- #1

- 51

- 0

## 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=R

_{kl}(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:

R

_{kl}(r)= C r

^{l}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 r

^{l}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 ?