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QM - delta function potential

  1. May 22, 2008 #1
    1. The problem statement, all variables and given/known data
    write the radial equation for a particle with mass m and angular momentum l=0 which is under the influence of the following potential:
    V(r)=-a*delta(r-R)
    a,R>0
    write all the conditions for the solution of the problem.


    2. Relevant equations

    Schroedinger's equation:
    Hu=Eu
    Hamiltonian: H=p/2m +V = pr/2m+L^2/2mr^2+V(r)

    3. The attempt at a solution

    since the angular momentum is zero, the radial equation appears as:
    (-hbar/2m)(d^2u/dr^2)-a*delta(r-R)u=Eu
    the conditions I can think of are:
    1) continuity of u
    2) u(infinity)= 0 (for u to be square integrable)
    3) from integration of Schroedinger's equation on the interval [R-epsilon, R+epsilon] the jump in the first derivative of u at r=R should be -2mau(R)/hbar^2

    but there is another condition according to the answers, that is, u(0)=0.
    where does this condition come from?
     
  2. jcsd
  3. May 23, 2008 #2
  4. May 23, 2008 #3

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    you mean why u(0) = 0 ?

    This is due to that [tex] \Psi (r) = \frac{u(r)}{r} [/tex] so [tex] u(r) [/tex] must go to zero faster than r, in order to have a bounded wave function [tex] \Psi (r) [/tex].
     
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