Partition function for position-independent hamiltonian

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SUMMARY

The discussion centers on the partition function for a position-independent Hamiltonian, specifically the free-particle Hamiltonian represented as H=p²/2m. In this case, the classical partition function for the canonical ensemble is expressed as Z(β)=∫dpdq e^{-βH(p,q)}. When the Hamiltonian lacks q-dependence, the integration over dq simplifies to the volume of the system, leading to the conclusion that the resulting partition function corresponds to the single particle partition function for the classical ideal gas, evaluated per unit volume.

PREREQUISITES
  • Understanding of classical mechanics and Hamiltonian dynamics
  • Familiarity with statistical mechanics concepts, particularly the canonical ensemble
  • Knowledge of partition functions and their role in thermodynamics
  • Basic mathematical skills for evaluating integrals
NEXT STEPS
  • Study the derivation of the canonical ensemble partition function in detail
  • Explore the implications of position-independent Hamiltonians in statistical mechanics
  • Learn about the properties of the classical ideal gas and its partition function
  • Investigate the role of volume in thermodynamic calculations
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Physicists, particularly those specializing in statistical mechanics and thermodynamics, as well as students seeking to deepen their understanding of Hamiltonian systems and partition functions.

Einj
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Hi everyone. Suppose I have an Hamiltonian which doesn't depend on the position (think for example to the free-particle one [itex]H=p^2/2m[/itex]). I know that the classical partition function for the canonical ensemble is given by:
$$
Z(\beta)=\int{dpdq e^{-\beta H(p,q)}}.
$$

What does it happen to the integration over [itex]dq[/itex] if there is no q-dependence in the Hamiltonian? Is it just the volume of the system?

Thank you
 
Physics news on Phys.org
yes,what you will get by this free particle hamiltonian is the single particle partition function for the classical ideal gas.You can evaluate things per unit volume.
 

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