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## Homework Statement

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

## Homework Equations

[/B]

Radial Schrodinger:

-((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)/(r

^{3}) *U(r/a)] - α(hbar)c/r

^{2}*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*a

^{2}) *U(x) = (E/xa)*U(x)

Then we replace E by ε=-2E/(α

^{2}*M*c

^{2}). This gives the final form (after some simplifying):

(d/d(ax))

^{2})U(x)=U(x)(ε/a

^{2}+ l(l+1)/(xa)

^{2}-2/x*a

^{2})

Then we're to check that (x

^{2})*e

^{(-(x2)}) is a solution to the equation.

Plugging that in gives

(d/d(ax))

^{2})(x

^{2})*e

^{-(x2)}=(x

^{2})*e

^{-(x2)}(ε/a

^{2}+ l(l+1)/(xa)

^{2}-2/x*a

^{2})

After taking the second derivative (which I got as (x

^{4}-5x

^{2}+2)*e

^{-(x2))}/a

^{2}), I ended up with:

e

^{-(x2)}(x

^{4}-5x

^{2}+2)=e

^{-(x2)}(εx

^{2}+ 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.