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vst98
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Homework Statement
Show that radial components of the continuum electron wave function
satisfies the radial equation:
[itex]{\left[\frac{-\hbar }{2m}\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{\hbar ^2l(l+1)}{2m
r^2}-\frac{Z e^2}{r}\right]R=E R}[/itex]
where:
E=(k*hbar)2/2m
R=Rkl(r)
Homework Equations
so, the radial component R is given as:
[itex]{R_{\text{kl}}(r)=\frac{C_{\text{kl}}}{(2l+1)!}(2{kr})^l\text{Exp}[-\text{ikr}]*F(i/k+l+1,2l+1;2\text{ikr})}[/itex]
The Attempt at a Solution
First I have rewritten R as:
Rkl(r)= C rl Exp[-ikr] F[i/k + l + 1,2l+1;2ikr]
where in C is everything that does not depend on r.
Great, I was thinking this will be pretty straightforward to show.
My reasoning was to insert R in the above radial equation, take those derivatives and at the end I will be able the factorize R out, so that I'm left with something like
HR=AR where A is the energy.
It did not work out like that
After taking the derivatives I'm left with a mess and I don't see how to factorize R out.
I mean, derivatives of Exp[-ikr] give me back the same function (multiplied by constant) so I can factorize that out but rl and F give me problems after derivation.
Also I was looking at the properties of these confluent hypergeometric functions F
and tried to adjust them back to original function but without success.
Can someone say whether I'm on the right track trying to solve the problem like that or
have I completely missed the point ?