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Introductory Physics Homework Help
Schrodinger Equation/verify solution
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[QUOTE="kwagz, post: 5422676, member: 589179"] [h2]Homework Statement [/h2] Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where. [h2]Homework Equations[/h2] [/B] Radial Schrodinger: -((hbar)[SUP]2[/SUP])/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ [h2]The Attempt at a Solution[/h2]We're first told to replace rψ(r) with U(r/a). For this I got the following: -((hbar)[SUP]2[/SUP])/2M * [(1/r)(d/dr)[SUP]2[/SUP](U(r/a)) - l(l+1)/(r[SUP]3[/SUP]) *U(r/a)] - α(hbar)c/r[SUP]2[/SUP] *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)[SUP]2[/SUP])/2M * [(1/xa)(d/dxa)[SUP]2[/SUP](U(x)) - l(l+1)/(xa)[SUP]3[/SUP]) *U(x)] - α(hbar)c/(x*a[SUP]2[/SUP]) *U(x) = (E/xa)*U(x) Then we replace E by ε=-2E/(α[SUP]2[/SUP] *M*c[SUP]2[/SUP]). This gives the final form (after some simplifying): (d/d(ax))[SUP]2[/SUP])U(x)=U(x)(ε/a[SUP]2[/SUP] + l(l+1)/(xa)[SUP]2[/SUP] -2/x*a[SUP]2[/SUP])Then we're to check that (x[SUP]2[/SUP])*e[SUP](-(x[SUP]2[/SUP])[/SUP]) is a solution to the equation. Plugging that in gives (d/d(ax))[SUP]2[/SUP])(x[SUP]2[/SUP])*e[SUP]-(x[SUP]2[/SUP])[/SUP]=(x[SUP]2[/SUP])*e[SUP]-(x[SUP]2[/SUP])[/SUP](ε/a[SUP]2[/SUP] + l(l+1)/(xa)[SUP]2[/SUP] -2/x*a[SUP]2[/SUP]) After taking the second derivative (which I got as (x[SUP]4[/SUP] -5x[SUP]2[/SUP] +2)*e[SUP]-(x[SUP]2[/SUP]))[/SUP]/a[SUP]2[/SUP]), I ended up with: e[SUP]-(x[SUP]2[/SUP])[/SUP](x[SUP]4[/SUP] -5x[SUP]2[/SUP]+2)=e[SUP]-(x[SUP]2[/SUP])[/SUP](εx[SUP]2[/SUP] + 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. [/QUOTE]
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Schrodinger Equation/verify solution
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