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What makes the wide/deep well and the shallow/narrow well, as Griffiths puts it the “two limiting cases of special interest”?(pg 80) What about the wide/shallow well and the narrow/deep wall??

The wide/deep well and the shallow/narrow well correspond with the endpoints of the graph on page 80 - the two cases you're asking about are sort of like “mixed” cases: it seems like if a potential was wide it would also be deep (for a generally “large” potential), or if it was shallow it would also be narrow (small). This is described by what Griffiths calls z0, which has terms for both a and square root of V0. It seems like as z0 and tan(z) intersect, only the product of V0 and a determine specific points, not one or the other. So in the limit of high or low z0, the well must be both wide and deep, or shallow and narrow, or a z0 with either a or V0 dominating the other.

Put another way, if you want to look at a well that is both wide and shallow, you have to decide - is it primarily wide, or primarily shallow? Which term dominates z0, the width or the depth? The question of a wide/shallow well is kind of ambiguous in that sense: in general, depending on which term is more important, the question can have more than one answer.

In discussion, Justin mentioned that the first excited state of the double delta potential we worked on only sometimes exists. I believe he said it was dependent on the spacing <math>a</math> (and <math>\alpha</math>?). How can you determine whether this second solution exists for a particular <math>a</math> and <math>\alpha</math>?

If you solve the wave function and apply the boundary conditions, the following relationship can be achieved:<math>\frac{\hbar^2k}{m\alpha}-1=\pm exp(-2ak)</math>. Then sketch the left-hand-side and right-hand-side respectively to solve k. Note k must be positive. When <math>\frac{\hbar^2}{m\alpha}<2a</math> (line a), there are two possible values for k, and in this condition the second solution exists. When <math>\frac{\hbar^2}{m\alpha}\ge2a</math> (line b or c), there is only one possible value for k, in this condition no excited state exists.

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