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Exam preparation question

  1. Apr 1, 2010 #1
    1. The problem statement, all variables and given/known data

    I was reviewing some homework problems and looking at the solutions. There is one problem with a tiny step I just cannot rationalize and I am hoping someone can point me in the right direction.

    I have a spherical finite well:

    [tex] V = {- V_{0}: 0 < r < a}[/tex],

    [tex] = {0: r \geq a} [/tex]

    [tex]- k_{2} = k_{1} cot (k_{1} a)[/tex] (1)

    Refining the notation,

    [tex]\alpha = a \sqrt{(2m(E + V_{0})}/hbar = k_{1} a[/tex]

    [tex]R = a \sqrt{(2m(V_{0})}/hbar[/tex] and [tex]k_{2} = \sqrt{(2m(V_{0})}/hbar[/tex]

    So (1) may be rewritten as [tex] \sqrt{R^{2} - \alpha^{2}} = - \alpha cot (\alpha)[/tex]

    2. Relevant equations

    From part 1.

    3. The attempt at a solution

    I don't understand how at [tex] R = \pi/2 [/tex] there are no bound states.

    Also, I am given this restriction: [tex] -V_{0} < E < 0 [/tex]

    How is this justified and how is the precise range of bound states determined?
  2. jcsd
  3. Apr 4, 2010 #2


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    When [itex]R=\pi/2[/itex], the only solutions are at [itex]\alpha=\pm\pi/2[/itex]. In these cases, you get k2=0.
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