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Homework Help: Using perturbation to calculate first order correction

  1. Mar 16, 2017 #1
    1. The problem statement, all variables and given/known data
    I'm trying to evaluate the following integral to calculate a first-order correction:
    $$\int_0^\infty R_{nl}(r)^* \delta \hat {\mathbf H} R_{nl}(r) r^2 dr$$
    The problem states that ##b## is small compared to the Bohr radius ##a_o##
    2. Relevant equations
    I've been given:
    $$R_{nl}(r)=\left( \frac {1} {a_0} \right)^{3/2} 2 e^{-r/a_0}$$ $$\int_0^x e^{-u} du = 1 - e^{-x} $$ $$\int_0^x u e^{-u} du = 1 - e^{-x} - xe^{-x} $$
    And I've calculated ##\delta \hat {\mathbf H} ## to be
    $$\delta \hat {\mathbf H} = -\frac {e^2} {4 \pi \varepsilon_0} \left( \frac {b} {r^2} - \frac 1 r \right)$$
    3. The attempt at a solution
    So far I've got:
    $$E_1^{(1)} = - \frac {e^2} {4 \pi \varepsilon_0 a_0^3} \left( \int_0^b b e^{ - \frac {2r} {a_0} } dr - \int_0^b r e^{ - \frac {2r} {a_0} } dr \right) $$
    And when I use the integrals given I get:
    $$E_1^{(1)} = - \frac {e^2} {4 \pi \varepsilon_0 a_0^3} \left( \frac {a_0b} {2} \left(1-e^{-b} \right) - \frac {a_0^2} {4} \left(1-e^{-b}-be^{-b} \right) \right) $$
    Could someone tell me if I've got this right so far?
  2. jcsd
  3. Mar 16, 2017 #2
    Is your integral in section 1 missing a factor of 4π?

    In the first equation in section 3 I think you've lost the factor of 2 from each Rnl(r). And when you put in the limits of integration, the exponents in your final equation should be -2b/a0.

    It's not clear to me what the perturbation is supposed to be; so I don't know if your δH is correct or if you should be integrating over r from 0 to b; but if all that is okay, you're basically on the right track.
    Last edited: Mar 16, 2017
  4. Mar 16, 2017 #3
    Thanks for your help John.

    I don't think its missing the factor 4##\pi##. The original state is given as ##\psi_{1,0,0} = R_{nl} \left( r \right) Y_{lm}\left( \theta, \phi \right) ##, and I made it that the spherical harmonics are normalized in the state: ##Y_{0,0}\left( \theta, \phi \right) = \frac 1 {\sqrt {4 \pi}}##.

    Ah, I can see it now, the missing factor 2 and I've forgotten to change the exponents back after substitution. Thanks for pointing those out.

    I confident the perturbation is correct and the question I've been given involves calculating the first-order correction for a modified Coulomb model of a hydrogen atom where ##V\left(r\right)=- \frac {e^2b^2} {4 \pi \varepsilon_0 r^2} ## where ## 0 \lt r \leq b ## and ##V\left(r\right)=- \frac {e^2} {4 \pi \varepsilon_0 r} ## where ## r \gt b##. So I think I have to integral from ##0## to ##b##.

    I'm quite new to this forum and I'm really amazed at how helpful and friendly everyone is here. I don't know where I'd be without PF!
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