Hydrogen's energy levels don't depend on l or m because

[awat]

Homework Statement

Why doesn't the energy of the hydrogen atom depend on quantum numbers l or m?

Homework Equations

E = (Z^2) 13.6 eV / n^2

The Attempt at a Solution

Regarding l:

I know that it has to do with the inverse-square law between the distance, r, of the e- and the nucleus' force exerted on the e-...

I also know that it has to do with the fact that there is only 1 e- orbiting H...

But can someone elaborate on exactly what happens when more e- are introduced?

Is it that another e- can interfere with the "pull" felt by the outer e- and exactly HOW much it interferes is dependent upon the angular momentum number, l?

Regarding m:

?

 

[Benefits]

The reason that the energy level depends only on n is that for just one electron the only force acting on it is the coulomb force between the nucleus and the electron. As such this is the only force that dictates the potential energy term in the Schrödinger equation. In this case the strength of the coulomb force between the hydrogen nucleus and the electron only depends on the distance between the two and the charge involved (2q in this case). The force is spherically symmetric (aka rotationally invariant) so it doesn't matter where around the hydrogen nucleus an electron is. As long as the distance between the two is constant the strength of the force is constant. Hence, the energy is constant.

Since l dictates the electron's angular momentum it has an affect on the electron's position but it doesn't dictate how far away from the nucleus it will be. The same is true for m. m roughly corresponds to the direction of the angular momentum vector and dictates how the electron will behave in the presence of an external magnetic field; since neither of these things have to do with distance between the electron and it's nucleus the energy level doesn't depend on it. It's worth noting that since l and m do affect position they help shape the atomic orbital clouds just not the energy levels.

On to atoms with multiple electrons:

"Is it that another e- can interfere with the "pull" felt by the outer e- and exactly HOW much it interferes is dependent upon the angular momentum number, l?"

That's basically right on. Consider a hydrogen atom with two electrons (let's pretend there's an inner and outer electron to differentiate the two even though this is unrealistic). The strength of the net force on the outer electron now depends on the attractive coulomb force between the nucleus and electron and the repulsive force between the inner and outer electrons. When the outer and inner electron are close to each other the potential energy (due to the repulsion) of the outer electron is high, so the energy level of the electron is higher. The result is that the energy level of the outer electron is lowest when the outer electron spends the least amount of time near the inner electron; this occurs when they travel in the same direction (which makes sense when you think about it). It turns out that for light atoms like this we can define a vector L which is the vector sum of the electrons' individual angular moments. When L is small (remember, vector sum) the e- are orbiting in opposite directions and spend more time near each other (hence higher energy level). When L is large the two electrons are traveling in the same direction, so they spend less time together, and the energy is lower.I'm still just an undergrad myself so take this all with a grain of salt, but I think I'm right. Hope this helps!

 

[awat]

Fantastic response, Benefits! I appreciate your thoroughness. Cheers!

 

[Benefits]

You're more than welcome! It's always nice to dust the cob webs off of old material and check that you still understand it qualitatively. Plus, I got to procrastinate from studying physics... by talking about physics. Basically a win-win.

On a side note I had to do a quick review of the energy levels in a multi-electron atom; I convinced myself you had to be right when you said "HOW much it interferes is dependent upon the angular momentum number, l?" but I couldn't quite nail down exactly how on my own so I checked it out on HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html, which is an awesome physics resource. It can be a little difficult to navigate exactly where you need to go (it's deceptively large) so I usually just google my question topic and add "hyperphysics" at the end. Regardless, you should definitely check it out if you haven't discovered it already. It's a fantastic reference for relevant equations and graphs, I still use it occasionally and I'm a senior.