Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2109.08033 · Physics Letters B

Bootstrap method in harmonic oscillator

Yu Aikawa, Takeshi Morita, Kota Yoshimura

Published Jul 14, 2022• 20 citations
arXivHTMLPDF

Abstract

Recently, an application of the numerical bootstrap method to quantum mechanics was proposed, and it successfully reproduces the eigenstates of various systems. However, it is unclear why this method works. In order to understand this question, we study the bootstrap method in harmonic oscillators. We find that the problem reduces to the Dirac's ladder operator problem and is exactly solvable analytically. Our result suggests that the bootstrap method may be regarded as a numerical version of the Dirac's approach and it may explain why it works in various systems.

DOI
10.1016/j.physletb.2022.137305

Related papers

Quantum Bootstrap in Microcanonical Ensembles: Computational Insights and Applications

Shujaatali Badami

This paper explores the quantum bootstrap method, a promising numerical approach rooted in self-consistency conditions, and its application in microcanonical ensembles. Initially developed in the 1960s and ’70s, the bootstrap technique has found applications in various fields, including quantum mechanics, lattice theory, conformal field theory, and matrix models. While extensively studied in quantum mechanics, this paper introduces a classical correspondence to the quantum bootstrap method, expanding its applicability to microcanonical ensembles. The study incorporates a novel approach, demonstrating stronger constraints by integrating additional information in phase space. Numerical examples, including the double well potential, harmonic oscillator, Coulomb potential, and non-relativistic Toda model, illustrate the method’s effectiveness in resolving unphysical energy regions and improving computational precision. These results highlight its potential as a powerful tool in computational physics and quantum computing.

International Conference on Pervasive Computational Technologies (ICPCT)•Feb 8, 2025•3 citations

Application of bootstrap to a θ term

Yu Aikawa, Takeshi Morita, Kota Yoshimura

Recently, a novel numerical computation on quantum mechanics by using a bootstrap method was proposed by Han, Hartnoll, and Kruthoff. We consider whether this method works in systems with a $\ensuremath{\theta}$-term, where the standard Monte-Carlo computation may fail due to the sign problem. As a starting point, we study quantum mechanics of a charged particle on a circle in which a constant gauge potential is a counterpart of a $\ensuremath{\theta}$-term. We find that it is hard to determine physical quantities as functions of $\ensuremath{\theta}$ such as $E(\ensuremath{\theta})$, except at $\ensuremath{\theta}=0$ and $\ensuremath{\pi}$. On the other hand, the correlations among observables for energy eigenstates are correctly reproduced for any $\ensuremath{\theta}$. Our results suggest that the bootstrap method may work not perfectly but sufficiently well, even if a $\ensuremath{\theta}$-term exists in the system.

arXiv:2109.02701•Physical review. D/Physical review. D.•Apr 25, 2022•25 citations

Unified Representation of Quantum Mechanics on One-dimensional Harmonic Oscillator

Yongqin Wang

A quantum state corresponds to a specific wave function. We adopt a new mathematical method [1] to improve Dirac's ladder operator method. A set of orthonormal wave functions will be used to associate the operator with the square matrix corresponding to it. These allow us to determine the matrix elements by using the operator relations without having to know the specific wave functions. As a result, we can get the direct results of matrix mechanics and wave mechanics on one-dimensional Harmonic oscillator and their descriptions will be also unified.

•Mar 7, 2013•1 citation

A Non-relativistic Approach to Relativistic Quantum Mechanics: The Case of the Harmonic Oscillator

Luis A. Poveda, Luis Grave de Peralta, Jacob Pittman +1 more

Abstract A recently proposed approach to relativistic quantum mechanics (Grave de Peralta, Poveda, Poirier in Eur J Phys 42:055404, 2021) is applied to the problem of a particle in a quadratic potential. The methods, both exact and approximate, allow one to obtain eigenstate energy levels and wavefunctions, using conventional numerical eigensolvers applied to Schrödinger-like equations. Results are obtained over a nine-order-of-magnitude variation of system parameters, ranging from the non-relativistic to the ultrarelativistic limits. Various trends are analyzed and discussed—some of which might have been easily predicted, others which may be a bit more surprising.

Foundations of Physics•Feb 1, 2022•11 citations

The Harmonic Oscillator–A Simplified Approach

L. R. Ganesan, M. Balaji

Among the early problems in quantum chemistry, the one dimensional harmonic oscillator problem is an important one, providing a valuable exercise in the study of quantum mechanical methods. There are several approaches to this problem, the time honoured infinite series method, the ladder operator method etc. A method which is much shorter, mathematically simpler is presented here.

Journal of Chemistry•Jan 1, 2008•0 citations

Bootstrap methods for digitized scalar field theory

Zane Ozzello, Yannick Meurice

General positivity constraints linking various powers of observables in energy eigenstates can be used to sharply locate acceptable regions for the energy eigenvalues, provided that efficient recursive methods are available to calculate the matrix elements. These recursive methods are derived by looking at the commutation relations of the observables with the Hamiltonian. We discuss how this self-consistent (bootstrap) approach can be applied to the study of digitized scalar field theory in the harmonic basis. Using known results, we develop the method by testing on quantum systems, including the harmonic and anharmonic oscillators. We report recent numerical results for up to four coupled anharmonic oscillators. From here, we consider the possibility of using the groundwork of this method as a means of studying phase transitions in 1+1 dimensions.

arXiv:2312.10559•International Symposium on Lattice Field Theory (LATTICE)•Dec 27, 2023•1 citation
Recently, an application of the numerical bootstrap method to quantum mechanics was proposed, and it successfully reproduces the eigenstates of various systems. However, it is unclear why this method works. In order to understand this question, we study the bootstrap method in harmonic oscillators. We find that the problem reduces to the Dirac's ladder operator problem and is exactly solvable analytically. Our result suggests that the bootstrap method may be regarded as a numerical version of the Dirac's approach and it may explain why it works in various systems.
This paper explores the quantum bootstrap method, a promising numerical approach rooted in self-consistency conditions, and its application in microcanonical ensembles. Initially developed in the 1960s and ’70s, the bootstrap technique has found applications in various fields, including quantum mechanics, lattice theory, conformal field theory, and matrix models. While extensively studied in quantum mechanics, this paper introduces a classical correspondence to the quantum bootstrap method, expanding its applicability to microcanonical ensembles. The study incorporates a novel approach, demonstrating stronger constraints by integrating additional information in phase space. Numerical examples, including the double well potential, harmonic oscillator, Coulomb potential, and non-relativistic Toda model, illustrate the method’s effectiveness in resolving unphysical energy regions and improving computational precision. These results highlight its potential as a powerful tool in computational physics and quantum computing.
Recently, a novel numerical computation on quantum mechanics by using a bootstrap method was proposed by Han, Hartnoll, and Kruthoff. We consider whether this method works in systems with a $\ensuremath{\theta}$-term, where the standard Monte-Carlo computation may fail due to the sign problem. As a starting point, we study quantum mechanics of a charged particle on a circle in which a constant gauge potential is a counterpart of a $\ensuremath{\theta}$-term. We find that it is hard to determine physical quantities as functions of $\ensuremath{\theta}$ such as $E(\ensuremath{\theta})$, except at $\ensuremath{\theta}=0$ and $\ensuremath{\pi}$. On the other hand, the correlations among observables for energy eigenstates are correctly reproduced for any $\ensuremath{\theta}$. Our results suggest that the bootstrap method may work not perfectly but sufficiently well, even if a $\ensuremath{\theta}$-term exists in the system.
A quantum state corresponds to a specific wave function. We adopt a new mathematical method [1] to improve Dirac's ladder operator method. A set of orthonormal wave functions will be used to associate the operator with the square matrix corresponding to it. These allow us to determine the matrix elements by using the operator relations without having to know the specific wave functions. As a result, we can get the direct results of matrix mechanics and wave mechanics on one-dimensional Harmonic oscillator and their descriptions will be also unified.
Abstract A recently proposed approach to relativistic quantum mechanics (Grave de Peralta, Poveda, Poirier in Eur J Phys 42:055404, 2021) is applied to the problem of a particle in a quadratic potential. The methods, both exact and approximate, allow one to obtain eigenstate energy levels and wavefunctions, using conventional numerical eigensolvers applied to Schrödinger-like equations. Results are obtained over a nine-order-of-magnitude variation of system parameters, ranging from the non-relativistic to the ultrarelativistic limits. Various trends are analyzed and discussed—some of which might have been easily predicted, others which may be a bit more surprising.
General positivity constraints linking various powers of observables in energy eigenstates can be used to sharply locate acceptable regions for the energy eigenvalues, provided that efficient recursive methods are available to calculate the matrix elements. These recursive methods are derived by looking at the commutation relations of the observables with the Hamiltonian. We discuss how this self-consistent (bootstrap) approach can be applied to the study of digitized scalar field theory in the harmonic basis. Using known results, we develop the method by testing on quantum systems, including the harmonic and anharmonic oscillators. We report recent numerical results for up to four coupled anharmonic oscillators. From here, we consider the possibility of using the groundwork of this method as a means of studying phase transitions in 1+1 dimensions.