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

International journal of high school research

Analysis and Discussion of Solutions to the Schrödinger Equation Under a Sawtooth Potential

Tao zhixuan

Published Jan 29, 2026• 0 citations
Publisher

Abstract

Sawtooth potentials-piecewise-linear potentials defined by alternating rising and falling slopes-have garnered interest for their roles in classical Brownian ratchets, quantum transport, and flat bands in ultracold-atom lattices.However, a general solution of the time-independent Schrödinger equation for an arbitrary asymmetric sawtooth potential and a study of its band structure remain absent from the literature.We derive the exact analytical eigenstates in terms of Airy functions, but due to numerical difficulty, we solved for the eigenstates using a piecewise constant "staircase approximation."We find localized low-energy states confined within individual wells and plane waves for high energies.We apply the obtained solutions to investigate the relationship between bandwidth and parameters of the sawtooth potentials, which could offer design principles for realizing flatbands in condensed-matter and ultracold-atom research.We find that the bandwidth of the lowest energy band is entirely independent of the sawtooth's asymmetry, while it remains inversely correlated with potential height and cell size.The methodologies presented here provide a toolkit for further exploration of sawtooth potentials.

DOI
10.36838/v8i2.7

Related papers

One-dimensional sawtooth and zigzag lattices for ultracold atoms

Ting Zhang, Gyu-Boong Jo

We describe tunable optical sawtooth and zigzag lattices for ultracold atoms. Making use of the superlattice generated by commensurate wavelengths of light beams, tunable geometries including zigzag and sawtooth configurations can be realised. We provide an experimentally feasible method to fully control inter- (t) and intra- (t') unit-cell tunnelling in zigzag and sawtooth lattices. We analyse the conversion of the lattice geometry from zigzag to sawtooth, and show that a nearly flat band is attainable in the sawtooth configuration by means of tuning the lattice parameters. The bandwidth of the first excited band can be reduced up to 2% of the ground bandwidth for a wide range of lattice setting. A nearly flat band available in a tunable sawtooth lattice would offer a versatile platform for the study of interaction-driven quantum many-body states with ultracold atoms.

arXiv:1510.02015•Scientific Reports•Nov 4, 2015•68 citations

Exact solution of the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential barrier

Wayne W. Lui, Masao Fukuma

A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.

Journal of Applied Physics•Sep 1, 1986•208 citations

Solving Two-Dimensional Schrödinger Equation for a Double Well Potential Using Analytical and Variational Methods

Kathryn Chain, Dylan Herrmann, Julio Navas

The Schrödinger equation serves as a pivotal differential equation, capturing the fundamental duality of particles as both matter and waves. Upon solving it, we derive a particle's wave function within a given potential, delineating the confines within which the particle operates. This wave function enables us to ascertain the "most probable" values for parameters such as position, momentum, and energy within the specified potential. In this study, we focus on the double-well potential in two dimensions, resembling a bowl-like structure. Specifically tailored equations, termed Equation 1 and Equation 2, stem from the general Schrödinger equation when applied to the prescribed conditions. This research aims to utilize these equations to describe a particle’s behavior within the ground and the first excited states of the double well potential, employing two distinct methodologies. The first methodology involves leveraging the variational method on Equations 1 and 2 to obtain approximate solutions, providing crucial inputs for our subsequent approach. The second methodology serves as entails an analytical solution of Equations 1 and 2. A comparative analysis of the outcomes from both methods is anticipated to ensure the coherence and validity of the variational method. This project integrates key principles and proficiencies from both physics and mathematics disciplines, offering a platform for honing practical skills in applying these methodologies within a real-world context.

•Mar 22, 2024•0 citations

Wave Function Penetration, Tunneling, and Quantum Wells

Uri Peskin

Scanning tunneling microscopy is introduced as an analytic tool for characterizing nanostructures and dynamics on the nanoscale. To analyze the underlying tunneling phenomenon, the Schrödinger equation is solved for a particle confined between finite potential energy steps. For a single potential energy well, the proper solutions to the Schrödinger equation are similar to those of a particle in an infinite box, except for the emerging “tails” of the stationary wave functions in the “classically forbidden” regions of the external confining potentials. For a symmetric double well potential, wave function penetration of the stationary solutions manifests in their even distribution among the wells. An attempt to localize the particle in one of them would lead to periodic oscillations between the wells, as if the particle can “tunnel” under the separating energy barrier. The wave function penetration and tunneling phenomena are discussed also in relation to energy band formation in periodic lattices.

Cambridge University Press eBooks•May 11, 2023•0 citations

An Exact Analytical Method for Arbitrary Matter-Waveform Generation in Optical Lattices

Pradosh Basu, Sriganapathy Raghav, Utpal Roy

Generation and control of waveforms is a versatile tool for advanced technologies. Quantum arbitrary waveform generator is also in place by exploiting the quantum state of coherent light. Motivated by the high coherence and tunability of the trapped ultracold matter waves, which may offer a platform for arbitrary waveform generation with atoms, we present an exact analytical approach for designing the first atomic arbitrary waveform generator. Our method considers the dynamics of an one-dimensional Bose-Einstein condensate in a quantum simulation environment with a moving bichromatic optical lattice. Functional expressions and parameter regimes are derived for engineering some of the fundamental waveforms, including sinusoidal, sawtooth, triangular and square profiles. In addition, compound waveforms, such as step-sawtooth, triangular-square, slanted-square, and slanted-step-square are also constructed. A numerical stability analysis is explicated in support of the experimental feasibility of the reported atomic waveform generator.

IEEE Journal of Quantum Electronics•Dec 22, 2025•0 citations

General solution of the Schrödinger equation with potential field quantization

Hasan Hüseyin Erbil

A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.

arXiv:1712.00610•TURKISH JOURNAL OF PHYSICS•Oct 12, 2018•2 citations
Sawtooth potentials-piecewise-linear potentials defined by alternating rising and falling slopes-have garnered interest for their roles in classical Brownian ratchets, quantum transport, and flat bands in ultracold-atom lattices.However, a general solution of the time-independent Schrödinger equation for an arbitrary asymmetric sawtooth potential and a study of its band structure remain absent from the literature.We derive the exact analytical eigenstates in terms of Airy functions, but due to numerical difficulty, we solved for the eigenstates using a piecewise constant "staircase approximation."We find localized low-energy states confined within individual wells and plane waves for high energies.We apply the obtained solutions to investigate the relationship between bandwidth and parameters of the sawtooth potentials, which could offer design principles for realizing flatbands in condensed-matter and ultracold-atom research.We find that the bandwidth of the lowest energy band is entirely independent of the sawtooth's asymmetry, while it remains inversely correlated with potential height and cell size.The methodologies presented here provide a toolkit for further exploration of sawtooth potentials.
We describe tunable optical sawtooth and zigzag lattices for ultracold atoms. Making use of the superlattice generated by commensurate wavelengths of light beams, tunable geometries including zigzag and sawtooth configurations can be realised. We provide an experimentally feasible method to fully control inter- (t) and intra- (t') unit-cell tunnelling in zigzag and sawtooth lattices. We analyse the conversion of the lattice geometry from zigzag to sawtooth, and show that a nearly flat band is attainable in the sawtooth configuration by means of tuning the lattice parameters. The bandwidth of the first excited band can be reduced up to 2% of the ground bandwidth for a wide range of lattice setting. A nearly flat band available in a tunable sawtooth lattice would offer a versatile platform for the study of interaction-driven quantum many-body states with ultracold atoms.
A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.
The Schrödinger equation serves as a pivotal differential equation, capturing the fundamental duality of particles as both matter and waves. Upon solving it, we derive a particle's wave function within a given potential, delineating the confines within which the particle operates. This wave function enables us to ascertain the "most probable" values for parameters such as position, momentum, and energy within the specified potential. In this study, we focus on the double-well potential in two dimensions, resembling a bowl-like structure. Specifically tailored equations, termed Equation 1 and Equation 2, stem from the general Schrödinger equation when applied to the prescribed conditions. This research aims to utilize these equations to describe a particle’s behavior within the ground and the first excited states of the double well potential, employing two distinct methodologies. The first methodology involves leveraging the variational method on Equations 1 and 2 to obtain approximate solutions, providing crucial inputs for our subsequent approach. The second methodology serves as entails an analytical solution of Equations 1 and 2. A comparative analysis of the outcomes from both methods is anticipated to ensure the coherence and validity of the variational method. This project integrates key principles and proficiencies from both physics and mathematics disciplines, offering a platform for honing practical skills in applying these methodologies within a real-world context.
Scanning tunneling microscopy is introduced as an analytic tool for characterizing nanostructures and dynamics on the nanoscale. To analyze the underlying tunneling phenomenon, the Schrödinger equation is solved for a particle confined between finite potential energy steps. For a single potential energy well, the proper solutions to the Schrödinger equation are similar to those of a particle in an infinite box, except for the emerging “tails” of the stationary wave functions in the “classically forbidden” regions of the external confining potentials. For a symmetric double well potential, wave function penetration of the stationary solutions manifests in their even distribution among the wells. An attempt to localize the particle in one of them would lead to periodic oscillations between the wells, as if the particle can “tunnel” under the separating energy barrier. The wave function penetration and tunneling phenomena are discussed also in relation to energy band formation in periodic lattices.
Generation and control of waveforms is a versatile tool for advanced technologies. Quantum arbitrary waveform generator is also in place by exploiting the quantum state of coherent light. Motivated by the high coherence and tunability of the trapped ultracold matter waves, which may offer a platform for arbitrary waveform generation with atoms, we present an exact analytical approach for designing the first atomic arbitrary waveform generator. Our method considers the dynamics of an one-dimensional Bose-Einstein condensate in a quantum simulation environment with a moving bichromatic optical lattice. Functional expressions and parameter regimes are derived for engineering some of the fundamental waveforms, including sinusoidal, sawtooth, triangular and square profiles. In addition, compound waveforms, such as step-sawtooth, triangular-square, slanted-square, and slanted-step-square are also constructed. A numerical stability analysis is explicated in support of the experimental feasibility of the reported atomic waveform generator.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.