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Position-momenta uncertainties in classical systems.

Dipesh K Singh1, P K Mohanty1

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Summary
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We developed a novel thermal bath preserving angular momentum, leading to a minimum uncertainty in particle position and momentum. This lower bound is directly linked to the mean angular momentum, offering new insights into quantum mechanics.

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Area of Science:

  • Statistical mechanics
  • Quantum mechanics
  • Classical mechanics

Background:

  • Understanding the behavior of systems in thermal baths is crucial in statistical mechanics.
  • The role of angular momentum in defining system properties requires further investigation.
  • Existing thermal baths often lead to zero mean angular momentum, limiting certain applications.

Purpose of the Study:

  • To design a thermal bath that can preserve or control the angular momentum of a system.
  • To investigate the implications of a non-zero mean angular momentum on particle uncertainties.
  • To establish a fundamental lower bound for position-momentum uncertainties related to angular momentum.

Main Methods:

  • Development of a specialized thermal bath.
  • Theoretical analysis of classical particles within this bath.
  • Derivation of position-momentum uncertainty relations.

Main Results:

  • The designed thermal bath maintains a Boltzmann energy distribution in the steady state.
  • Particles in this bath exhibit position-momentum uncertainties with a strictly positive lower bound.
  • This lower bound is proportional to the absolute value of the mean angular momentum.
  • A dimensionless constant 'c' characterizes this proportionality, universally bounded by unity.
  • For particles in central potentials, c precisely equals 1/2.

Conclusions:

  • The study introduces a novel thermal bath with implications for statistical and quantum mechanics.
  • A direct relationship between angular momentum and fundamental particle uncertainties is established.
  • The findings suggest new theoretical frameworks for systems with conserved or controlled angular momentum.