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Related Concept Videos

State Space Representation01:27

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
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¹H NMR: Interpreting Distorted and Overlapping Signals

Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
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Discrete Fourier Transform01:15

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The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...
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Free-energy diagrams, or reaction coordinate diagrams, are graphs showing the energy changes that occur during a chemical reaction. The reaction coordinate represented on the horizontal axis shows how far the reaction has progressed structurally. Positions along the x-axis close to the reactants have structures resembling the reactants, while positions close to the products resemble the products.  Peaks on the energy diagram represent stable structures with measurable lifetimes, while other...
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Time and frequency -Domain Interpretation of Phase-lead Control

Phase-lead controllers are commonly used in various control systems to enhance response speed and stability. Adjusting the brightness on a television screen offers a practical example of phase-lead control. When contrast is enhanced, a phase-lead controller is employed. Mathematically, phase-lead control is identified when the first parameter is smaller than the second.
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Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to occupy...

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Uncovering Hidden Dynamics of Natural Photonic Structures Using Holographic Imaging
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Decoding the dynamical information embedded in highly excited vibrational eigenstates: state space and phase space

Paranjothy Manikandan1, Aravindan Semparithi, Srihari Keshavamurthy

  • 1Department of Chemistry, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016, India.

The Journal of Physical Chemistry. A
|February 13, 2009
PubMed
Summary

Researchers developed new methods to identify and analyze quantum eigenstates in complex systems. These techniques help distinguish between classical and quantum properties in highly mixed states, advancing our understanding of quantum chaos.

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Area of Science:

  • Quantum mechanics
  • Spectroscopy
  • Chemical physics

Background:

  • Studying highly excited eigenstates in strongly coupled multimode systems is challenging due to complexity and chaos.
  • Classical-quantum correspondence techniques face difficulties with multiple degrees of freedom and chaotic classical phase space.

Purpose of the Study:

  • To develop effective methods for identifying and characterizing highly excited eigenstates in complex quantum systems.
  • To establish a connection between quantum eigenstates and the classical resonance web.
  • To differentiate between predominantly classical and quantum mixing in highly mixed quantum states.

Main Methods:

  • Utilized a parametric variation technique to identify sequences of localized and delocalized states.
  • Introduced a novel wavelet-based local time-frequency approach to map quantum eigenstates onto the classical resonance web.
  • Analyzed spectroscopic Hamiltonians for CDBrClF and CF(3)CHFI as illustrative examples.

Main Results:

  • Successfully identified sequences of localized states interspersed with delocalized states using parametric variation.
  • The wavelet-based lifting procedure revealed dominant nonlinear resonances influencing the eigenstates.
  • Demonstrated consistency between state space and phase space perspectives of eigenstates for the studied molecules.

Conclusions:

  • The developed methods provide clear insights into the nature of highly excited eigenstates in strongly coupled systems.
  • The approach effectively links quantum properties to classical resonances, aiding in the assignment of complex states.
  • Distinguishing between classical and quantum mixing in highly mixed states is achievable through detailed analysis.