# Planck distribtion

## Homework Statement

I cant find the proof of "the ratio of the number of oscillators in their (n+1) )th quantum state of excitation to the number in nth quantum state is:
k is boltzman costant
N_(n+1)/N_(n)=exp(-hω/2π(kT)"

## The Attempt at a Solution

:-( I dont have any idea

Last edited:

malawi_glenn
Homework Helper
Can you write down the planck distribution and explain it in words?

for obtaining planck distribution first we use this equation

N_(n+1)/N_(n)=exp(-hω/2π(kT))

and then the fraction of total number of oscilators in nth quantum stae is

N_n/∑_(s=0)^∞▒ N_s =exp(-hω/2π(kT))/∑_(s=0)^∞▒〖exp(-shω/2π(kT))〗

<n>=∑_(s=0)^∞▒〖s exp(-shω/2π(kT))〗/∑_(s=0)^∞▒〖exp(-shω/2π(kT))

<n>=1/[exp(-hω/2π(kT))-1] " n" is average excitation quantum number of an oscillator

But I dont know how can get this equation" N_(n+1)/N_(n)=exp(-hω/2π(kT))"

Matterwave
Gold Member
Have you learned about Boltzmann factors? This is basically a direct application of Boltzmann factors: $$\frac{n_i}{n_j}=e^{\frac{-\Delta E_{ij}}{kT}}$$

Do you know how to get the Boltzmann factors? (Hint: it has to do with entropy)

Last edited:
I've seen boltzmann factor, but I dont know how I can proove it,could u tell me some hints?

malawi_glenn
Homework Helper
prove and prove, why do you need to prove the boltzman factor?

Sorry for my hint going via the plank distribution, working with boltzmann factors are much easier ;-)

thanx any way :-)

Matterwave
Gold Member
So, the Boltzmann factor can be proved using Entropy of a reservoir and a particle in state i. The gist of it is, if you change the state of the particle, you change the energy of the particle and the entropy (multiplicities) of the reservoir. If you use some entropy and multiplicity relations, you can get the Boltzmann factor.

I don't remember the exact proof, but it's provided here:

http://www.physics.thetangentbundle.net/wiki/Statistical_mechanics/Boltzmann_factor [Broken]

Edit: oops, I realize I forgot a - sign in my first post. I've fixxed it.

Last edited by a moderator:
for the derivation of Boltzmman factor you can see the book SEARS AND SALINGER of thermodynamics.

In this book much simpler method is used.

malawi_glenn