Planck Distribution Homework Solution

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In summary, the homework statement is saying that 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, and N_(n+1)/N_(n)=exp(-hω/2π(kT))
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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 boltzman costant
N_(n+1)/N_(n)=exp(-hω/2π(kT)"

Homework Equations


The Attempt at a Solution

:-( I don't have any idea
 
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  • #2
Can you write down the Planck distribution and explain it in words?
 
  • #3
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))"
 
  • #4
Have you learned about Boltzmann factors? This is basically a direct application of Boltzmann factors: [tex]\frac{n_i}{n_j}=e^{\frac{-\Delta E_{ij}}{kT}}[/tex]

Do you know how to get the Boltzmann factors? (Hint: it has to do with entropy)
 
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  • #5
I've seen Boltzmann factor, but I don't know how I can proove it,could u tell me some hints?
 
  • #6
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 ;-)
 
  • #7
thanx any way :-)
 
  • #8
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.
 
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  • #9
for the derivation of Boltzmman factor you can see the book SEARS AND SALINGER of thermodynamics.

In this book much simpler method is used.
 
  • #10
derivation of boltzman factor is done in almost any book/resource on statistical thermodynamics..
 

1. What is the Planck Distribution equation?

The Planck Distribution equation, also known as the Planck's Law, describes the spectral radiance of an ideal black body at a given temperature. It is given by B(λ, T) = (2hc²/λ⁵) * 1/(e^(hc/λkT) - 1), where B is the spectral radiance, λ is the wavelength, T is the temperature, h is the Planck constant, c is the speed of light, and k is the Boltzmann constant.

2. How is the Planck Distribution used in physics and astronomy?

The Planck Distribution is used to explain the emission of radiation from a black body at different temperatures. It is also used in physics and astronomy to study the properties of stars and other celestial objects, as well as to understand the cosmic microwave background radiation.

3. What are the assumptions made in the Planck Distribution equation?

The Planck Distribution equation assumes that the body emitting the radiation is an ideal black body, which absorbs all incident radiation and emits radiation at all wavelengths. It also assumes that the body is in thermal equilibrium and that the radiation is a result of thermal energy.

4. Can the Planck Distribution be applied to real objects?

While the Planck Distribution is based on ideal conditions, it can still be applied to real objects with some modifications. This is because most objects can be approximated as black bodies at certain wavelengths, and the equation can be adjusted to account for any deviations from ideal conditions.

5. How can the Planck Distribution be used to calculate the peak wavelength of emission?

The peak wavelength of emission, also known as the Wien's displacement law, can be calculated using the Planck Distribution equation by finding the maximum value of the spectral radiance. This corresponds to the wavelength at which the slope of the curve is zero, and can be determined using calculus or by plotting the equation on a graph.

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