Partition function for position-independent hamiltonian

In summary, The classical partition function for the canonical ensemble is given by Z=∫dpdq e^(-βH(p,q)). If the Hamiltonian does not depend on the position, the integration over dq becomes the volume of the system. This results in the single particle partition function for the classical ideal gas, allowing for the evaluation of things per unit volume.
  • #1
Einj
470
59
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
 
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  • #2
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.
 

Related to Partition function for position-independent hamiltonian

1. What is the partition function for a position-independent Hamiltonian?

The partition function for a position-independent Hamiltonian is a mathematical quantity that is used to describe the statistical mechanics of a system with a Hamiltonian that does not depend on position. It is a sum of all possible states of the system, weighted by their Boltzmann factors.

2. How is the partition function related to the thermodynamic properties of a system?

The partition function is directly related to the thermodynamic properties of a system as it is used to calculate quantities such as free energy, entropy, and internal energy. These properties can then be used to make predictions about the behavior of the system.

3. What is the relationship between the partition function and the probability distribution of states?

The partition function is related to the probability distribution of states through the Boltzmann distribution. The probability of a state is proportional to the Boltzmann factor of that state divided by the partition function. This allows us to calculate the relative probabilities of different states.

4. How is the partition function affected by changes in temperature and energy?

The partition function is directly affected by changes in temperature and energy. As temperature increases, the partition function also increases, and as energy increases, the partition function decreases. This is because higher temperatures and energies lead to a larger number of accessible states.

5. Can the partition function be used for all types of Hamiltonians?

No, the partition function is specifically used for position-independent Hamiltonians. For Hamiltonians that do depend on position, a different partition function must be used, which takes into account the spatial distribution of particles in the system.

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