On the quantum theory of radiation

[A_B]

Hi,

In section 3 of Einstein's "On the quantum theory of radiation" Einstein says equations (8) and (9) follow from (7) and Wien's displacement law. I don't see how that is. For example, if we replace (8) by
[tex]
\frac{A_m^n}{B_m^n} = \alpha \frac{\nu}{T} \nu^3
[/tex]
All conditions still seem to hold. What am I missing?

Also, can somebody point me to a derivation of Wien's displacement law that does not assume Planck's formula?


Thank you,

A_B

 

[Bill_K]

He says A and B are constants which are "characteristic for the combination of indices considered". The indices referred to are ε_m and ε_n, the initial and final energy of the state, and in particular the energy difference ε_m - ε_n, that is, the frequency ν of the emitted radiation. They cannot depend on the temperature T. So the numerator A/B must be a function of ν only, and from Wien's Law it must be proportional to ν³, giving us Eq 8.

Likewise from Wien's Law the denominator must be a function of ν/T, which implies Eq 9.