(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In the Drude model the probability of an electron having a collision in an infinitesimal time interval dt is given by dt/[tex]\tau[/tex].

(a) Show that an electron picked at random at a given moment will have no collisions during the next t seconds with probability e^{-t/[tex]\tau[/tex]}.

(b) Show that the probability that the time interval between two successive collisions of an electron falls in the range between t and t + dt is (dt/[tex]\tau[/tex])e^{(-t/[tex]\tau[/tex])}

(c) Show as a consequence of a) that at any moment the mean time up to the next collision averaged over alll electrons is [tex]\tau[/tex].

(d) Show that as a consequence of b) that the mean time between successive collisions is [tex]\tau[/tex].

2. Relevant equations

Probability of a collision per unit time = t/[tex]\tau[/tex]

Poisson Distribution of Random Variables, Poisson(k,[tex]\lambda[/tex])= ([tex]\lambda[/tex]^{k}e^{-dt/[tex]\tau[/tex]})/k!

3. The attempt at a solution

So I proved part (a) by using the Poisson Distribution of RV's. Part (b) I tried to do the same thing as part (a), but for the time interval I used (t+dt)-t which gave me a lambda of dt/[tex]\tau[/tex]. Then I used k = 1 and went from there and it worked until the exponent where I got e^{-dt/[tex]\tau[/tex]}rather than e^{-t/[tex]\tau[/tex]}. Part (c) and (d) are where I get lost and have no clue of what to do.

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# Homework Help: Poisson Statistics in Solid State Physics

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