Help understanding molecular vs atomic electron quantum states

[Bob65]

I am a retired electrical engineer, now able to get back to studying what I really enjoy - mathematics and physics.

As a genuine old geezer, my modern physics knowledge, which was never very deep, is now way out of date. I purchased a copy of "Modern Physics", by Kenneth Krane, and have been studying it - extremely interesting, and very helpful. I had no idea how ignorant I really am.

My questions regarding molecular versus atomic quantum states for the electrons in a molecule are very simple compared to other questions I see on this forum, and are as follows:

1. Are there more than the four "normal" electron atomic quantum numbers (principle quantum number, usually designated n; orbital quantum number, usually designated by the letter l; magnetic quantum number, usually designated m_l; and spin quantum number, usually designated m_s), associated with the electrons in a molecule?

2. Has the Schrödinger equation been solved analytically for any simple molecules (say, diatomic hydrogen)? If so, where may I find a reference to these solutions?

3. Does the Pauli exclusion principle apply to all of the electrons in a molecule, or just to the "outer" electrons?

4. Where does application of the Pauli exclusion principle stop? Obviously, it doesn't apply to two isolated hydrogen atoms. But as the atoms are brought closer together, I suspect that it "begins to apply."

5. (Related to question 4 above) I have read about white dwarf stars, where degeneracy pressure, associated with the Paul principle, keeps the star from further collapse. Is there an effective "range" (physical length) over which the Pauli principle applies to electrons in a dwarf star?

Can anyone make a suggestion for a modern physics text that would go more deeply than Krane's book (but not so deeply that I couldn't understand it, having no more background than Krane's book supplies), and that might contain the answers to the above questions? I'm like to do further self-study, and would appreciate suggestions.

Thanks for any help offered.

Bob65

 

[Simon Bridge]

1. the normal four quantum numbers apply.

2. see H2 molecule, does that count?)

3. PEP applies to all electrons - indeed to all Fermions.

4. The PEP does not stop - no two fermions can have the same quantum state.
A hydrogen atom is not a Fermion so PEP won't apply to the overall atomic states.

However - it does apply to the electrons bound to each nuclei.
Imagine a Universe with only two H nuclei in it. When you solve the Schrödinger equation, you get a different set of energy levels for that case that for when you are only considering one H nucleus.
When you compare them - you get the same basic "ladder" pattern, but with two "rungs" close together where you only got one before.
The separation depends on how close the nuclei are to each other - outside a few angstroms the rungs basically overlap so we can treat the nuclei as if they were isolated.

5. Everywhere.

Suggested texts: depends on your area of interest. "modern physics" is a large field.
Above Krane's level it gets very mathematical... but you d't seem to be bothered by the math.
"Modern" physics usually starts with special relativity and wave mechanics
Goes as high as general relativity and field theory (then string theory and such beasties)
At some point you need to revisit classical mechanics in the form of lagrange mechanics.

I'd suggest looking at the college lecture series that are online if you are serious.
Leonard Susskind's lectures are online - and the MIT Open Courseware has a good rep.
http://ocw.mit.edu/courses/physics/
http://www.newpackettech.com/Resources/Susskind/

 

[dipole]

If you want a good book on quantum mechnics, Sakurai is a good choice. As an electrical engineer, you should be familiar with most of the mathematics - mostly just linearly algebra and differential equations. (Sakurai doesn't get into some of the very gritty mathematical subleties involved with QM, that most of the time are ignored in practice).

Understanding QM is the bedrock for understanding modern physcs. However, you should definitely read up on classical mechanics, especially Hamiltonian dynamics, to see the link between classical and quantum theory.

 

[DrDu]

2. see H2 molecule, does that count?)

Even for H2 we have no analytic solution. For H2+ we have one and even this is not simple but involves very complicated functions.

 

[DrDu]

There is a very crisp and accurate description of bonding by W. Kutzelnigg, "The physical mechanism of the chemical bond", which is one of my favourite citations:
http://onlinelibrary.wiley.com/doi/10.1002/anie.197305461/abstract

 

[Ken G]

5. Although it is true the the PEP always applies, it doesn't have much effect until the average kinetic energy of the electrons (which you can get from the gravitational energy of the ions, it's of the same order by the virial theorem) starts to get "large." By "large", I mean that if you convert that energy into a "box" size by saying that's the kinetic energy the electron would need to fit in that "box" (this is a surprise of quantum mechanics, it takes kinetic energy to get a particle to fit in a small box, you can get that energy from the Heisenberg uncertainty principle where the momentum is of order h/x, where x is the distance across the box), then the volume of that box is of order the size of the box each electron is actually in (i.e., a volume of the inverse electron density).

You see, normally, the particles in a gas have an energy that is much larger than the energy they'd need to fit in the "boxes" they are in by virtue of their density. But as you extract heat from a bound system, it generally turns out that the kinetic energy the particles actually have eventually becomes comparable to the energy they need just to fit in the "boxes" they are in. (Either the kinetic energy drops, if the total volume stays fixed, or more often, the system shrinks and the kinetic energy actually rises, but the volume gets so small that the minimum energy rises even faster and overtakes the actual energy.) We call that situation a "quantum mechanical ground state." Then the PEP says you can't get any more heat out, and for some rather strange reason, that state of affairs in astronomy often gets called "degeneracy pressure", though the term is a bit counterproductive.

The point is, you can estimate when that happens by looking at the physics of how much kinetic energy the particles have, versus how much do they need to fit in the boxes they are in, and when the latter starts to get as large as the former (often the former is rising but the latter even faster), then you start to have to worry about "degeneracy." It's easier to say your system is approaching its ground state, i.e., it is becoming what in physics is referred to as "condensed matter." But it's not always a sudden transition-- there's no phase change from a gravitationally bound ideal gas to a degenerate gas, for example.