-known box-counting algorithm for Minkowski dimension calculation.
Experiments on real datasets show that the suggested approach based on two methods combination is powerful and effective.
arXiv Detail & Related papers (2021-07-08T15:35:54Z) - Quantum double aspects of surface code models [77.34726150561087]
We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry.
We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
arXiv Detail & Related papers (2021-06-25T17:03:38Z) - Random quantum circuits anti-concentrate in log depth [118.18170052022323]
We study the number of gates needed for the distribution over measurement outcomes for typical circuit instances to be anti-concentrated.
Our definition of anti-concentration is that the expected collision probability is only a constant factor larger than if the distribution were uniform.
In both the case where the gates are nearest-neighbor on a 1D ring and the case where gates are long-range, we show $O(n log(n)) gates are also sufficient.
arXiv Detail & Related papers (2020-11-24T18:44:57Z) - Mapping the charge-dyon system into the position-dependent effective
mass background via Pauli equation [77.34726150561087]
This work aims to reproduce a quantum system composed of a charged spin - $1/2$ fermion interacting with a dyon with an opposite electrical charge.
arXiv Detail & Related papers (2020-11-01T14:38:34Z) - The periodically driven electron in a quantum well with two
characteristic curvatures -- redux [0.0]
I develop the solution to the problem of an electron confined in a composite quadratic well subject to a simple, external periodic force.
The method of solution illustrates several of the basic techniques useful in formally solving the one-dimensional, time-dependent Schrodinger equation.
arXiv Detail & Related papers (2020-10-20T17:26:04Z) - Schrodinger's original quantum-mechanical solution for hydrogen [0.0]
In his first paper, Erwin Schrodinger solved the Schrodinger equation using the Laplace method.
We show how the Laplace method can be used to solve for the quantum-mechanical energy eigenfunctions of the hydrogen atom.
arXiv Detail & Related papers (2020-07-24T21:23:26Z)
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.