In summary, the average energy in the microcanonical ensemble can be calculated by taking the sum of the non-interacting energy and the interaction energy for each state, and then dividing by the total number of states. This is due to the decoupling of the non-interacting part from the interactions.
The average energy in the microcanonical ensemble is given by the expression:E_avg = \frac{1}{N}\sum_{i=1}^N E_i where N is the number of states and E_i is the energy of the i-th state. In our case, the Hamiltonian is H = H_0 + V, where H_0 is the non-interacting part and V is the interaction term. Therefore, the energy of each state is given by E_i = E_0(i) + V(i), where E_0(i) is the energy of the i-th state of the non-interacting part and V(i) is the energy of the i-th state due to interactions.Therefore, we can rewrite the average energy as:E_avg = \frac{1}{N}\sum_{i=1}^N [E_0(i) + V(i)] which can be further simplified to:E_avg = \frac{1}{N}\sum_{i=1}^N E_0(i) + \frac{1}{N}\sum_{i=1}^N V(i) Since the non-interacting part is decoupled from the interactions, the two sums can be computed separately and we obtain:E_avg = \frac{1}{N}\sum_{i=1}^N E_0(i) + \frac{1}{N}\sum_{i=1}^N V(i) which is the average energy in the microcanonical ensemble.