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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

    BvU

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    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
    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

    BvU

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    My pleasure.
     
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