# Help in calculating Partition function and choosing the ensemble

I've been given this situation

"A surface contains $$N$$ identical atoms in a fixed position. Every atom can occupy one of two states with energies $$E_1$$ or $$E_2$$ and the temperature is $$T$$."

For the solution, I am uncertain what kind of ensemble to use seeing as it is a surface, I would imagine it to be a 2-dimensional surface of a classical continuous system on the form of a canonical ensemble. Which would take the form

$$Z=\frac{1}{h^2}\int d^2 r \int d^2 p\exp{(-\beta\cdot E(r,p))}$$ For $$h=\Delta x\Delta p$$

However, the verbal communication with the professor indicates a solution on the form of $$Z=Z_1^N=(e^{-\beta\cdot E_1}+e^{-\beta\cdot E_2})^N$$

I do not really the professor's solution, the first partition function would give a different result than his answer. Can anyone help?

In general the partition function has the form $$Z = \sum_{\gamma\in\Gamma} e^{-\beta E_\gamma}$$ where $$\Gamma$$ is the set of possible microstates of the system and $$E_\gamma$$ are the corresponding energies. If the possible microstates can be characterised by a set of continuous variables, then the sum turns in to an integral of the form you wrote down in your question. In this case however we have a discrete set of variables, so the sums remain discrete.
The set of sates we are dealing with can be characterised by listing the states of each of $$N$$ atoms, for example $$\gamma = (1,1,2,1,2,1...)$$ and the corresponding energies are given by adding up the energies for each atom $$E_\gamma = \sum_{n=1}^N E^{(n)}$$ where $$E^{(n)}$$ is $$E_1$$ if atom $$n$$ is in state $$1$$ and $$E_2$$ is it is in state $$2$$.
From here the thing to notice is that since the energy of each atom is independant of the state of any other atom, we can pull appart the sum over $$\Gamma$$ into $$N$$ sums over the state of each indivdual atom $$\sum_{\gamma\in\Gamma} = \sum_{\gamma_1\in\{1,2\}}\sum_{\gamma_2\in\{1,2\}}...\sum_{\gamma_N\in\{1,2\}}$$