IalChange of variables/verifying solution

[kwagz]

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)/(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.

 

[mighty2000]

After reviewing your attempt at the solution, I believe the mistake lies in your substitution of x=r/a. When you make this substitution, you should also change the differential operator d/dr to d/dx, which will change the second derivative term in your final equation. The correct form should be:

(d/dx)^2 U(x) = U(x)(ε/a^2 + l(l+1)/x^2 - 2/ax^2)

After making this correction, you should get the following expression for the second derivative:

(d/dx)^2 (x^2 e^(-x^2)) = (x^2 e^(-x^2))(ε/a^2 + l(l+1)/x^2 - 2/ax^2)

When you simplify this expression, you should get an equivalent expression on the left and right, confirming that (x^2)e^(-x^2) is indeed a solution to the Schrodinger equation. I hope this helps!