# A system has non-degenerate energy levels with energy

1. Oct 19, 2014

### Eats Dirt

1. The problem statement, all variables and given/known data
A system has non-degenerate energy levels with energy$$\epsilon=(n+1/2)\hbar\omega$$ where h-bar*omega=1.4*10^-23J and n positive integer zero what is the probability that it is in n=1 state with a heat bath of temperature 1K
2. Relevant equations
$$Z=\exp^\frac{-E_i}{k_b T} \\ p_r=\frac{\exp^\frac{-E_i}{k_b T}}{\sum^N_j \exp^\frac{-E_j}{k_b T}}$$

3. The attempt at a solution
I'm not really sure what to do now, I dont know how to sum the total number of states to get the fraction of states in the n=1 state

Last edited by a moderator: Oct 19, 2014
2. Oct 19, 2014

### Staff: Mentor

The sum is a sum over q^i for some q<1, this has an analytic result. You can just plug in all values and calculate the result.

3. Oct 19, 2014

### Eats Dirt

Ok I think I might have gotten it, to deal with the infinite sum use a geometric series,

$$\sum_0^\inf e^\frac{-(n+\frac{1}{2})}{k_b T} \\ =e^\frac{-\hbar\omega}{2k_b T}\sum_0^\inf e^\frac{-n}{k_b T}\\ =\frac{e^\frac{-\hbar\omega}{2k_b T}}{1-e^\frac{-\hbar\omega}{k_b T}$$

then evaluate using the pr as stated before.

Also I don't know why my LaTeX is not displaying correctly.

4. Oct 20, 2014

### Staff: Mentor

Some error, probably with brackets.
Yes the approach is good.