Related Experiment Video
Updated: Aug 22, 2025
09:23
Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
Published on: May 30, 2014
14.6K
Quantum Information Entropy of Hyperbolic Potentials in Fractional Schrödinger Equation
R Santana-Carrillo1, Jesus S González-Flores1, Emilio Magaña-Espinal1
1Centro de Investigación en Computación, Instituto Politécnico Nacional, UPALM, Ciudad de Mexico 07738, Mexico.
Entropy (Basel, Switzerland)
|November 11, 2022
Summary
This study explores Shannon information entropy in fractional quantum systems. Decreasing fractional derivative number localizes position entropy and delocalizes momentum entropy, while Fisher entropy increases with potential depth.
Area of Science:
- Quantum Mechanics
- Information Theory
- Mathematical Physics
Background:
- The fractional Schrödinger equation extends quantum mechanics to non-integer orders of spatial derivatives.
- Information entropy quantifies uncertainty in quantum systems, with position and momentum entropies being key measures.
- Hyperbolic potentials are relevant in various physical contexts, including atomic and molecular physics.
Purpose of the Study:
- To investigate the behavior of Shannon information entropy for hyperbolic single-well potentials within the fractional Schrödinger equation.
- To analyze the impact of the fractional derivative order (n) and potential depth (u) on position and momentum entropy.
- To examine the validity of the Beckner-Bialynicki-Birula-Mycieslki (BBM) inequality and the Fisher entropy under these conditions.
Main Methods:
- Calculation of position and momentum entropy for hyperbolic potentials using the fractional Schrödinger equation.
- Analysis of the Shannon entropy's dependence on the fractional derivative number (n) and potential depth (u).
- Verification of the Beckner-Bialynicki-Birula-Mycieslki (BBM) inequality and computation of Fisher entropy.
Main Results:
- As the fractional derivative number (n) decreases, the wave function localizes towards the origin, leading to more localized position entropy density.
- A decrease in n results in more delocalized momentum probability density.
- The Shannon entropies consistently satisfy the BBM inequality, which shows a gradual change with increasing potential depth (u).
- Fisher entropy increases with potential depth (u) and decreases with the fractional derivative number (n).
Conclusions:
- The study demonstrates a clear relationship between fractionalization, potential characteristics, and information-theoretic quantities in quantum systems.
- Findings highlight how changes in the fractional derivative order significantly alter the localization properties of wave functions and their associated entropies.
- The results confirm the robustness of the BBM inequality and provide insights into the behavior of Fisher entropy in fractional quantum mechanics.
More Related Videos
07:56A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference
Published on: September 5, 2019
8.6K
08:04Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids
Published on: May 27, 2020
8.5K
Related Concept Videos
02:45The Quantum-Mechanical Model of an Atom
42.8K
Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra.
42.8K
02:39Entropy
30.7K
Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
30.7K
01:26Thermodynamic Potentials
945
Thermodynamic potentials are state functions that are extremely useful in analyzing a thermodynamic system. They have dimensions of energy. The four important thermodynamic potentials are internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. These thermodynamic potentials can be expressed using two of the following variables: pressure, volume, temperature, and entropy. These two variables are expressed as the rate of change of the thermodynamic potential with respect to other...
945
01:20Entropy and the Second Law of Thermodynamics
3.0K
The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
The relation between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
3.0K
04:08The Uncertainty Principle
23.6K
Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He...
23.6K
02:38Third Law of Thermodynamics
19.3K
A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
19.3K