Where am I going wrong in my radial equation substitution derivation?

[MxwllsPersuasns]

Homework Statement

Essentially we are describing the ODE for the radial function in quantum mechanics and in the derivation a substitution of u(r) = rR(r) is made, the problem then asks you to show that {(1/r^2)(d/dr(r^2(dR/dr))) = 1/r(d^(2)u/dr^2)

Homework Equations

The substitution: u(r) = rR(r)
The ODE: {(1/r^2)(d/dr(r^2(dR/dr))) = 1/r(d^(2)u/dr^2)

The Attempt at a Solution

So first I decided to try differentiation u once; I got du/dr = d{rR(r)}/dr = R(r) + r{dR(r)}/dr which didn't look promising and then differentiating again I get: dR(r)/dr + {dR(r)/dr + d^(2)R(r)/dr^2} = 2dR(r)/dr + d^(2)R(r)/dr^2 this also didn't lead anywhere.

Next I tried starting from the ODE and solving for R(r), hoping to find that R(r) = u(r)/r
Starting with {(1/r^2)(d/dr(r^2(dR/dr))) = 1/r(d^(2)u/dr^2) I first multiplied by r^2 to find...
- d/dr{r^2(dR/dr)} = r{d^(2)u/dr^2} Then I integrated wrt r...
- r^(2)dR/dr = {r^(2)/2}du/dr The r^2's cancel and we find R = u(r)/2 which is off by a factor of (1/r)

I'm assuming I must be losing a factor of (1/r) somewhere along the way but can't quite see where.. I think I remember some little "trick" with the whole (1/r^2)(d/dr(r^2(dR/dr)) term from classical mechanics but can't quite remember how to manipulate it. Any insight into where I'm going wrong is GREATLY appreciated. Cheers!

 

[BvU]

No trick. Straightforward
$$u'= rR'+ R \quad \Rightarrow\quad u'' = R'+ r R'' + R'\quad \Rightarrow\quad {u''\over r} = R'' + 2{R'\over r}$$

(Your

this also didn't lead anywhere

But it's exactly what you want:
$$ {1\over r^2 } {d\over dr} \left (r^2 R'\right ) = {1\over r^2 } \left (2rR' + r^2R'' \right )$$ Bingo

 

[MxwllsPersuasns]

Hey thanks! I actually found the error in my thinking as I was waiting for a response but your response confirmed it for me and I thank you for your prompt and friendly response BvU. Have a nice day :)

 

[BvU]

My pleasure.