- #1

danyull

- 9

- 1

## Homework Statement

Find the average energy ##\langle E \rangle## for

(a) an n-state system in which a given state can have energy 0, ε, 2ε, 3ε... nε.

(b) a harmonic oscillator, in which a state can have energy 0, ε, 2ε, 3ε... (i.e. with no upper limit).

## Homework Equations

Definition of temperature: ##β = \frac 1 {K_BT} = \frac {d lnΩ(E)} {dE}##

Boltzmann distribution: ##P(ε) ∝ e^{-εβ}##

## The Attempt at a Solution

Since the energy here takes on discrete values, the average is found by taking the weighted sum of the probabilities, $$\langle E \rangle = \sum_{n=0}^n nε~e^{-nεβ}$$

and in the case of part (b), the sum goes to infinity. My problem is I don't know how to evaluate these sums. Any help would be appreciated, thanks!