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Titration Calculations: Strong Acid - Strong Base02:28

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Calculating pH for Titration Solutions: Strong Acid/Strong Base
A titration is carried out for 25.00 mL of 0.100 M HCl (strong acid) with 0.100 M of a strong base NaOH. The pH at different volumes of added base solution can be calculated as follows:
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Strong eigenstate thermalization within a generalized shell in noninteracting integrable systems.

Takashi Ishii1, Takashi Mori2

  • 1Department of Physics, Graduate School of Science, University of Tokyo, Kashiwa 277-8574, Japan.

Physical Review. E
|September 11, 2019
PubMed
Summary

Integrable quantum systems do not follow the strong eigenstate thermalization hypothesis (ETH). Instead, a generalized ETH proves these systems reach a generalized Gibbs ensemble (GGE) steady state.

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Area of Science:

  • Quantum mechanics
  • Statistical mechanics
  • Condensed matter physics

Background:

  • Integrable systems are a key area of study in quantum mechanics and statistical mechanics.
  • The eigenstate thermalization hypothesis (ETH) explains thermalization in isolated quantum systems.
  • Integrable systems are known not to obey ETH, suggesting a different relaxation mechanism.

Purpose of the Study:

  • To investigate the thermalization properties of integrable systems.
  • To determine if a generalized version of ETH applies to these systems.
  • To establish the conditions under which integrable systems reach a generalized Gibbs ensemble (GGE).

Main Methods:

  • Proving a generalized version of the strong ETH for noninteracting, translationally invariant integrable systems.
  • Analyzing the behavior of local observables and correlations between energy eigenstates.
  • Demonstrating relaxation to a GGE without assuming the cluster decomposition property for initial states.

Main Results:

  • A generalized strong ETH is proven to hold for noninteracting integrable systems.
  • This generalized ETH shows that energy eigenstates with similar local conserved quantities exhibit similar local observable behavior.
  • The study confirms that integrable systems relax to a GGE for initial states with subextensive fluctuations of local conserved quantities.

Conclusions:

  • The findings provide a theoretical basis for understanding the steady states of integrable quantum systems.
  • This work relaxes previous assumptions required for GGE derivations.
  • The generalized ETH offers a new perspective on thermalization in systems that do not obey standard ETH.