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Derivation of Planck Function



After having taught college level physics for the past five years, I am now teaching physics at the secondary level.  My lesson plan for the “introduction to quantum mechanics” involves an illustration of Max Planck’s quantum mechanics explanation of cavity radiation.  I set the stage by having the students use their graphing calculators to play with the function y=x^3/((e^cx)-1) where c=0.95, 0.85, …, 0.45.  By doing this, they establish for themselves that the family of curves are bounded (i.e., no ultraviolet catastrophe). 

 

I then schematically derive the Planck Function by introducing the concept of the number of energy modes of cavity radiation per unit frequency proportional to frequency cubed, and then the number of modes per unit volume per frequency range.  I next discuss Planck’s rejection of equal energy participation over all modes—which leads to the “catastrophe”—and discuss how Planck worked on the assumption that the energy existed only in integral multiples of some lowest “quantum” amount that was proportional to the frequency. 

 

It is when one next gets to the concept of the spectral density per frequency interval and the average number of photons per mode that I find really difficult to explain to the students; i.e., the statistical mechanics of the problem.  Of course, one gets out of this the average number of photons per mode being 1/(e^ (beta h nu)-1). 

 

If anyone has come up with a way to best present these statistical mechanics ideas to second level students in a precise but relatively uncomplicated manner, I would be most happy to hear from you.

 

Best regards,

 

William J. Lutschak
The Park at White Oak
Houston, Texas 77009