Discussion
A system consists of two indistinguishable bosons. Each particle can occupy only two energy levels \( E = \epsilon \) and \( E = 2\epsilon \). The canonical partition function for the system is :
Answer: A
The partition function is \( Z = e^{-4\beta\epsilon} + e^{-3\beta\epsilon} + e^{-2\beta\epsilon} \).
For indistinguishable bosons a state is fixed by how many particles are in each level, and any number may share a level. With two particles and two levels there are three states:
- both in \( \epsilon \): total energy \( 2\epsilon \);
- one in each level: \( 3\epsilon \), counted once, since the particles cannot be told apart;
- both in \( 2\epsilon \): \( 4\epsilon \).
Summing \( e^{-\beta E} \) over these three states gives the keyed result. The other options keep a single Boltzmann factor, as if the system had only one state.
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