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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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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...
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Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
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H-Theorem in an Isolated Quantum Harmonic Oscillator.

Che-Hsiu Hsueh1,2, Chi-Ho Cheng3, Tzyy-Leng Horng4

  • 1Department of Optoelectric Physics, Chinese Culture University, Taipei 111, Taiwan.

Entropy (Basel, Switzerland)
|August 26, 2022
PubMed
Summary
This summary is machine-generated.

Adding a barrier potential to a quantum harmonic oscillator induces thermalization, aligning microscopic systems with macroscopic thermodynamics. Without the barrier, the system does not thermalize, demonstrating the crucial role of potential in quantum systems.

Keywords:
H-theoremShannon entropybarrier potentialdecoherencequantum harmonic oscillatorthermalization

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Area of Science:

  • Quantum mechanics
  • Statistical mechanics
  • Thermodynamics

Background:

  • The H-theorem describes the approach to equilibrium in classical systems.
  • Investigating quantum systems' approach to equilibrium is crucial for understanding thermodynamics at microscopic scales.

Purpose of the Study:

  • To investigate the H-theorem in an isolated quantum harmonic oscillator.
  • To determine the effect of potential on entropy production and thermalization.

Main Methods:

  • Solving the time-dependent Schrödinger equation for a quantum harmonic oscillator.
  • Analyzing the impact of a barrier potential on system thermalization.
  • Calculating Shannon entropy to quantify disorder and thermalization.

Main Results:

  • A harmonic trap with a barrier potential leads to system thermalization.
  • A harmonic trap alone does not cause thermalization.
  • Shannon entropy increases during thermalization, confirming macroscopic thermodynamics laws.
  • Coherent mechanical energy transforms into incoherent thermal energy, causing decoherence.
  • Equilibrium state shows density distributions fitting a microcanonical ensemble.

Conclusions:

  • Potential barriers are key to inducing thermalization in quantum harmonic oscillators.
  • Microscopic quantum systems adhere to macroscopic thermodynamic laws.
  • The study demonstrates quantum decoherence and energy thermalization in a harmonic oscillator model.