Partition function for position-independent hamiltonian

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In a position-independent Hamiltonian, such as that of a free particle, the classical partition function simplifies significantly. The integration over the position variable, dq, effectively results in the volume of the system since the Hamiltonian does not depend on q. This leads to the conclusion that the partition function represents the single particle partition function for a classical ideal gas. Evaluating the partition function per unit volume is a common approach in this context. Thus, the integration over dq confirms that it contributes to the overall volume in the partition function calculation.
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 H=p^2/2m). 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 dq if there is no q-dependence in the Hamiltonian? Is it just the volume of the system?

Thank you
 
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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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