Hydrogen atom: potential well and orbit radii

• shallowbay
In summary, Sah states that the electron orbit radius is half the well radius at the energy level E_n. The orbit radius is determined by the formula r_n=\frac{4*\pi*ε_0*\hbar^2*n^2}{mq^2} and the potential well is represented by V(r_n)=\frac{-q^4*m}{(4*\pi*ε_0)^2*\hbar^2*n^2}. This is because the orbit radius must be confined within the potential well, and this ratio ensures that the well is twice the Bohr radius. A diagram provided by Sah further illustrates this concept.
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

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 = ∞.

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.

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1. What is a potential well in a hydrogen atom?

A potential well in a hydrogen atom refers to the region of space around the nucleus where the electron experiences an attractive force due to the positive charge of the nucleus. This well represents the lowest potential energy state for the electron, and it is where the electron is most likely to be found.

2. How does the potential well in a hydrogen atom affect the electron's orbit?

The potential well determines the size and shape of the electron's orbit around the nucleus. The deeper the well, the more tightly bound the electron is to the nucleus, resulting in a smaller orbit radius. Conversely, a shallow potential well leads to a larger orbit radius for the electron.

3. What factors influence the size of the potential well in a hydrogen atom?

The size of the potential well in a hydrogen atom is primarily determined by the charge of the nucleus and the distance between the nucleus and the electron. A higher nuclear charge or a smaller distance between the two will result in a deeper potential well.

4. Can the electron escape the potential well in a hydrogen atom?

Yes, the electron can escape the potential well if it gains enough energy to overcome the attractive force of the nucleus. This can happen through absorption of a photon or collision with another particle.

5. How is the orbit radius of the electron in a hydrogen atom calculated?

The orbit radius of the electron in a hydrogen atom can be calculated using the Bohr model equation, which takes into account the charge of the nucleus, the mass of the electron, and Planck's constant. It is also affected by the electron's angular momentum and the energy level of the electron.

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