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Kronig-Penney potential as spacing -> infinity

  1. Nov 29, 2008 #1
    Kronig-Penney potential as spacing --> infinity

    1. The problem statement, all variables and given/known data

    Show that in the limit that the atomic sites of the Kronig-Penney potential become far removed from each other (b-->infinity), energies of the more strongly bound electrons (E<<V) become the eigenenergies k1a=n*Pi of a 1D box of width a.

    (This is problem 16 in chapter 8 of Liboff, if you have this text).

    2. Relevant equations

    The b in the problem refers to the spacing of the atomic sites - if d is the spacing of the sites and a is the width of the well, a+b=d.

    There are only two equations in this whole section that even use b, so far as I can tell, and these are the "dispersion relations" that Liboff gets to determine the eigenenergies. There is one for the case of E>V and E<V. I believe the latter is the pertinent one for this problem (correct me if I'm wrong) and this is:

    E< V ==> cos(k1*a)cosh(k*b) - [(k1^2 - k^2)/2*k1*k]* sin(k1*a)sinh(k*b) = cos(k*d).

    3. The attempt at a solution

    Here is my problem. I am a bit confused in general about why this is a dispersion relation, and worse, the limits confuse me: cosh(b*k) and sinh(b*k) both blow up when b approaches infinity, so as far as I can tell the left hand side of the so-called dispersion relation becomes infinity-infinity which is undefined.

    If I ignore mathematics for a minute and play the hand-waving game I could maybe say that since the LHS goes to infinity but the RHS is finite, that the sin(k1*a) must** equal 0 and this would give us the necessary condition of k1*a=n*Pi... but that would only make the sine term 0, not the cosine term, and then I have something left like (-1)^n * infinity = cos(k*d).... I have NO idea what that means... so yeah... I am clearly doing something wrong.

    ** I know 0*infinity is no good either, but this is a tactic my professor has used on the board numerous times. I don't like it, but it seems to be a physics hand-wavy thing - maybe would work here??
  2. jcsd
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