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IalChange of variables/verifying solution

  1. Mar 26, 2016 #1
    1. The problem statement, all variables and given/known data

    Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where.

    2. Relevant equations

    Radial Schrodinger:

    -((hbar)2)/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ

    3. The attempt at a solution

    We're first told to replace rψ(r) with U(r/a). For this I got the following:

    -((hbar)2)/2M * [(1/r)(d/dr)2(U(r/a)) - l(l+1)/(r3) *U(r/a)] - α(hbar)c/r2 *U(r/a) = (E/r)*U(r/a)

    The next step is to use x=r/a to change variables to x. a=hbar/α*M*c This leads me to:

    -((hbar)2)/2M * [(1/xa)(d/dxa)2(U(x)) - l(l+1)/(xa)3) *U(x)] - α(hbar)c/(x*a2) *U(x) = (E/xa)*U(x)

    Then we replace E by ε=-2E/(α2 *M*c2). This gives the final form (after some simplifying):

    (d/d(ax))2)U(x)=U(x)(ε/a2 + l(l+1)/(xa)2 -2/x*a2)

    Then we're to check that (x2)*e(-(x2)) is a solution to the equation.

    Plugging that in gives

    (d/d(ax))2)(x2)*e-(x2)=(x2)*e-(x2)(ε/a2 + l(l+1)/(xa)2 -2/x*a2)

    After taking the second derivative (which I got as (x4 -5x2 +2)*e-(x2))/a2), I ended up with:

    e-(x2)(x4 -5x2+2)=e-(x2)(εx2 + l(l+1) -2x)

    I'm pretty sure this means I went wrong somewhere, as I think I should have an equivalent expression on the left and right. If anyone can see where I might have made a mistake, it'd be very helpful.
  2. jcsd
  3. Mar 31, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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