Question

On the basis of Eq,(6.4g) find the number of free electrons in a metal at `T=0` as a function of de Broglie wavelengths.

Answer 1

Using the formula of the previous section
`dn(v)=(m^(3))/(pi^(2)ħ^(3))v^(2)dv`
We put `mv=(2piħ)/(lambda)`, where `lambda=` Broglie wavelength
`mdv(2piħ)/(lambda^(2))d lambda`
Taking account of the fact that `lambda` decreases when `v` increase we write
`db(lambda)= -dn(v)=((2pi)^(3)dlambda)/(pi^(2)lambda^(4)).=(8pi)/(lambda^(4))d lambda`