Number of States in a 1D Simple Harmonic Oscillator

[Ang Han Wei]

Homework Statement

A system is made of N 1D simple harmonic oscillators. Show that the number of states with total energy E is given by $\Omega$(E) = $\frac{(M+N-1)!}{(M!)(N-1)!}$

Homework Equations

Each particle has energy ε = $\overline{h}$$\omega$(n + $\frac{1}{2}$), n = 0, 1

Total energy is given by E = [$\frac{N}{2}$ + M]$\overline{h}$$\omega$, M is an integer

The Attempt at a Solution

If the entire system is in the ground state, n = 0 for all particles and the energy will be $\frac{N}{2}$$\overline{h}$$\omega$

So M must be the number of excited particles having the value of n = 1

Hence, the total number of states should be the number of ways that I can choose M particles to be "excited" out of N particles.
$\Omega$ = $\frac{N!}{(M!)(N-M)!}$ but that is not the case.

I am not sure where I have gone wrong.

 

[vela]

Your mistake is in assuming the excited oscillators have to be in the n=1 state. For example, say you had three oscillators. If two are in the n=1 state and one is in the n=0 state, the system has the same energy as one with one oscillator in the n=2 state and two oscillators in the n=0 state.