Question

Consider an idolated system which is comprised of four identical indistinguishable particles that can access to four different energy level:

E_1 = 1ev
E_2 = 2eV
E_3 = 3eV
E_4 = 4ev

a) Calculate the total number of possible microstates

b) How many microstates can the system access if it is isolated and has total energy 9eV?

Answer 1

Well if treat it as a Einstein solid you know

\[\Omega(N,q) = \left(\begin{matrix}q + N-1 \\ q\end{matrix}\right)\]

Answer 2

However the wording of four indistinguishable particles and not oscillators makes me wary of treating it as an Einstein solid unless I am over thinking it :P

Answer 3

Where q is energy units and N is the number of oscillators.

Answer 4

The number of ways to arrange 4 identical particles amongst 4 energy levels is given by 7!/(3!*4!) which comes to 35.
In general, the number of ways to put N identical particles into M energy levels is (N+M-1)!/(N!*(M-1)!) Why ? The problem is equivalent to finding the number of ways that you can order N identical particles and M-1 identical partitions.

By going thru the possible states and calculating the energy of each I found 4 states that had an energy of 9ev.

Answer 5

Imagine Q as dots representing energy and N-1 partitions you just choose how many are dots.