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How to use extra term in radial equation

  1. Apr 28, 2007 #1
    1. The problem statement, all variables and given/known data

    Due to the modification of inner-shell electrons of a multi-electron system,

    the outer shell electron can feel an effective electrostatic potential as

    V(r)=-e^2/(4*π*eo*R)-lambda*(e^2/(4*π*eo*R^2)) ; 0<lambda≤1

    Find the energy eigenvalues and wavefunctions of the outer shell electron

    and compare to those of the hydrogen atom

    2. Relevant equations

    Radial equation

    -hbar^2/2*m(d^2u/dr^2)+[(V(r)+(hbar^2/2*m)*l(l+1)/R^2]

    Eigenenergies for hydrogen atom

    En=-[(m/2*hbar^2)*(e^2/4*π*e0)^2]*(1/n^2)=E1/n^2



    3. The attempt at a solution

    i plugged the effective potential into the radial equation, divided by E

    (E=-hbar^2*k^2/(2*m)) where k=sqrt(2mE)/hbar

    and get an equation with the same general solutions as in my book

    u(rho)=(rho^(l+1))*(e^(-rho))*(v(rho))

    still, how does this term effect the energy compared to that with hydrogen's

    eigenenergies? i assume a const term with lambda in it finds it way into the

    eigenenergies, but how to solve this is beyond me!


    how do i calculate the new energy eigenvalues and wave functions?

    how do they compare to that of hydrogen?

    cheers
    nate
     
  2. jcsd
  3. Apr 28, 2007 #2
    My feeling is that you could rescale the r-axis to get this problem solved.
    The additional term in the potential is similar to the angular momentum term.
    Therefore, the effect should be equivalent to the same problem with r rescaled and V(r) rescaled correspondingly.
    But that is only my guess ...
    Hope it helps.
     
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