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Potential value , Bound state

  1. Nov 27, 2013 #1
    Hi,
    1. The problem statement, all variables and given/known data

    A particle of mass m has a potential
    V(r)= -Vo r<a
    0 r>a

    Find the minimum value of Vo for which there's a bound state of energy and angular momentum are zero by solving shrodinger equation for E<0 and taking the limit E-> 0

    2. Relevant equations



    3. The attempt at a solution
    I want to know what does l represent in the radial shrodinger equation ? Is it the angular momentum ?
    For now, I considered l=0 and solved Shrodinger equation
    I got
    r<a
    u(r)=C*sin(βr) with β=sqrt(2m(Vo+E)/h²)
    r>a
    u(r)=Bexp(-kr) with k= sqrt(-2mE/h²)

    With boundary conditions we have
    -β/k=tan(βa)

    Where am I supposed to use E->0 ..?
    Is it just in β=sqrt(2m(Vo+E)/h²)
    so that Vo=β²h²/2m ?
     
    Last edited: Nov 27, 2013
  2. jcsd
  3. Nov 28, 2013 #2

    TSny

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    You need to let E → 0 in both β and k.
     
  4. Nov 28, 2013 #3
    So we have that k=0 and then ? I dont Know what i have to do then
     
  5. Nov 28, 2013 #4

    TSny

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    What is the limit of β/k as E → 0? What does that tell you about the quantity βa in tan(βa)?
     
  6. Nov 28, 2013 #5
    We have tan(aβ)= -β/k=-sqrt(-1-Vo/E)
    so when E-> 0-
    tan(aβ)=-infinity
    so βa=-pi/2 ?
     
    Last edited: Nov 28, 2013
  7. Nov 28, 2013 #6

    TSny

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    Yes, except for the sign. β is a positive quantity.
     
  8. Nov 28, 2013 #7
    OK Thank you !
     
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