Kronig-Penney potential as spacing -> infinity

[quasar_4]

Kronig-Penney potential as spacing --> infinity

Homework Statement

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).

Homework 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).

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??

 

[Jhero]

Thank you for your post. I can definitely help you with this problem. The Kronig-Penney potential is a periodic potential that describes the potential energy experienced by an electron in a crystal lattice. As you mentioned, the spacing between the atomic sites is denoted by b and the width of the well is denoted by a. In the limit where b approaches infinity, the potential becomes a series of infinitely deep and infinitely wide wells, with no overlap between them. This is known as the "tight-binding" approximation.

To understand why the eigenenergies of the Kronig-Penney potential approach the eigenenergies of a 1D box as b approaches infinity, we can look at the dispersion relation you provided. As you correctly noted, the LHS of the dispersion relation becomes undefined as b approaches infinity. However, we can take the limit of the LHS as b approaches infinity as follows:

lim cos(k1*a)cosh(k*b) - [(k1^2 - k^2)/2*k1*k]* sin(k1*a)sinh(k*b) = lim cos(k1*a)*lim cosh(k*b) - [(k1^2 - k^2)/2*k1*k]* lim sin(k1*a)*lim sinh(k*b)

As b approaches infinity, both cosh(k*b) and sinh(k*b) approach infinity, but their ratio approaches 1. Similarly, as b approaches infinity, both sin(k1*a) and sinh(k*b) approach 0. This means that we can rewrite the dispersion relation as:

lim cos(k1*a)*lim cosh(k*b) - [(k1^2 - k^2)/2*k1*k]* lim sin(k1*a)*lim sinh(k*b) = lim cos(k*d)

Now, we can use the trigonometric identity cos^2(x) = 1-sin^2(x) to rewrite the LHS as:

lim [1 - sin^2(k1*a)]*lim cosh(k*b) - [(k1^2 - k^2)/2*k1*k]* lim sin(k1*a)*lim sinh(k*b) = lim cos(k*d)

We can then expand the LHS and use the fact that cosh(x) = cosh(-x) to get:

lim cosh(k1*a) - sin^2(k1*a)*lim cosh(k*b) - [(k1