. (b)
At very high frequency, the electrons in the metal are essentially oscillated
by the electric field without any drifting. Write down the equation of
motion of an electron and then show that 
.
2. Conductivity for free electrons at high frequency.
(a) Conductivity is defined as = j/E. Since j and E are not necessary in
phase, can be complex. Let the conductivity of a metal be () = '() + i"(),
where '() and "() are the real part and imaginary part of the function
respectively. Use Kramers-Kronig relation, show that at high frequency
, . (b)
At very high frequency, the electrons in the metal are essentially oscillated
by the electric field without any drifting. Write down the equation of
motion of an electron and then show that .
(c) Further prove that 
.
(d) The electron concentration in a copper sample is 8x1022
cm-3, mean free path is ~400, and the Fermi velocity is 1.6x108
cm s-1. Give a measure on the meaning of "high frequency" for
the above results to be valid.
Hint. In case if you forget Kramer-Kronig relation, you can derive
it by calculating the integral 
,
and compare real and imaginary parts. Assume () has no pole in the upper
half of the complex plane, and '() is even and "() is odd.
Solution:
