Hey everyone,
This is my first time posting on PF!
I want to model the photons ejected from a blackbody source at temperature T.
The question I want answered is: given a photon is detected, what is the probability of the photon having a wavelength λ? This amounts to just attaining the probability density function for the different wavelengths.
We normally talk about blackbodies in terms of intensity, or more specifically spectral radiance, but I want to talk about the individual photon wavelength distribution.
My current thoughts are:
Any thoughts on whether this is right?
Peter.
This is my first time posting on PF!
I want to model the photons ejected from a blackbody source at temperature T.
The question I want answered is: given a photon is detected, what is the probability of the photon having a wavelength λ? This amounts to just attaining the probability density function for the different wavelengths.
We normally talk about blackbodies in terms of intensity, or more specifically spectral radiance, but I want to talk about the individual photon wavelength distribution.
My current thoughts are:
Quote:
|
Planks law for spectral radiance as a function of wavelength may be written as : \begin{eqnarray} \label{plank1} I(\lambda,T) = \frac{2 h c^2 }{\lambda^5} \frac{1 }{e^{\frac{hc}{\lambda k_B T}}-1} \end{eqnarray} where I is the spectral radiance, the power radiated per unit area of emitting surface in the normal direction per unit solid angle per unit frequency by a black body at temperature T, h is Planks constant, c is the speed of light in a vacuum and k is the Boltzmann constant. The energy produced by a single wavelength of light is \begin{eqnarray} \label{plank15} E=N_\lambda h c / \lambda \end{eqnarray} where N is the number of photons at this wavelength produced by the blackbody. It follows from Planks law that the number of photons produced by a blackbody with a certain wavelength follows the proportionality: \begin{eqnarray} \label{plank2} N_\lambda(\lambda,T) \propto \frac{1}{\lambda^4} \frac{1 }{e^{\frac{hc}{\lambda k_B T}}-1} \end{eqnarray} Then the probability density of a blackbody emitting a photon of wavelength λ is given by normalising: \begin{eqnarray} \label{plank3} Prob(\lambda)= \frac{\frac{1}{\lambda^4} \frac{1 }{e^{\frac{hc}{\lambda k_B T}}-1}} {\int_{0}^{\infty} \frac{1}{\lambda^{\prime^4}} \frac{1 }{e^{\frac{hc}{\lambda^\prime k_B T}}-1} d\lambda^\prime} \end{eqnarray} |
Peter.