Science > Physics > Link between single particle states and full wavefunction?
| Topic: |
Science > Physics |
| User: |
"Anton81" |
| Date: |
08 Aug 2005 08:58:12 AM |
| Object: |
Link between single particle states and full wavefunction? |
Hi!
In quantum mechanics I learned about the wavefunction
Psi(r1,s1,r2,s2,r3,s3,...) [r is position and s is spin] and the Hartree
approximation.
In another course they talked about Hund's rules and single particle states
with quantum numbers l,l_z,s_z being filled.
Now I wonder how to rigorously justify the use of s,p,d,... orbitals and
spin up/down states for electrons in crystals and molecules?!
I heard the hydrogen orbitals form a complete set and that some kind of
spinor can always be split into spin-1/2 spinors. I've also seen some group
theory suggesting that orbitals in that sense are connected with
irreducible representations. I don't know much about these topics, but a
proper of explanation of this kind is what I'm looking for.
Any hints or even references?
Anton
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| User: "Gregory L. Hansen" |
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| Title: Re: Link between single particle states and full wavefunction? |
09 Aug 2005 10:07:43 AM |
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In article <dd7odk$akq$1@gwdu112.gwdg.de>, Anton81 <berrybear@gmx.net> wrote:
Hi!
In quantum mechanics I learned about the wavefunction
Psi(r1,s1,r2,s2,r3,s3,...) [r is position and s is spin] and the Hartree
approximation.
In another course they talked about Hund's rules and single particle states
with quantum numbers l,l_z,s_z being filled.
Now I wonder how to rigorously justify the use of s,p,d,... orbitals and
spin up/down states for electrons in crystals and molecules?!
I heard the hydrogen orbitals form a complete set and that some kind of
spinor can always be split into spin-1/2 spinors. I've also seen some group
theory suggesting that orbitals in that sense are connected with
irreducible representations. I don't know much about these topics, but a
proper of explanation of this kind is what I'm looking for.
Any hints or even references?
Anton
You're kind of all over the place there.
The s,p,d,... orbitals form a complete set. That means they can be used
to describe a particle in a harmonic potential, a free particle, neutrons
diffracting from a crystal, whatever you like. Take any wavefunction,
apply orthogonality rules, and you can reexpress it in the basis of
hydrogen orbitals. That they form a complete set *is* the rigorous
justification. Any complete basis could be used.
The not-so-rigorous justification is that in crystals and molecules the
orbitals of the valence electrons are only slightly modified, and the core
electrons basically not modified at all. When you apply perturbation
theory you should prefer small perturbations over large ones.
I thought Harrison's book (available as a cheap Dover reprint) did a good
job of justifying approximation schemes, even if I didn't think it was a
good primary textbook.
--
"The result of this experiment was inconclusive, so we had to use
statistics." (Overheard at international physics conference)
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