What is the Perturbation in the Modified Coulomb Model of a Hydrogen Atom?

[Leechie]

Homework Statement

Suppose there is a deviation from Coulomb's law at very small distances, with the mutual Coulomb potential energy between an electron and a proton being given by:
$$V_{mod}(r)= \begin{cases} - \frac {e^2} {4 \pi \varepsilon_0} \frac {b} {r^2} & \text {for } 0 \lt r \leq b \\ - \frac {e^2} {4 \pi \varepsilon_0} \frac {1} {r} & \text {for } r \gt b \end{cases}$$
where $e$ is the magnitude of the electon charge, $\varepsilon_0$ is the permittivity of free space, $r$ is the electron-proton separation and $b$ is a constant length that is small compared to the Bohr radius but large compared to the radius of a proton. Throughout this question, the perturbed systen, with $V(r)$ replaced by $V_{mod}(r)$, will be called the modified Coulomb model.

a) Specify the perturbation for the modified Coulomb model of a hydrogen atom relative to the unperturbed Coulomb model.

b) Use this perturbation to calculate the first-order correction, $E_1^{(1)}$ to the fround-state energy of a hydrogen atom in the modified Coulomb model, givesn that the fround-state energy eigenfunction for the unperturbed Coulomb model is:
$$\psi_{1,0,0} \left( r,\theta,\phi \right) = \left( \frac {1} {\pi a_0^3} \right)^{1/2} e^{-r/a_0} $$
c) Show that your answer to part (b) can be approximated by
$$E_1^{(1)} \approx - \frac {4b^2} {a_0^2} E_R$$ where $E_R = {e^2} / 8 \pi \varepsilon_0 a_0$ is the Rydberg energy. Hence deduce the largest value of $b$ that would be consistent with the fact that the ground-state energy of a hydrogen atom agrees with the predictions of the Coulomb model to one part in a thousand. Express your answer as a numerical multiple of $a_o$.

Homework Equations

$$\int_0^x e^{-u} du = 1-e^{-x}$$ $$\int_0^x u e^{-u} du = 1-e^{-x}-xe^{-x}$$ for $x \ll 1$,$$e{-x}=1-x+ \frac {x^2} {2} $$

The Attempt at a Solution

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

b)
$E_1^{(1)}= - \frac {e^2} {\pi \varepsilon_0 a_0^3} \left( \frac {a_0b} {2} \left(1-e^{-2b/a_0} \right) - \frac {a_0^2} {4} \left(1-e^{-2b/a_0}-\frac {2b} {a_0} e^{-2b/a_0} \right) \right)$

c)
This is where I'm having problems. I can get $E_1^{(1)} \approx - \frac {4b^2} {a_0^2} E_R$ by setting $e^{-2b/a_0}=1$ since $b \ll a_0$ and then integrate in the same way I did to get to answer (b), but should I be using my answer to part (b) to show $E_1^{(1)} \approx - \frac {4b^2} {a_0^2} E_R$ because I can't make it do that.

Also, I'm not sure how to proceed from here to deduce the largest value of $b$, and I'm a bit unclear to what the question means by "agrees with the predictions of the Coulomb model to one part in a thousand".

Can anyone offer and advice with this please.

 

[vela]

I got a different result for (b). Can you show us your calculations?

 

[Leechie]

This is how I got to (b):
$$\begin{align} E_1^{(1)} & = \int_0^\infty R_{nl}^*(r) \delta \hat {\mathbf H} R_{nl}(r) r^2 dr \nonumber \\ & = \int_0^b \left( \frac {1} {a_0} \right)^{3/2} 2e^{-r/a_0} \left( - \frac {e^2} {4 \pi \varepsilon_0} \right) \left( \frac {b} {r^2} - \frac 1 r \right) \left( \frac {1} {a_0} \right)^{3/2} 2e^{-r/a_0} r^2 dr \nonumber \\ & = \int_0^b \left( - \frac {e^2} {4 \pi \varepsilon_0} \right) \left( \frac {b} {r^2} - \frac 1 r \right) \frac {4} {a_0^3} e^{-2r/a_0} r^2 dr \nonumber \\ & = - \frac {e^2} {\pi \varepsilon_0 a_0^3} \int_0^b \left( \frac {b} {r^2} - \frac 1 r \right) e^{-2r/a_0} r^2 dr \nonumber \\ & = - \frac {e^2} {\pi \varepsilon_0 a_0^3} \int_0^b \left( b - r \right) e^{-2r/a_0} dr \nonumber \\ & = - \frac {e^2} {\pi \varepsilon_0 a_0^3} \left( \int_0^b b e^{-2r/a_0} dr - \int_0^b r e^{-2r/a_0} dr \right) \nonumber \end{align} $$
Then I used the given integrals to get to:
$$\int_0^b b e^{-2r/a_0} dr = \frac {a_0 b} {2} \int_0^{2b/a_0} e^{-u} du = \frac {a_0 b} {2} \left( 1 - e^{-2b/a_0} \right) $$
And
$$\int_0^b r e^{-2r/a_0} dr = \frac {a_0^2} {4} \int_0^{2b/a_0} u e^{-u} du = \frac {a_0^2} {4} \left( 1 - e^{-2b/a_0} - \frac {2b} {a_0}e^{-2b/a_0} \right) $$
Then I substituted back into get my answer for (b). Have I messed this up somewhere?

 

[vela]

Oops, my mistake, I simplified your answer incorrectly. You have some terms that will cancel out, and you should end up with
$$\frac{k e^2}{a}\left(1-\frac{2b}{a} - e^{-2b/a}\right),$$ where $k = \frac{1}{4\pi\varepsilon_0}$. Now expand the exponential term to second order, and you'll get the result you seek.

 

[Leechie]

Thanks vela. I see how that all fits together now.

I'm still a little confused with this bit though:

Hence deduce the largest value of $b$ that would be consistent with the fact that the ground-state energy of a hydrogen atom agrees with the predictions of the Coulomb model to one part in a thousand. Express your answer as a numerical multiple of $a_o$.

I know how to find the largest value of $b$ using a derivative (if that's what the questions is asking), but I'm not sure I understand the bit about 'one part in a thousand'. I'm thinking the idea behind this is to find the point where the Coulomb model first matches the ground state as $b$ increases from $0$, which would be the maximum value $b$?

 

[vela]

The question is asking when do the unperturbed and perturbed energies differ by less than 0.1%.

 

[Leechie]

I think I know where I'm heading now, thanks.