# Rare Gases

Hi!

I saw something on my lecture notes that I don't really understand. It reads "Rare gases have filled s and p-sub-shells, which leads to a spherically symmetric charge distribution. Since electrons are indistinguishable they take on a common wavefunction. The point is that this results in a higher binding energy for each one of the electrons."

I don't really understand the last bit, especially why having a common wavefunction leads to a higher binding energy. Does anyone care to explain?

Thanks!

Simon Bridge
Homework Helper
Its quantum. I don't think there is a classicalesque analogy for it.

Basically, you can pack electrons in tighter when there is a lot of symmetry.
Look up the wavefunctions of Helium for a "simple" example. Compare with Hydrogen and Lithium.
But it is not really cause and effect - the "common wavefunction" and the higher binding energy go hand-in-hand.

I am guessing the first point is due to exchange symmetry? I am not sure how to visualise this... is there a mathematical proof to this?

Simon Bridge
Homework Helper
You cannot prove an empirical truth by mathematics alone ... but there is a demonstration that the regular schrodinger formulation for fermions results in these symmetries.
It's part of the usual mathy description.

Astronuc
Staff Emeritus
Hi!

I saw something on my lecture notes that I don't really understand. It reads "Rare gases have filled s and p-sub-shells, which leads to a spherically symmetric charge distribution. Since electrons are indistinguishable they take on a common wavefunction. The point is that this results in a higher binding energy for each one of the electrons."

I don't really understand the last bit, especially why having a common wavefunction leads to a higher binding energy. Does anyone care to explain?
I believe the correct statement should be that none of the 6 electrons has a different or distinguishable binding energy compared to the others. The high binding energy has to do with paired electrons with opposite spin. The binding energy of a given electron has to with the Z and probable 'distance' from the nucleus. Note that the group I elements S1 are readily ionized or give up one electron, while group IV readily attract one electron to fill the outer shell.

Note that He has a filled S shell, 1s2. Ne has 1s2 2s2 2p6.

The theory for multi-electron atoms is found in Hartree-Fock theory.
http://en.wikipedia.org/wiki/Hartree–Fock_method
http://www.eng.fsu.edu/~dommelen/quantum/style_a/hf.html

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

DrDu