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Related Concept Videos

State Space Representation01:27

State Space Representation

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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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Gaussian Elimination: Problem Solving01:30

Gaussian Elimination: Problem Solving

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Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...
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State Space to Transfer Function01:21

State Space to Transfer Function

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The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
The transformation process begins with the state-space representation, characterized by the state equation and the output equation. These equations are typically represented as:
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Transfer Function to State Space01:23

Transfer Function to State Space

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State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
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Random Variables01:09

Random Variables

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A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
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The Quantum-Mechanical Model of an Atom02:45

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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.
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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Hilbert Space Geometry of Random Matrix Eigenstates.

Alexander-Georg Penner1, Felix von Oppen1, Gergely Zaránd2,3

  • 1Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany.

Physical Review Letters
|June 10, 2021
PubMed
Summary
This summary is machine-generated.

We derived the probability distribution for the quantum geometric tensor in random matrix ensembles. This provides exact joint distributions for Fubini-Study metrics and Berry curvature in quantum systems.

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Area of Science:

  • Quantum mechanics
  • Statistical physics
  • Condensed matter physics

Background:

  • The geometry of quantum states is crucial for understanding quantum dynamics and critical phenomena.
  • Parameter-dependent quantum systems, such as those studied in quantum quenches and critical point characterization, exhibit complex Hilbert space geometries.

Purpose of the Study:

  • To derive the complete probability distribution of the quantum geometric tensor for parameter-dependent random matrix ensembles.
  • To analyze the Fubini-Study metric and Berry curvature distributions within the Gaussian unitary ensemble.
  • To explore connections between quantum geometry and Levy stable distributions.

Main Methods:

  • Analytical derivation of the probability distribution for the quantum geometric tensor.
  • Focus on the Gaussian unitary ensemble (GUE) of random matrices.
  • Comparison with numerical simulations of random matrix ensembles and electrons in random magnetic fields.

Main Results:

  • The exact joint probability distribution function for the Fubini-Study metric and Berry curvature was obtained.
  • Analytical results reveal connections to Levy stable distributions.
  • The findings were validated through numerical simulations.

Conclusions:

  • The study provides a comprehensive analytical framework for understanding the Hilbert space geometry of quantum states in random matrix ensembles.
  • The derived distributions offer precise tools for characterizing quantum systems and their dynamics.
  • The results have implications for diverse fields, including quantum information and condensed matter theory.