Photon wavelength probability distribution for blackbody

[Orthoptera]

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:

Plank’s 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 Plank’s 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 Plank’s 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}

Any thoughts on whether this is right?

Peter.

 

[Charles Link]

That is almost correct. The probability that the wavelength $\lambda$ obeys $\lambda_1 < \lambda \leq \lambda_1 +\Delta \lambda$ is given by $Prob(\lambda_1) \Delta \lambda$ for small $\Delta \lambda$, and where $Prob(\lambda)$ is as you defined it above.