# Using perturbation to calculate first order correction

## Homework Statement

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##

## Homework 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
$$\delta \hat {\mathbf H} = -\frac {e^2} {4 \pi \varepsilon_0} \left( \frac {b} {r^2} - \frac 1 r \right)$$

## 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?
Thanks

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:
Leechie