# Boltzmann Distribution

## Homework Statement

Compute the mean energy and mean square energy of a free particle using the Boltzmann distribution.

## The Attempt at a Solution

I calculated the mean energy to be 3/2 kT by computing the mean square velocity directly from the Boltmann distribution function and then multiplying it by 1/2 m and then again multiplying by 3 to get the answer in three dimensions.

However, I cannot figure out how to compute the mean square energy as E^2 is proportional v^4 and computing it in one dimension does not allow me to simply multiply by 3 to get the answer in 3 dimensions.

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I just tried <E^2> = <(Ex+Ey+Ez)^2> = 9<Ex^2> since <Ex>=<Ey>=<Ez> and each of the three components are independent of the others.

My result was <E^2> = 27/8 k^2 T^2.

Does this make sense?

Last edited:
dextercioby
Homework Helper
No. Why 9 and not 3 ?

No. Why 9 and not 3 ?

Cross terms. (x+y+z)^2 = x^2+y^2+z^2 +2xz +2xy +2yz

dextercioby
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