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Solid State Physics [Ashcroft] Chapter 1, Question 1a!

  1. Oct 4, 2006 #1
    Hi all,

    Edit: Here's the question, in case you guys don't have the book:

    In the Drude moel the probability of the electron suffering a collision in any infinintesimal interval dt is just dt/T.

    a) show that an electron picked at random at a given moment had no collision during the preceding t seconds with the probability exp[-t/T]. Show that it will have no collision during the next t seconds with the same probability.


    I'm trying to figure out how to get the exp[-t/T] part, 'cause nowhere in the chapter does it show a probability as XX exp[-t/T].

    Please just help me get started. Any reference equations would be good...

    Thanks!
     
  2. jcsd
  3. Oct 4, 2006 #2
    Lets say our randomly picked electron was selected at time [itex]t_0[/itex], and we want to find the probability that this electron suffered no collisions from [itex]t=0[/itex] to [itex]t=t_0[/itex].
    We know that the probability an electron to be scattered in an infintesimal interval [itex]dt[/itex] is

    [tex]
    P_{c}(t,t+dt) = \frac{dt}{\tau}.
    [/itex]

    Now the probability for the electron to suffer a collision, and the probability for it not to suffer a collision in this interval must sum to [itex]1[/itex]. So

    [tex]
    P_c(t,t+dt) + P_{nc}(t,+dt) = 1
    [/tex]

    Hence,
    [tex]
    P_{nc}(t,t+dt) = 1 - \frac{dt}{\tau}.
    [/tex]

    Now I will split the time interval [itex][0,t_0][/itex] into [itex]M[/itex] discrete intervals, such that

    [tex]
    t_0 = \sum_{n=1}^{M}dt = Mdt.
    [/tex]

    If I label the probability to not scatter in the interval [itex][0,ndt][/itex] as [itex]P_n[/itex], then I have

    [tex]
    P_1 = (1-\frac{dt}{\tau}),
    [/tex]

    and

    [tex]
    P_2 = P_1\times (1-\frac{dt}{\tau}) = (1-\frac{dt}{\tau})^2
    [/tex]

    The probability to not scatter from [itex]t=0[/itex] to [itex]t=Mdt=t_0[/itex] is

    [tex]
    P_M = \ldots
    [/tex]

    Once you have your expression for [itex]P_M[/itex], eliminate from this expression [itex]dt[/itex] using the definition of [itex]t_0[/itex], and notice that

    [tex]
    \lim_{dt\to 0} \equiv \lim_{M\to\infty}
    [/tex]

    Then you will have the limit of some function to evaluate...
     
    Last edited: Oct 4, 2006
  4. Oct 4, 2006 #3
    Hey,

    Thanks for that! So if I divide up the t0 time into intervals, I can use the infinitesimal probability:

    P(t, t+t0) = P(t, t+ndt) {n--> infinity}

    so:
    P(t, t+ndt) = (1-dt/T)^n, now we sub the definition of t0
    P(t, t+t0) = (1 - t0/nT)^n, taking the lim n--> infinity
    P(t, t+t0)=exp[-t0/T]

    How do I distinguish between "before" and "after" t0 seconds? It looks like it's the same both ways, but how do I be rigorous about it?

    Thanks!!
     
  5. Oct 4, 2006 #4
    Just use the same logic as before. You have an expression for the particle not to have a collision from t=0 up to t=t_0 . Consider the probability for not scattering from t=0 to t=2t_0 (using the expression you derived). Express this in terms of the probability of not scattering from t=0 to t=t_0, and the probability not to scatter from t=t_0 to t=2t_0.
     
  6. Oct 7, 2006 #5
    Or you could use a Poisson distribution to evaluate the probability. Then you get the exponential function naturally from the poisson distribution.
     
  7. Mar 15, 2007 #6

    berkeman

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    Staff: Mentor

    Welcome to the PF, Ruy. We do not provide answers here on the PF, we provide tutorial help. If you have a specific problem that you need help on, please start a thread with the question, using the Homework Help template, and show us what work you've done so far.
     
  8. Mar 27, 2007 #7

    fes

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    Somebody would have the subject 1c 1d 1e

    1c)show as consequece of (a) ....
    1d) show as consequence of (b) ....
    1e) part (c0 implies that.....
    Thanks
     
  9. Mar 27, 2007 #8

    berkeman

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    Staff: Mentor

    I'm not sure what you are asking. What is your question? It is probably best for you to start a new thread here in the Homework Help forums, use the Homework Help question template that is provided, and show your work so far so that we can help you in a tutorial way.
     
  10. Feb 18, 2010 #9
    Hi every body, I am new comer and be interested in Solid State Physics [Ashcroft]. Has anyone have solution for the book. chapter 9, Problem 2(Density of levels...)?
    I am confused by intergarating delta function with separated variable as shown in.
    thank you
     
  11. Feb 18, 2010 #10

    berkeman

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    Staff: Mentor

    Welcome to the PF, Repetit. You should start a new thread with your question. Be sure to use the Homework Help Template that you are given when starting new homework question threads, and fill out the sections on the Relevant Equations and show your Attempt at a Solution.

    We certainly don't do your schoowork problems for you, but we can usually offer tutorial hints that will help you to do the work yourself.
     
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