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When analyzing the motion of falling objects, it is essential to consider not only the force of gravity but also the opposing force of air resistance. A practical example involves releasing a heavy test weight during a safety check on a ship. As the weight falls from rest, gravity accelerates it downward while air resistance exerts an upward force that increases with velocity. This dynamic interplay of forces is well described by differential equations, which provide a mathematical framework...
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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.
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The eigenvalue problem in phase space.

Leon Cohen1

  • 1Department of Physics, Hunter College of the City University, New York, New York, 10065.

Journal of Computational Chemistry
|July 28, 2017
PubMed
Summary
This summary is machine-generated.

This study introduces a quantum phase space eigenvalue equation using the Wigner distribution. It identifies proper and improper solutions, defining psi-representability conditions for accurate quantum mechanics analysis.

Keywords:
Weyl operatorWigner distributioneigenvalue problemquantum phase space

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Area of Science:

  • Quantum Mechanics
  • Quantum Phase Space Theory

Background:

  • The standard quantum mechanical eigenvalue problem is typically formulated in Hilbert space.
  • Quantum phase space representations offer an alternative framework for analyzing quantum systems.

Purpose of the Study:

  • To formulate the quantum mechanical eigenvalue problem in quantum phase space.
  • To investigate the nature of solutions obtained in this phase space formulation.
  • To establish criteria for physically valid phase space distributions.

Main Methods:

  • Formulation of the eigenvalue equation in quantum phase space.
  • Utilizing the Wigner distribution function to represent quantum states.
  • Development and application of psi-representability conditions.

Main Results:

  • An eigenvalue equation involving the c-function (quantum operator correspondence) was derived.
  • The existence of both proper and improper solutions to the phase space eigenvalue equation was identified.
  • Conditions for psi-representability were established, ensuring the extraction of correct phase space eigenfunctions.
  • The phase space eigenvalue equation was generalized for arbitrary phase space distributions.

Conclusions:

  • The quantum phase space eigenvalue equation provides a valuable alternative formulation.
  • Distinguishing between proper and improper solutions is crucial for physical interpretation.
  • Psi-representability conditions are essential for ensuring the validity of phase space quantum mechanics.