Using perturbation to calculate first order correction

[Leechie]

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

 

[John Park]

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 R_nl(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.

 

[Leechie]

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!