Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

3.8K
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
3.8K
Calculations of Electric Potential II01:27

Calculations of Electric Potential II

2.1K
An electric dipole is a system of two equal but opposite charges, separated by a fixed distance. This system is used to model many real-world systems, including atomic and molecular interactions. One of these systems is the water molecule, but only under certain circumstances. These circumstances are met inside a microwave oven, where electric fields with alternating directions make the water molecules change orientation. This vibration is equivalent to heat at the molecular level.
Consider a...
2.1K
Equipotential Surfaces and Field Lines01:29

Equipotential Surfaces and Field Lines

4.5K
Electric potential can be pictorially represented as a three-dimensional surface. On such a surface, the electric potential is constant everywhere. The equipotential surface is always perpendicular to the electric field lines, and while it is three-dimensional, it can be treated as an equipotential line in a two-dimensional case. These equipotential lines are also always perpendicular to electric field lines. The term equipotential is often used as a noun, referring to an equipotential line or...
4.5K
Electric Field of Two Equal and Opposite Charges01:30

Electric Field of Two Equal and Opposite Charges

6.7K
Atoms generally contain the same number of positively and negatively charged particles, protons, and electrons. Hence, they are electrically neutral. However, the centers of the positive and negative charges do not always coincide. In such a scenario, the electric field of an atom may not be zero.
A separation of the positive and negative charges can lead to a weak, remnant effect of the positive and negative charges. The expectation is that the more the distance between the positive and...
6.7K
Electrostatic Boundary Conditions01:16

Electrostatic Boundary Conditions

772
Consider an external electric field propagating through a homogeneous medium. When the electric field crosses the surface boundary of the medium, it undergoes a discontinuity. The electric field can be resolved into normal and tangential components. The amount by which the field changes at any boundary is given by the difference between the field components above and below the surface boundary.
The surface integral of an electric field is given by Gauss's law in integral form and is related to...
772
Potential Due to a Polarized Object01:29

Potential Due to a Polarized Object

602
A neutral atom consists of a positively charged nucleus surrounded by a negatively charged electron cloud. When placed in an external electric field, the external electric force pulls the electrons and nucleus apart, opposite to the intrinsic attraction between the nucleus and the electrons. The opposing forces balance each other with a slight shift between the center of masses of the nucleus and the electron cloud, resulting in a polarized atom. On the other hand, a few molecules, like water,...
602

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

Research on a Regional Availability Evaluation Model for Road-Area High-Entropy Energy Based on Synergy Factors.

Entropy (Basel, Switzerland)·2026
Same journal

Atmospheric Turbulence Channel Modeling and Performance Analysis of a CO-ZP-OFDM Coherent Optical Communication System for UAV Air-to-Ground Scenarios.

Entropy (Basel, Switzerland)·2026
Same journal

Information Geometry and Asymptotic Theory for SMML Estimators.

Entropy (Basel, Switzerland)·2026
Same journal

Correlation Entropy and Power-Law Kinetics.

Entropy (Basel, Switzerland)·2026
Same journal

Research on the Contagion of Systemic Financial Risk Under the Impact of Climate Risks-From the Perspective of Complex Networks and Machine Learning.

Entropy (Basel, Switzerland)·2026
Same journal

The Statistical-Mechanical Meaning of the Wave Function of Quantum Mechanics.

Entropy (Basel, Switzerland)·2026

Related Experiment Video

Updated: Nov 22, 2025

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method
05:51

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method

Published on: July 19, 2019

6.5K

PT -Symmetric Potentials from the Confluent Heun Equation.

Géza Lévai1

  • 1Institute for Nuclear Research (Atomki), P. O. Box 51, H-4001 Debrecen, Hungary.

Entropy (Basel, Switzerland)
|January 6, 2021
PubMed
Summary

Researchers derived exactly solvable potentials with PT symmetry from the confluent Heun equation. The symmetrical form is better for implementing PT symmetry, yielding potentials with tunable terms.

Keywords:
PT symmetryconfluent Heun differential equationsolvable potentials

More Related Videos

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.4K
Setting Limits on Supersymmetry Using Simplified Models
07:46

Setting Limits on Supersymmetry Using Simplified Models

Published on: November 15, 2013

8.8K

Related Experiment Videos

Last Updated: Nov 22, 2025

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method
05:51

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method

Published on: July 19, 2019

6.5K
Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.4K
Setting Limits on Supersymmetry Using Simplified Models
07:46

Setting Limits on Supersymmetry Using Simplified Models

Published on: November 15, 2013

8.8K

Area of Science:

  • Quantum mechanics
  • Mathematical physics

Background:

  • The confluent Heun equation is a complex differential equation with applications in physics.
  • Parity-time (PT) symmetry is a crucial concept in quantum mechanics, leading to unique physical properties.

Purpose of the Study:

  • To derive exactly solvable potentials from the confluent Heun equation.
  • To determine the conditions for these potentials to exhibit PT symmetry.
  • To compare different canonical forms of the Heun equation for PT symmetry implementation.

Main Methods:

  • Formal solutions of the confluent Heun equation were used to construct potentials.
  • Analysis of parameter dependencies for energy, coordinate scaling, variable transformation, and potential terms.
  • Comparison with potentials derived from the non-symmetrical canonical form.

Main Results:

  • Exactly solvable potentials were derived, dependent on twelve parameters.
  • The symmetrical canonical form of the Heun equation is more suitable for PT symmetry.
  • The derived potentials include Natanzon-class potentials as special cases.
  • Conditions for unbroken and broken PT symmetry were identified.

Conclusions:

  • The study provides a systematic method for obtaining PT-symmetric potentials from the confluent Heun equation.
  • The symmetrical canonical form offers advantages for implementing PT symmetry.
  • The derived potentials offer a rich framework for exploring quantum mechanical systems with PT symmetry.