1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Radial Equation substitution

  1. Dec 8, 2016 #1
    1. The problem statement, all variables and given/known data
    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)

    2. Relevant 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)

    3. 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!
  2. jcsd
  3. Dec 8, 2016 #2


    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    No trick. Straightforward
    $$u'= rR'+ R \quad \Rightarrow\quad u'' = R'+ r R'' + R'\quad \Rightarrow\quad {u''\over r} = R'' + 2{R'\over r}$$
    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
  4. Dec 8, 2016 #3
    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 :)
  5. Dec 8, 2016 #4


    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    My pleasure.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Radial Equation substitution