Planck Distribution Homework Solution

[najima]

Homework Statement

I can't 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 Boltzmann costant
N_(n+1)/N_(n)=exp(-hω/2π(kT)"

Homework Equations

The Attempt at a Solution

:-( I don't have any idea

 

[malawi_glenn]

Can you write down the Planck distribution and explain it in words?

 

[najima]

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 don't know how can get this equation" N_(n+1)/N_(n)=exp(-hω/2π(kT))"

 

[Matterwave]

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)

 

[najima]

I've seen Boltzmann factor, but I don't know how I can proove it,could u tell me some hints?

 

[malawi_glenn]

prove and prove, why do you need to prove the Boltzmann factor?

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

 

[najima]

thanx any way :-)

 

[Matterwave]

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

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

 

[guruji]

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]

derivation of Boltzmann factor is done in almost any book/resource on statistical thermodynamics..