Hydrogen atom: potential well and orbit radii

[shallowbay]

Hello,

I happened to open up an old book by Sah, and in it he says:

"it is evident that the electron orbit radius is half the well radius at the energy level $E_n$"

The orbit radius is $r_n=\frac{4*\pi*ε_0*\hbar^2*n^2}{mq^2}$ and the potential well $V(r_n)=\frac{-q^4*m}{(4*\pi*ε_0)^2*\hbar^2*n^2}$

Of course the orbit radius has to be confined in the well, but it's not obvious to me why it should be exactly half the well radius? This isn't something I recall seeing before either.

Thanks

 

[jtbell]

What does "well radius" even mean in this context? I've never seen anyone talk of the "radius" or "width" of a 1/r potential well; it extends from r = 0 to r = ∞.

 

[shallowbay]

He's speaking of the width of the potential well due to the nucleus at the specific energy levels $E_n$. So that apparently $r_1$ of the electron is half of the width of the potential well itself at $E_1$, or the well would be twice the Bohr radius.

Attached diagram he uses where he has drawn the orbit radius to be half that of the well. When I just add it to the post it is far too small to be useful.