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Derive delta potential bound states from finite square well

  1. Oct 25, 2008 #1
    1. The problem statement, all variables and given/known data
    I have to show that the delta function bound state energies can be derived from the finite square well potential.


    2. Relevant equations
    The wave functions in the three regions for the finite square well. (See wikipedia)


    3. The attempt at a solution
    1. I start from the wave functions and apply the boundary conditions.
    2. The coefficient to the Sin portion in the well would vanish and so(B=G=H in the wiki article).
    3. Now I am stuck. I try to use the normalization condition but there is not just E (in the form of alpha or k but also the coefficient.
     
  2. jcsd
  3. Oct 25, 2008 #2
    Don't start from boundary conditions etc.: you already have the exact solutions of the finite square well. All you have to do is take limits in the right way so as to turn a finite square well into a delta-function potential. How would you do that?
     
  4. Oct 25, 2008 #3
    I tried that as well. For example I started with Griffiths 2.153:
    [tex] E_n + V_o = (n^2 \pi^2 (h/2\pi)^2)/(2 m (2a^2)) [/tex] but not only that [tex]V_o[/tex] is approaching infinity but also that a is approaching 0. My attempt to make it so that the product of 'a' and [tex] V_o[/tex] become a constant have been fruitless.
     
  5. Oct 25, 2008 #4
    But that is the correct way, V_0 to infinity, a to zero, with the product of a and V_0 fixed...to what?
     
  6. Oct 25, 2008 #5
    The energy you mention is wrong though, that is not the equation for the finite square well.
     
  7. Oct 25, 2008 #6
    Now I need help :)
     
  8. Oct 25, 2008 #7
    Since you have Griffiths' book, you can look up the whole solution for the finite square well, and take the right limits.
     
  9. Apr 5, 2009 #8
    I need help. :)
    given that
    V(x) = 0 , lxl < a
    = V_o , lxl >a

    with V_o > 0

    Calculate the energy levels and plot the eigen functions for the three bound states of this system when V_oa^2=(6h-bar)/m.

    I duno what to do with this question, anybody who can tell me what should I do??
     
  10. Apr 5, 2009 #9
    correction(sory)
    given that
    V(x) = 0 , lxl < a
    = V_o , lxl >a

    with V_o > 0

    Calculate the energy levels and plot the eigen functions for the three bound states of this system when V_oa^2=(6h-bar^2)/m.

    I duno what to do with this question, anybody who can tell me what should I do??
     
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