Bohr Radius and Ground State Energy

[davesface]

Homework Statement

Given the potential energy $$V(r)=-\frac{1}{4\pi \epsilon_0}\frac{e^2}{r}$$ (where e is the unit charge), use the uncertainty principle $$\Delta x \Delta p \geq \hbar$$ to find the Bohr radius $$r_B$$ for a hydrogen atom and the ground state energy $$E_0$$.

Hint: write down the kinetic energy in terms of $$r_B$$ using the uncertainty principle.

Homework Equations

-TISE: $$-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2}+[V+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}]u=Eu$$, where $$u(r)=r*R(r)$$ and R(r) is the radial component of the separable wave function $$\Psi$$.
-Alternatively, $${V}_{eff} =V(r)+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}$$

The Attempt at a Solution

The hint is really what throws me off. Finding the ground state energy is done out entirely in the textbook, but there is nothing about the modified uncertainty relation or writing the kinetic energy in terms of the Bohr radius before finding it.

I tried doing that, so $$T=\frac{p^2}{2m}=\frac{\hbar^2}{2mx^2}$$ if we assume equality in the uncertainty relation. Then I plugged that into an equation for the defined quantity $$\kappa=\frac{\sqrt{-2mE}}{\hbar}=\frac{i}{x}$$. Do I just plug in $$r_B$$ for x now? I just don't see at all how doing that, if it's even correct, is leading closer to finding $$r_B$$.

EDIT: As an added note, I tried reverse engineering the solution: $$r_B=\frac{4\pi \epsilon_0\hbar^2}{m_e e^2}=\frac{-\hbar^2}{m_e r V(r)}$$, but that didn't make the problem any clearer to me.

 

[vela]

Hmm, I'm not sure how you can find the Bohr radius without an additional piece of information. If you use the virial theorem, which says <T> = -<V>/2, you can then solve for a₀.

 

[davesface]

Our book only mentions the special cases of the virial theorem for stationary states ,$$2<T>=<x\frac{dV}{dx}>$$, and stationary states of the harmonic oscillator, $$<T>=<V>$$.

Even if I use your version of it, though, I don't see how that helps.

 

[vela]

For a potential of the form rⁿ, the virial theorem reduces to what I said.

http://en.wikipedia.org/wiki/Virial_theorem

In your expression for T after you substituted with the uncertainty principle, set x to the Bohr radius. Similarly, in the potential, set r to the Bohr radius. Then solve for the Bohr radius.

 

[davesface]

Wow, that theorem made it a much easier problem to solve. Thanks very much.