- #1

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## Homework Statement

find the average energy of a system with n energy states (0, 1E, 2E, 3E...nE)

## Homework Equations

P(E) = e

^{-BE}/Z - where B=1/KbT and Z= ∑e(

^{-BE})

^{n}

<E>=∑(nE* (e

^{-BE})

^{n}) /Z

## The Attempt at a Solution

i feel like ive gone down the correct path - that is finding result of the sums.

Z - if S=R

^{0}+R

^{1}+...+R

^{n}i do S-RS and get S=R

^{0}-R

^{n+1}/1-R ... so Z= e

^{0}-e

^{-BE(n+1)}/1-e

^{BE}

now it gets tricky, i tried to evaluate the sum ∑(nE* (e

^{-BE})

^{n}) in a similar way,

so S

_{2}=0R

^{0}+1R

^{1}+2R

^{2}...

and RS

_{2}=0R+1R

^{2}+2R

^{3}...

so S

_{2}-RS

_{2}=S-1 (remember S=Z; "-1" cause R

^{0}is missng)

so S

_{2}=(1-e

^{-BE(n+1)}/(1-e

^{-BE})

^{2}-1/(1-e

^{-BE}).

substituting into average fomula and simplifying my answer is: (1/1-e

^{-BE}) - (1/1-e

^{BE(n+1)})

is this correct?