# How are different sites in a Fock state related?

I'm trying to numerically construct a Hamiltonian of the form $$H = \hat n_1+ \hat n_2$$ for a two site system in some Fock space which I will truncate to allow a maximum of $N$ particles per site. These operators act on the Fock states as $$\hat n_1 | n_1,n_2 \rangle = n_1 | n_1,n_2 \rangle$$ $$\hat n_2 | n_1,n_2 \rangle = n_2 | n_1,n_2 \rangle$$ so they do nothing to the site that they're not acting on. Is it then correct to assume that the sites are related through a tensor product? i.e. is a Fock state interpreted as the tensor product of individual site occupancy states:$$| n_1,n_2 \rangle = | n_1 \rangle \otimes | n_2 \rangle$$ and thus $$\hat n_1 = \hat n_1^{(1)} \otimes \mathbb 1^{(2)}$$ or are the sites related through some other relation such as a direct sum?

• Yes, it is a tensor product. All your equations are correct. – Adam May 29 '18 at 22:13