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How do i find minimum for this potential?

  1. Dec 12, 2005 #1
    how do i find minimum for this potential:

    u(r)=L^2/2*m*r^2 - b*exp(-ar)/r ?

    b,a - constants
    L - angular momentum
    m - mass

    u`(r)=L^2/m+b*exp(-ar)*r(a*r-1)

    (u`(r*):=0 => r*=r*(a,b,L,m) )

    i plotted the function for some values of b,a,L and m and i see that for certain values there is a minimum but i just can't figure out how do i find the right r.

    can anyone help/give some hint?
     
  2. jcsd
  3. Dec 12, 2005 #2

    siddharth

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    If
    [tex] U(r) = \frac{L^2}{2mr^2} - \frac{b e^{-ar}}{r} [/tex]
    then, what you have written for U'(r) is not correct.
     
  4. Dec 12, 2005 #3
    ok, the correct expression for u`(r):

    u`(r)=L^2/m-b*r*(a*r+1)exp(-a*r)

    still i don't quite know how to proceed...
     
  5. Dec 12, 2005 #4

    siddharth

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    How is that the correct expression?
    What is
    [tex] \frac{d}{dr} (\frac{1}{r^2}) [/tex]?

    and do you know how to find
    [tex] \frac{d}{dr} (\frac{e^{-ar}}{r}) ? [/tex]
     
    Last edited: Dec 12, 2005
  6. Dec 12, 2005 #5
    siddhart , so sorry
    i simply already did some rearrangemnet:

    [tex]
    \frac{du}{dr}=\frac{L^2}{mr^3} + \frac{b(are^{-ar} + e^{-ar})}{r^2}
    [/tex]

    the expression i wrote is when i do
    u`(r):=0
    and i multiply by r^3
    so sorry for confusion
     
    Last edited: Dec 12, 2005
  7. Dec 12, 2005 #6

    Integral

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    Wormhole,
    To maximize the readability of your equations spend a few minutes reading this LaTex tutorial
     
  8. Dec 12, 2005 #7

    siddharth

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    Ok, by solving U'(r)=0, you can find the critical points and then you can find the points of minima with the second derivative test.


    On second thought, the equation you have to solve to find r looks really tough :surprised . I can find no obvious way to solve it.

    For values of (ar)<<1, you could approximate by using the series expansion of [tex] e^{ar} [/tex]
     
    Last edited: Dec 12, 2005
  9. Dec 12, 2005 #8
    i'll try to do that...
    from the plots i did the r where u(r) is at minimum is very close to zero
    i have to take only first term in [itex]e^{-ar}[/itex] expansion series otherwise i get an third order equation
     
    Last edited: Dec 12, 2005
  10. Dec 12, 2005 #9

    siddharth

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    Can you post your solutions for r when you get it? This is an interesting problem and I want to check that I got the right answers so I can show my classmates.
     
    Last edited: Dec 12, 2005
  11. Dec 16, 2005 #10
    i asked my classmates about the solution and some say that there is no need to use expansion series...
    i'm not sure what they did is right..so when i get the official solution(next week) i will give you a link to it
     
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