Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where.
-((hbar)2)/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ
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.