- #1
Leechie
- 19
- 2
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.