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Relation between angular momentum and total engery

  1. Dec 1, 2009 #1
    1. The problem statement, all variables and given/known data
    a) Express the total energy of an electron in the Coulomb potential of proton through the electron's angular momentum L and the shortest distance a between the proton and the electron's orbit. Hint: The electron's velocity is perpendicular to it's position vector whenever it is distance a away from the proton.

    b) For a fixed L, minimize the expression found in (a) with respect to a. Show that the minimum corresponds to the case of a circular orbit. State the minimum value of the total energy for fixed L.


    2. Relevant equations
    L=sqrt(l2+l)ћ
    Lz=mlћ
    L=r x mv=Iω
    En=-mee4/[2(4πε0)2ћ2n2]
    n > l > |ml| > 0
    KE=.5Iω2

    3. The attempt at a solution
    E=U+KE => E= .5L*v/a - e^2/[4πε0a]
    I have little confidence in that as a solution. Either I've stopped short or I'm going in the wrong direction, I'm not sure. I really only want help with part (a)... Once I get that far I should be handle (b) somewhat easily, I just wanted to provide context for the problem.
     
  2. jcsd
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